Minimal projections onto spaces of polynomials on  real euclidean   spheres]Minimal projections onto spaces of polynomials
on  real euclidean   spheres

Defant]A. Defant

Institut für Mathematik,
Carl von Ossietzky Universität,
26111 Oldenburg,

Galicer]D. Galicer
Departamento de Matemática,
Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and IMAS-CONICET. Ciudad Universitaria, Pabellón I
(C1428EGA) C.A.B.A., Argentina

Mansilla]M. Mansilla
Departamento de Matemática,
Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and IAM-CONICET. Saavedra 15 (C1083ACA) C.A.B.A., Argentina

Mastyło]M. Mastyło
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu
Poznańskiego 4,
61-614 Poznań, Poland

Muro]S. Muro
FCEIA, Universidad Nacional de Rosario and CIFASIS, CONICET, Ocampo $\&$ Esmeralda, S2000 Rosario, Argentina

The research of the fourth author was supported by the National Science Centre (NCN), Poland, Project 2019/33/B/ST1/00165; the second, third and fifth author were supported by CONICET-PIP 11220200102336 and PICT 2018-4250. The research of the second and fifth author is additionally supported by 20020220300242BA and  ING-586-UNR, respectively.

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics  and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We  establish a connection between these quantities and certain weighted $L_1$-norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions and  asymptotically accurate estimates  for them.  The real case we address is considerably more nuanced than its complex counterpart.

[2020]Primary: 33C55, 33C45, 46B06, 46B07. Secondary: 43A75, 46G25

§ INTRODUCTION

Spherical harmonics and spherical polynomials have become indispensable tools across scientific disciplines, providing elegant solutions for problems defined on the unit sphere and beyond. From solving partial differential equations to enhancing numerical integration methods, their versatility meets the demands of diverse applications.

Exploring approximation techniques and numerical methods over spheres reflects the growing interest in efficient computational algorithms.
From a functional analysis perspective, particularly within the realm of Banach space theory, it is crucial to investigate the projection constant for Banach spaces of spherical harmonics and spherical polynomials.

The concept of the projection constant is foundational in the field of Banach spaces and their local theory. Its origins trace back to the examination of complemented subspaces within Banach spaces.

Consider $X$ as a complemented subspace of a Banach space $Y$. The relative projection constant of $X$ in $Y$ is defined as follows:

\begin{align*}
\boldsymbol{\lambda}(X, Y) & =  \inf\big\{\|P\|: \,\, P\in \mathcal{L}(Y, X),\,\, P|_{X} = Id_X\big\}\\
&  = \inf\big\{c>0: \,\,\text{$\forall\, T \in \mathcal{L}(X, Z)$ \,\, $\exists$\,\, an extension\,
$\widetilde{T}\in \mathcal{L}(Y, Z)$\, with $\|\widetilde{T}\| \leq c\,\|T\|$}\big\}\,,
\end{align*}

where $Id_X$ denotes the identity operator on $X$, and $\mathcal{L}(U,V)$ represents the Banach space of all bounded linear operators between Banach spaces $U$ and $V$ with the uniform norm. Here, $\mathcal{L}(U):=\mathcal{L}(U,U)$, and we adopt the convention that $\inf \varnothing = \infty$.

The (absolute) projection constant of $X$ is expressed as:

\[
\boldsymbol{\lambda}(X) := \sup \,\,\boldsymbol{\lambda}(I(X),Y)\,,
\]

where the supremum is taken over all Banach spaces $Y$ and isometric embeddings $I\colon X \to Y$. If $X$ is a finite-dimensional Banach space and $X_1$ is a subspace of some $C(K)$-space isometric to $X$, then (see, e.g. <cit.>):

\begin{align}\label{eq: l_infty inyective}
\boldsymbol{\lambda}(X) = \boldsymbol{\lambda}(X_1, C(K))\,.
\end{align}

Thus, determining $\boldsymbol{\lambda}(X)$ is equivalent to finding the norm of a minimal projection from $C(K)$ onto $X_1$.

It is worth noting that projections play an important role in approximation theory. In fact, if $Y$ is a Banach space and $P$ a projection from $Y$ onto a subspace $X$, then the error $\|y - Py\|_Y$ of approximation of an element $y\in Y$ by $Py$ satisfies

\[
\|y- Py\|_Y \leq \|\id_Y - P\| \, \text{dist}(y, X) \leq (1 + \|P\|)\, \text{dist}(y, X)\,,
\]

where $\text{dist}(y, X) =\inf\{\|y-x\|_Y \,: \, x\in X\}$. This estimate motivates the problem of minimizing $\|P\|$, and any
projection  $P_0\colon Y \to Y$ onto $X$ such that $\|P_0\| = \boldsymbol{\lambda}(X, Y)$, is said to be a minimal projection of $Y$ onto $X$.

In $L^2(\mathbb{S}^{n-1})$, the space of square-integrable functions on the unit sphere in $\mathbb{R}^n$, each function can be decomposed into an orthogonal sum of homogeneous spherical harmonics, serving as fundamental building blocks. Understanding the projection constant in this context aids in evaluating the efficacy of a partial approximation for a given function.

The initial inspiration for this work stems from the seminal contributions of Ryll and Wojtaszczyk [30]. Solving a problem posed by Waigner, they demonstrated that the inclusion $H_\infty(B_{\ell_2^n(\mathbb{C})}) \hookrightarrow H_1(B_{\ell_2^n(\mathbb{C})})$ is not compact, where $B_{\ell_2^n(\mathbb{C})}$ denotes the unit ball of the complex $n$-dimensional Hilbert space $\ell_2^n(\mathbb{C})$ and $n>1$. As noted in [30], this result is intimately related to a question raised by Rudin in his monograph [28]: Does there exist an inner function on the open unit ball of the Hilbert space $\ell_2^n(\mathbb{C})$, $n>1$? A function $f \in H_\infty(B_{\ell_2^n(\mathbb{C})})$ is said to be inner if almost all its radial boundary values have modulus $1$. For detailed information, we refer to [26] and [33].

Much of this deep cycle of ideas  relies on the following concrete formula for the projection
constant  of the Banach space $\mathcal{P}_{d}(\ell_2^n(\mathbb{C}))$ of all  $d$-homogeneous polynomials on $\ell_2^n(\mathbb{C})$:

\begin{align} \label{fascinating}
\boldsymbol{\lambda}\big(\mathcal{P}_d(\ell_2^n(\mathbb{C})\big) = \frac{\Gamma(n +d) \Gamma(1 + \frac{d}{2})}{\Gamma(1 + d)\Gamma(n + \frac{d}{2})}, \quad\, d,n\in \mathbb{N}\,
\end{align}

(see [30], and also  <cit.>
and [26])).

A simple calculation yields,

\begin{align} \label{surprise}
\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^n(\mathbb{C})\big) \leq 2^{n-1}, \quad\, d,n\in \mathbb{N}\,.
\end{align}

The case $n=2$ is of particular interest. Indeed, in the mentioned paper [30] the authors noticed
the surprising fact that the sequence $\big(X_d\big)_{d\geq 1}$ with $X_d:= \mathcal{P}_{d}(\ell_2^2(\mathbb{C}))$ forms the first  known example of a sequence of finite-dimensional Banach spaces for which $\lim_{d \to \infty} \dim X_d =\infty$ although

\[
\sup_d \boldsymbol{\lambda}(X_d) <~\infty\,.
\]

It seems interesting to note that the formula (<ref>)
combined with the well-known limit shown further in Equation (<ref>) gives that

\begin{align} \label{surpriseB}
\lim_{d\to \infty} \boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^n(\mathbb{C})\big) =2^{n-1}\,.
\end{align}

Thus, more precisely, we have $\sup_d \boldsymbol{\lambda}(X_d)=2$. It is worth noting here that Bourgain [5] gave an affirmative solution to a problem considered in [30],
showing that the sequence $\big(\text{dist}\big(X_d, \ell_\infty^{\dim X_d}(\mathbb{C})\big)\big)_d$ of Banach Mazur distances is bounded.

What about real Hilbert spaces $\ell_2^n(\mathbb{R})$? By Rutovitz [29]  we know that

\begin{align} \label{beginning}
\boldsymbol{\lambda}\big(\ell_2^n(\mathbb{R})\big)
%=  n \int_{\mathbb{S}^{n-1}} |\eta_1|\,d\sigma_n(\eta)
= \frac{2}{\sqrt{\pi}}   \frac{\Gamma(\frac{n+2}{2})}{\Gamma(\frac{n+1}{2})}\,,
\end{align}

which is different from the complex case $d=1$ in

\begin{align} \label{Ruto}
\boldsymbol{\lambda}\big(\ell_2^n(\mathbb{C})\big)
%= n \int_{\mathbb{S}_{\mathbb{C}}^{n-1}} |\eta_1|\,d\sigma_n(\eta)
= \frac{\sqrt{\pi}}{2}   \frac{n!}{\Gamma(n + \frac{1}{2})}\,.
\end{align}

Among others, motivated by the fascinating formula from (<ref>), our main goal here is to study the projection constant of $\mathcal{P}_{d}(\ell_2^n(\mathbb{R}))$.

Contrary to initial expectations, the real case unfolds with intricate subtleties, ultimately revealing nuanced differences from its complex counterpart.
To illustrate one of these differences, we note that by (<ref>) the sequence
$\big(\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^n(\mathbb{C}))\big)\big)_d$  is bounded for each fixed $n$,
whereas we are going to show that in the real case  for each fixed $n\geq 2$ the sequence $\big(\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^n(\mathbb{R}))\big)\big)_d$ increases  to infinite.

Our focus in this article lies on finding
suitable extensions of (<ref>) for various spaces of multivariate polynomials on finite dimensional real Hilbert spaces  - more precisely, we intend  to present  concrete
formulas or computable expressions for the projection constants of various classes of polynomials defined on finite dimensional real Hilbert spaces or their  spheres.

To accomplish this, we introduce a framework that correlates the projection constant with an average integral. Given the similarity in the required effort between handling the sphere or other compact spaces
with a certain group structure, we opt to present an abstract approach which can be useful for other contexts.

Utilizing averaging techniques, we demonstrate that the optimal projection for minimizing the norm coincides with the canonical projection from $L_2(\mathbb{S}^{n-1})$. This insight enables us to furnish precise formulas, computable expressions, or asymptotically accurate estimates. Specifically, we present integral formulas for the projection constants we deal, expressed in terms of certain Jacobi polynomials.

In particular, we show that the projection constants of the spaces $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$, $\mathcal{P}_{ d}(\mathbb{S}^{n-1})$, and $\mathcal{H}_{d}(\mathbb{S}^{n-1})$ for $n>2$ behave asymptotically as $d^{\frac{n-2}{2}}$ for large $d$, despite their considerable dimensional differences. These spaces represent the Banach spaces of degree-$d$ polynomials, $d$-homogeneous polynomials, and $d$-homogeneous spherical harmonics, respectively, endowed with the uniform norm on the real unit sphere $\mathbb{S}^{n-1}$. Consequently, we observe that the first two of these Banach spaces do not meet the classical Kadets-Snobar bound. This estimate asserts that the projection constant of any finite-dimensional space is bounded by the square root of its dimension (see, for instance, <cit.>).
We also analyze the behavior of the ratio of their projection constant and $d^{\frac{n-2}{2}}$ as a function of the dimension $n$ when $d$ is sufficiently large.
The case $n=2$ is also addressed. Specifically, we prove that the projection constant of $\mathcal{P}_{\leq d}(\mathbb{S}^{1})$ and $\mathcal{P}_{d}(\mathbb{S}^{1})$ behaves as $\log d$ as $d\to \infty$, whereas the one for $\mathcal{H}_{d}(\mathbb{S}^{1})$ remains constant.

Some of the techniques or theory we present here are recognized by specialists in Banach space theory, while others are known to those engaged in approximation or function theory on the unit sphere.
In order for the article to be comprehensible to a diverse audience, we opted to delve into detail and incorporate certain results or demonstrations, which might be familiar to one community or the other. However, we suspect that the entirety of the material may not be familiar to either community, as was the case for the authors at the outset of this research endeavor.

Concluding this introduction, we provide a list of recent literature relevant to the study of the projection constant:
[4, 6, 8, 9, 7, 11, 12, 14, 13, 16, 17, 19, 20, 18, 21]. It's important to note that this compilation is not exhaustive, but rather serves as a starting point for exploring latest advancements in this topic and the current interest in the field.

§ PRELIMINARIES
We recommend consulting Wojtaszczyk's book  [33] for matters related to Banach space theory and Atkinson and Han's monograph [3] for topics concerning spherical harmonics and approximation theory.

We  denote the Lebesgue measure on $\mathbb{R}^n$ by $\lambda_n$ and the surface (Borel) measure on $\mathbb{S}^{n-1}$ by $s_n$. Recall one of the simplest definitions of surface measure.  If $A$ is a Borel set in $\mathbb{S}^{n-1}$, then

\[
s_n(A) := n\,\lambda_n(\{rx: \, r\in (0, 1),\, \, x\in A\})\,.
\]

Since $\pmb{\tau}_n : = \lambda_n(B(0,1))= \frac{\pi^{n/2}}{\Gamma(n/2 + 1)}$, it follows that

\begin{equation} \label{omega}
\pmb{\omega}_n :=s_n(\mathbb{S}^{n-1}) = \frac{2 \pi^{n/2}}{\Gamma(n/2)}\,.
\end{equation}

Throughout the paper, the measure $\frac{1}{\pmb{\omega}_n} s_n$ is called the normalized surface measure
on $\mathbb{S}^{n-1}$ and is denoted, as usual, by $\sigma_n$.

The orthogonal group acting on  $\mathbb{S}^{n-1}$ is denoted by $\mathcal{O}_n$. For a compact set $K$, we denote by $C(K)$ the Banach space of all $\mathbb{C}$-valued continuous functions defined on $K$ endowed with the supremum norm.

Recall that the normalized area measure $\sigma_n$ is invariant under orthogonal transformations. Thus, it can also be obtained by the useful formula

\[
\int_{\mathbb{S}^{n-1}} f(\xi)\,d\sigma_n(\xi) = \int_{\mathcal{O}_n} f(U\xi_0)\,d\textbf{m}(U), \quad\, f\in C(\mathbb{S}^{n-1})\,,
\]

where $\textbf{m}$ is the normalized Haar measure on $\mathcal{O}_n$ and $\xi_0$  some fixed vector in $\mathbb{S}^{n-1}$.

We write  $\mathcal P (\mathbb{R}^n)$ for the linear space of all  polynomials $f\colon \mathbb{R}^n \to \mathbb{C}$ of the form

\begin{align} \label{repr}
f\left(x\right)=\sum_{\alpha \in J} c_\alpha\,x^\alpha, \quad\, x\in \mathbb{R}^n\,,
\end{align}

where  $J \subset  \mathbb{N}_0^n$ is finite, and $(c_\alpha)_{\alpha\in J}$ are  complex coefficients.
Since $c_\alpha=\partial^\alpha f(0)/\alpha!$ for $\alpha \in J$, it is immediate that the preceding representation  is unique.
For $f \in \mathcal P (\mathbb{R}^n)$ we call

\[
\text{deg}(f)
:= \max_{\alpha \in J} |\alpha|
\]

the degree of $f$,
where as usual $|\alpha|:=\sum \alpha_i$
for $\alpha \in \mathbb{N}^n_0$.

Given  $d \in \mathbb{N}_0$, we write
$\mathcal P_d (\mathbb{R}^n)$  for all $d$-homogeneous polynomials in $\mathcal P (\mathbb{R}^n)$
(that is, $f$ has a representation like in (<ref>) with coefficients $ c_\alpha \neq 0$ only if $|\alpha|= k $),
and $\mathcal P_{\leq d}(\mathbb{R}^n)$ for all polynomials of
degree $\leq d$. Due to the linear independence of monomials $x^\alpha$, any degree $d$-polynomial $f \in \mathcal P (\mathbb{R}^n)$ may be uniquely represented as a sum $f= \sum_{k=0}^d f_k$ of $k$-homogeneous polynomials $f_k$.

\begin{equation} \label{klm1}
\mathcal P_{\leq d}(\mathbb{R}^n)
\spa_{k \leq d} \mathcal P_{k}(\mathbb{R}^n)
\quad \text{and} \quad
\mathcal P(\mathbb{R}^n)
\spa_k \mathcal P_k(\mathbb{R}^n)\,.
\end{equation}

Recall that

\begin{equation}\label{dimformuA}
\dim \mathcal{P}_d(\mathbb{R}^n)
=  \binom{n+d-1}{n-1}=  \binom{n+d-1}{d}\,.
\end{equation}

The  linear spaces $\mathcal{P}_{d} (\mathbb{R}^n)$ and $\mathcal{P}_{\leq d}(\mathbb{R}^n)$ equipped  with the uniform norm over the real unit euclidean ball $B_{\ell_2^n(\mathbb{R})}$ form complex Banach spaces, which we  denote  by
and $\mathcal{P}_{\leq d}(\ell_2^n(\mathbb{R}))$ respectively.

We write $\mathcal{P}(\mathbb{S}^{n-1})$ for the linear space of all restrictions  $f|_{\mathbb{S}^{n-1}}$ of polynomials
$f \in \mathcal{P}(\mathbb{R}^n)$, and $\mathcal{P}_d(\mathbb{S}^{n-1})$ or $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$, whenever  only restrictions of
$d$-homogeneous or degree-$d$ polynomials are considered.
Together with the uniform norm taken on $\mathbb{S}^{n-1}$,
both spaces $\mathcal{P}_d(\mathbb{S}^{n-1})$ or $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$
form  finite dimensional subspaces of $C(\mathbb{S}^{n-1})$.
Clearly,  by (<ref>)

\begin{equation}\label{ludoABAB}
\mathcal P_{\leq d} (\mathbb{S}^{n-1})
\spa_{k \leq d} \mathcal P_{k}(\mathbb{S}^{n-1})
\quad \,\text{and} \,\quad
\mathcal P ( \mathbb{S}^{n-1})=\spa_{k}\mathcal{P}_k (\mathbb{S}^{n-1})\,.
\end{equation}

Note that for each $d$ the restriction map

\begin{equation}\label{aprilo}
\mathcal{P}_{d}(\ell_2^n(\mathbb{R}))  \to \mathcal{P}_{d}(\mathbb{S}^{n-1})\,, \quad\,
f \mapsto f_{|\mathbb{S}^{n-1}}
\end{equation}

is an isometric bijection; indeed, by homogeneity for each $f \in \mathcal{P}_{d}(\ell_2^n(\mathbb{R}))$

\[
\sup_{x\in B_{\ell_2^n(\mathbb{R})}} |f(x)| = \sup_{x \in \mathbb{S}^{n-1}} |f(x)|\,.
\]

On the other hand,
since $\sum_{k=1}^n x_k^2=1$ on $\mathbb{S}^{n-1}$, the surjective restriction map from $\mathcal{P}_{\leq d}(\ell_2^n(\mathbb{R}))$
to $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$
is  non-injective for $d \geq 2$, implying
that $ \mathcal{P}_{\leq d}(\ell_2^n(\mathbb{R}))
\,\neq\,\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})
\,.$

A polynomial $f \in  \mathcal{P}(\mathbb{R}^n)$ is said to be harmonic, whenever $\Delta f=0$, where as usual

\[
\Delta : = \sum_{j=1}^n \frac{\partial^2}{\partial x_{j}^2} \colon
\,\, \mathcal P (\mathbb{R}^n) \to \mathcal P (\mathbb{R}^n)
\]

denotes  the Laplace operator. We  write $\mathcal{H}(\mathbb{R}^n)$ for the subspace of all harmonic polynomials in
$\mathcal{P}(\mathbb{R}^n)$, and $\mathcal{H}_k (\mathbb{R}^n)$ for all $k$-homogeneous, harmonic polynomials.
Similarly,  we define $\mathcal H_{\leq d}(\mathbb{R}^n)$.

Both spaces  $\mathcal{H}(\mathbb{R}^n)$ and $\mathcal{H}_d(\mathbb{R}^n)$ are $\mathcal{O}_n$-invariant in the sense that
for all $O \in \mathcal{O}_n$, we have
$f \circ O \in \mathcal{H}(\mathbb{R}^n) $ for  all $f \in \mathcal{H}(\mathbb{R}^n)$ as well as  $f \circ O \in \mathcal{H}_d(\mathbb{R}^n) $ for  all $f \in \mathcal{H}_d(\mathbb{R}^n)$.

Observe that

\begin{equation}\label{ludo1} \mathcal H_{\leq d}(\mathbb{R}^n) = \spa_{k \leq d} \mathcal H_{k}(\mathbb{R}^n) \quad \text{and} \quad
\mathcal H(\mathbb{R}^n) = \spa_k \mathcal H_k(\mathbb{R}^n)\,.
\end{equation}

In fact, if $f\in \mathcal{H} (\mathbb{R}^n)$ has degree $d$, then $f= \sum_{k=0}^d f_k$ for some $f_k \in \mathcal{P}_k(\mathbb{R}^n)$. But since $\triangle f_k$ for $k\geq 2$ is $(k-2)$-homogeneous (resp., $0$-homogeneous whenever for $k<2$), it follows by the unique homogeneous decomposition  $\triangle f = \sum_{k=0}^d\triangle f_k =0$  that $\triangle f_k =0$ for each $0 \leq k \leq d$.

Much of what follows is based on the following well-known decomposition of $\mathcal{P}_{d}(\mathbb{R}^n)$ into
harmonic subspaces (see, e.g.  <cit.>).

For each $d\geq 2$ and  $n\geq 1$

\begin{equation*}
\mathcal{P}_d(\mathbb{R}^n) =\mathcal{H} _d(\mathbb{R}^n) \oplus \|\cdot\|_2^2\,\, \mathcal{P}_{d-2}(\mathbb{R}^n)\,,
\end{equation*}

that is, every $f \in \mathcal{P}_d(\mathbb{R}^n)$
has a unique decomposition

\[
f(x ) = g(x) + \|x\|_2^2 \,h(x)\,,\quad x \in \mathbb{R}^n
\]

with $g \in \mathcal{H}_d(\mathbb{R}^n)$
and $h \in \mathcal{P}_{d-2}(\mathbb{R}^n)$ .

All restrictions of
polynomials  in $\mathcal{H}(\mathbb{R}^n)$ to  $\mathbb{S}^{n-1}$
are denoted by
\mathcal{H}(\mathbb{S}^{n-1})\,,
and they are called spherical harmonics on $\mathbb{S}^{n-1}$.

We write   $\mathcal{H}_d(\mathbb{S}^{n-1})$ for the
space collecting all  restrictions of polynomials from  $\mathcal{H}_d(\mathbb{R}^n)$ to $\mathbb{S}^{n-1}$,
and define $\mathcal{H}_{\leq d}(\mathbb{S}^{n-1})$ similarly.
Endowed with the supremum norm taken
on $\mathbb{S}^{n-1}$, both spaces
form finite dimensional
$\mathcal{O}_n$-invariant subspaces of the Banach space $C(\mathbb{S}^{n-1})$.

Iterating Lemma <ref>, leads to the following well-known decomposition theorem of spherical harmonics.

We have

\begin{equation}\label{LH9}
\mathcal H_{\leq d} (\mathbb{S}^{n-1})= \bigoplus_{k \leq d}\mathcal{H}_k ( \mathbb{S}^{n-1})
\quad \text{and}  \quad
\mathcal H ( \mathbb{S}^{n-1})= \bigoplus_k \mathcal{H}_k ( \mathbb{S}^{n-1})\,,
\end{equation}

\begin{equation}\label{klm2}
\mathcal P_d(\mathbb{S}^{n-1})
=  \bigoplus_{j=0}^{\lfloor d/2\rfloor} \,\mathcal{H}_{d-2j}(\mathbb{S}^{n-1})\,,
\end{equation}

where all  decompositions are orthogonal in $L_2( \mathbb{S}^{n-1})$. In particular,

\begin{equation}\label{ludoAA}
\mathcal H_{\leq d} ( \mathbb{S}^{n-1})=\mathcal P_{\leq d} ( \mathbb{S}^{n-1})
\quad \text{and} \quad
\mathcal H (\mathbb{S}^{n-1})=\mathcal P ( \mathbb{S}^{n-1})\,.
\end{equation}

Looking at (<ref>), we immediately see (<ref>).
Moreover,  since $ \|x\|_2^2=1$
for every  $x \in \mathbb{S}^{n-1} $, we deduce from Lemma <ref> that (<ref>) holds algebraically.
Together with (<ref>) this implies  (<ref>).
For the simple argument (based on Green's identity) showing that  $\mathcal{H}_{k}(\mathbb{S}^{n-1})$ and $\mathcal{H}_{\ell}(\mathbb{S}^{n-1})$ are
orthogonal in $L_2( \mathbb{S}^{n-1})$
see, e.g. <cit.>.

As an immediate consequence of Lemma <ref>
and (<ref>), we see that for all    $n \geq 2$ and $d \geq 1$

\begin{equation} \label{dimfor}
N_{n,d} := \dim \mathcal{H}_d(\mathbb{S}^{n-1})
=  \frac{(n+ 2d-2)(n+d-3)!}{d!(n-2)!}\,.
\end{equation}

The following density theorem is fundamental for our coming purposes and again well-known.

$\mathcal H ( \mathbb{S}^{n-1})=\spa_{k}\mathcal{H}_k ( \mathbb{S}^{n-1})$ is dense in $C(\mathbb{S}^{n-1})$.

For the proof of this, in a first step, one deduces from the   Stone-Weierstrass theorem  that $\mathcal P(\mathbb{S}^{n-1})$ is dense
in $C(\mathbb{S}^{n-1})$. Then the conclusion follows from (<ref>) and (<ref>).

We will use throughout the following notation: given two sequences of positive numbers $(a_{n,d})_{n,d \in \mathbb{N}}$ and $(b_{n,d})_{n,d \in \mathbb{N}}$, we write

\[ a_{n,d} \sim_{c(n)} b_{n,d}, \]

if there exist constants $c_1(n), c_2(n) > 0$ that depend solely on $n$ (and not on $d$) such that

\[ c_1(n) a_{n,d} \leq b_{n,d} \leq c_2(n) a_{n,d} \]

for all $n, d \in \mathbb{N}$. The notation $a_{n,d} \sim_{c(d)} b_{n,d}$ is defined analogously.

In concluding this preliminary section, it is imperative to emphasize that all elements within the aforementioned function spaces have values in the complex plane $\mathbb{C}$. For a treatment on real-valued function, we refer to Section <ref>.

§ RUDIN TRIPLES

Below, we delineate an abstract framework that we had already applied to other ostensibly disparate scenarios (such as functions on Boolean cubes [8],  Dirichlet series [7] or trace class operators [9]) for computing projection constants of finite-dimensional subspaces within $C(K)$ in a somewhat systematic manner.

The approach outlined here relies on the compact space $K$ having some form of group structure and emerges from isolating techniques developed by Rudin in the study of minimal projections through averaging techniques, as stated in Rudin's works [25, 27]. However, the theory and method presented are not explicitly detailed in any of our previously mentioned articles. This systematic development may prove beneficial in various other contexts.

We consider  triples  $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$, where

* $K$ is a compact space together with a Borel probability measure $\mu$ on $K$ ,

* $G$ is a compact  group with  its Haar measure  $\mathrm{m}$ ,

* and  $\varphi$ a multiplicative and continuous mapping from $G$ into
$\text{Hom}(K)$, the space  of all  homeomorphisms from $K$ onto $K$, that is,

\[
\varphi: G \longrightarrow \text{Hom}(K)\,, \quad g \mapsto \varphi_g\,,
\]

is such that    $\varphi_{gh}= \varphi_h\circ \varphi_g$ for all $g,h\in G $ and   the mapping  $G \times K \ni (g, x) \mapsto \varphi_gx \in K$ is continuous.

A subspace $S$ of $C(K)$ is said to be $(G,\varphi)$-invariant
($\varphi$-invariant for short), whenever $ f \circ \varphi_g \in S $ for all $g \in G$ and $f \in S$.

We call $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ a  Rudin triple whenever

\forall \,\,\,x, \, y \in K \,\,\,\exists  \,\,\, g \in G \colon \quad \varphi_g x = y

$\forall \,\,\,g \in G \,\,\,\forall \,\,\,f \in C(K) \colon \quad
\int_{K}  f \circ \varphi_g d \mu = \int_{K} f d \mu\,

There is a family $(S_j)_{j \in I}$ of finite dimensional $\varphi$-invariant subspaces in $C(K)$ with dense union.

The following lemma shows that for any $y \in K$ the push forward measure under the mapping from $G$ to $K$ given by $G \ni g \mapsto \varphi_g(y)\in K$ coincides with

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple. Then for all $f \in C(K)$ and $y \in K$

\[
\int_K f d\mu = \int_G  f\big(\varphi_gy\big)\, d\mathrm{m}(g) \,.
\]

Let $y \in K$ be given. Since the mapping
$G \times K \longrightarrow K\,, \,\, (g,x) \mapsto f(\varphi_gx)$  is continuous,  Fubini's theorem combined with the condition $(R2)$ yields the following

\[
\int_K \int_G f\big(\varphi_gx\big)\,  d\mathrm{m}(g)d\mu(x)=\int_G \int_K f\big(\varphi_gx\big)\, d\mu(x)d\mathrm{m}(g) = \int_K f(x)\,d\mu(x)\,.
\]

Fixing $x\in K$, there is $h_x \in G$ such that $\varphi_{h_x}y = x$, and hence by condition $(R1)$, the multiplicativity of $\varphi$,
and the invariance of the Haar measure $\mathrm{m}$

\[
\int_G f\big(\varphi_gx\big)\, d\mathrm{m}(g) = \int_G f\big(\varphi_g(\varphi_{h_x}y)\big)\, d\mathrm{m}(g)
= \int_G f\big(\varphi_{h_x g}y\big)\, d\mathrm{m}(g) = \int_G f\big(\varphi_gy\big)\, d\mathrm{m}(g)\,,
\]

and so

\[
\int_K \int_G f\big(\varphi_gx\big) \, d\mathrm{m}(g)d\mu(x) = \int_G f\big(\varphi_gy\big) \, d\mathrm{m}(g)\,. \qedhere
\]

For future reference, it is important to observe that the preceding lemma extends its applicability to a continuous function $f$ on $K$ with values in a Banach space $X$, wherein the conventional integrals are to be replaced by the corresponding Bochner integrals. To establish this, apply the earlier scalar formula to all functions $x^\ast \circ f$, where $x^\ast \in X^\ast$, and invoke the Hahn-Banach theorem for the proof.

Later on, we will explore various scenarios where the aforementioned framework can be effectively applied.

§.§ Reproducing kernels

In this section we adapt some well-known facts
on reproducing kernels to our setting of Rudin triples.
All arguments are rather standard and here repeated for the sake of completeness.

For $L_2(\mu)$-closed subspaces $S$ of $L_2(\mu)$ we denote by $\pi_S$  the orthogonal
projection from $L_2(\mu)$ onto $S$.
The following lemma shows that the orthogonal projection $\pi_S$ commutes with the action of $G$ on $C(K)$.

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple and let $S$ be a
$\varphi$-invariant, finite dimensional subspace
of   $C(K)$. Then $\pi_S$ commutes with the action which
$\varphi$ induces on $C(K)$, that is,
for all $f \in C(K)$ and $g \in G$

\begin{equation*}
\pi_S(f\circ \varphi_g) = \pi_S(f)\circ \varphi_g\,.
\end{equation*}

Note that  $S^\perp$ is also $\varphi$-invariant, and fix  $ g\in G$ and
$f\in L_2(\mu)$. Then

\[
\text{$(I\!d-\pi_S)(f)\circ \varphi_g\in S^\perp$
\,\,\,\, and \,\,\,\,$f\circ\varphi_g=\pi_S(f)\circ \varphi_g+(I\!d-\pi_S)(f)\circ \varphi_g$\,,}
\]

and hence

\begin{equation*}
\pi_S(f\circ \varphi_g)=\pi_S(\pi_S(f)\circ \varphi_g)+\pi_S((I\!d-\pi_S)(f)\circ \varphi_g)=\pi_S(f)\circ \varphi_g\,.
\qedhere
\end{equation*}

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple and  $S$ a $\varphi$-invariant, finite dimensional subspace of $C(K)$.
Then there is a unique continuous function $\mathbf{k}_S: K\times K \to \mathbb C$ such that
for all $f\in L_2(\mu)$  and $x \in K$

(\pi_Sf)(x)=\big\langle f, \mathbf{k}_S(x, \cdot)\big\rangle_{L_2(\mu)}\,.

Moreover, the following properties hold:

$\mathbf{k}_S(x, \cdot)\in S$  for all $x \in K$ ,

$\mathbf{k}_S(x, y)=\overline{\mathbf{k}_S(y, x)}$   for all $x,y \in K$ ,

\mathbf{k}_S(x,y)
for all $g \in G$ and $x  \in K$ ,

$\mathbf{k}_S(x,x) = \dim S$    for all  $x \in K$ ,

$\big(\int_K |\mathbf{k}_S(x, y)|^2 d\mu(y)\big)^{\frac{1}{2}}= \sqrt{\dim S}$   for all  $x \in K$ .

Choose an orthonormal basis $(f_j)_{j=1}^{\dim S}$
of $S$.
Extending it to an orthonormal basis of $L_2(\mu)$, we see
that for all

\[
\pi_Sf = \sum_{j=1}^{\dim S}
\big\langle f,f_j \big\rangle f_j\,.
\]

Then for almost all $x \in K$

\[
\pi_Sf(x) =
\sum_{j=1}^{\dim S}
\Big(\int_K f(y) \overline{f_j(y)}\, d\mu(y)\Big) f_j(x)
\int_K f(y)\,
\Big(\sum_{j=1}^{\dim S}
\overline{f_j(y)} f_j(x)  \Big)\,d\mu(y)
\,.
\]

This shows that the function $\mathbf{k}_S\colon  K\times K \to \mathbb C$ given by

\begin{align} \label{ker-def}
\text{
$\mathbf{k}_S(x,y) := \sum_{j=1}^{\dim S}
\overline{f_j(x)} f_j(y)$,   \quad\, $x,y\in K$,}
\end{align}

satisfies the required equation in $(i)$.  Since the uniqueness of this kernel as well as $(ii)$ and $(iii)$ now are obvious, it remains to check $(iv)$ and $(v)$.

Recall by Lemma <ref> that the orthogonal projection $\pi_S$ commutes with the action of $G$ on $C(K)$ (that is, with  $\varphi_g$ for all $g \in G$).
Thus, using  this and condition  $(R2)$ from Definition <ref>, one gets for all $g \in G$, $x  \in K$, and $f \in C(K)$

\begin{align*}
\big\langle f, \mathbf{k}_{S}(\varphi_gx, \cdot)\big\rangle
\pi_S(f \circ \varphi_g)(x)
\\&
\int_K (f \circ \varphi_g)(\zeta) \, \overline{\mathbf{k}_S(x,\zeta)}\,d\mu(\zeta)
\\
\int_K (f \circ \varphi_g)(\varphi_g^{-1}\zeta) \,
\overline{\mathbf{k}_S(x,\varphi_g^{-1}\zeta)}\,
\big\langle f, \mathbf{k}_S(x,\varphi_g^{-1}(\cdot))\big\rangle\,.
\end{align*}

This proves that
$\mathbf{k}_{S}(\varphi_gx, y)=
\mathbf{k}_S(x,\varphi_g^{-1}y)
for all $x,y \in K$,
which is the conclusion.

$(v)$ For  $x,y \in K$, choose according to  condition $(R1)$ from Definition <ref>
some $g \in G$ such that $x = \varphi_g y$.
Then by $(iv)$

\begin{eqnarray*}
\mathbf{k}_S(x, x)
\mathbf{k}_S(\varphi_gy, \varphi_gy)
\mathbf{k}_S(y,y)\,,
\end{eqnarray*}

and consequently

\[
\mathbf{k}_S(x, x)
\int_K \sum_{j=1}^{\dim S}
|f_j(x)|^2 d\mu(x)
= \sum_{j=1}^{\dim S}
\|f_j\|_2^2 \,d\mu(x) = \dim S\,.
\]

Using (<ref>) and orthogonality,  we for all  $x \in K$ have

\begin{equation*}
\int_K |\mathbf{k}_S(x, y)|^2\, d\mu(y)
\sum_{j=1}^{\dim S} |f_j(x)|^2 = \mathbf{k}_S(x, x)\,,
\end{equation*}

and  hence the claim is an immediate  consequence of (vi).
This completes the proof.

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple and let $S$ be a $\varphi$-invariant, finite dimensional subspace $S$ of  $C(K)$. Then for all $x\in K$ we have

\[
\big\|\pi_S:C(K) \to S\big\|=\int_{K }|\mathbf{k}_S(x,y) |\, d\mu(y) \,.
\]

By the Riesz representation theorem
we have that

\begin{align*}
\big\|\pi_S:C(K) \to S\big\|
\sup_{\|f\|_\infty \leq  1} \sup_{x \in K}
\Big|   \int_{K } f(y) \overline{\mathbf{k}_S(x,y)}\, d \mu(y) \Big|
\\&
\sup_{x \in K}
\sup_{\|f\|_\infty \leq  1}
\Big|   \int_{K } f(y) \overline{\mathbf{k}_S(x,y)}\,d \mu(y) \Big|
\sup_{x \in K}
\int_{K }|\mathbf{k}_S(x,y) |\,d\mu(y)\,.
\end{align*}

We use  Lemma <ref> which shows that, for all $x,z \in K$,

\[
\int_{K }|\mathbf{k}_S(x,y) |\,d\mu(y)
\int_{K }|\mathbf{k}_S(z,y) |\,d\mu(y) \,;
\]

indeed, choose by condition  $(R1)$ from  Definition <ref> some $g \in G$ such that $\varphi_gz=x$. Then we deduce from condition  $(R2)$ of Definition <ref> and Theorem <ref>, (iii) that

\begin{align*}
\int_{K }|\mathbf{k}_S(x,y) |\,d\mu(y) =
\int_{K }|\mathbf{k}_S({\varphi_gz},y) |\,d\mu(y)
= \int_{K }|\mathbf{k}_S (z \circ \varphi_g^{-1},y) |\, d\mu(y) = \int_{K }|\mathbf{k}_S(z,y)|\,d\mu(y)\,,
\end{align*}

and this completes the proof.

§.§ Accessability

Given a Rudin triple $\big((K,\mu),
(G, \mathrm{m}),\varphi\big)$, we  say that the
$\varphi$-invariant, finite dimensional subspace $S$ of $C(K)$ is accessible whenever the restriction of  the orthogonal projection $\pi_S\restrict{C(K)}$ is the unique projection that in the following sense 'harmonizes' with the group representation which $\varphi$
induces on $K$.

Let $\big((K,\mu),
(G, \mathrm{m}),\varphi\big)$ be a Rudin triple.
A finite-dimensional subspace $S$ of $C(K)$ is termed  accessible if it is $\varphi$-invariant and  $\pi_S\restrict{C(K)}$ is the unique projection $\mathbf{Q}$ on $C(K)$ onto $S$ that commutes with all  $\varphi_g, \, g \in G$, that is,
$\mathbf{Q} (f \circ \varphi_g) = \mathbf{Q}(f) \circ \varphi_g$
for all $f \in S, g \in G$.

The following result is one of our main abstract tools. It shows how to compute the projection constant of a subspace $S$ in terms of $L_1$-norm of  the kernel $\mathbf{k}_S(x,\cdot)$ for  $x\in K$.

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple and  $S$  a $\varphi$-invariant, finite dimensional accessible subspace $S$ of $C(K)$. Then
for all $x \in K$ we have
\[
\boldsymbol{\lambda}(S) =  \big\|\pi_S:C(K) \to S\big\|=\int_{K }|\mathbf{k}_S(x,y) |\,d\mu(y) \,.
\]

The proof uses the  standard technique
of averaging projections - see, e.g.
[25] or <cit.>.

Take any projection $\mathbf{Q}$ of $C(K)$ onto $S$, and for every $g\in G$ consider the composition operator $\phi_g \in \mathcal{L}(C(K))$ given by

\[
\phi_g(f) := f\circ \varphi_{g}, \quad\, f\in C(K)\,.
\]

Clearly, $\phi_e$ is the identity on $C(K)$ (where $e$ denotes the unity in $G$), and $\phi_{gh} = \phi_g \phi_h$ for all $g, h \in G$. Furthermore,

\begin{equation*}
G \ni g \mapsto \phi_{g^{-1}}\mathbf{Q} \, \phi_g \in \mathcal{L}(C(K))
\end{equation*}

is Bochner integrable, as it is bounded by the Banach-Steinhaus theorem and measurable due to its weak measurability. Then  the
Bochner integral

\begin{equation*}\label{equation rudy}
\int_{G} \phi_{g^{-1}}\mathbf{Q}\,\phi_g\,d\mathrm{m}(g)
\in \mathcal{L}(C(K))
\end{equation*}

is a projection from $C(K)$ that commutes with all $\varphi_g, \, g \in G$. Since $S$ is assumed to be accessible, we have that

\begin{equation*}
\pi_S= \int_{G} \phi_{g^{-1}}\mathbf{Q} \,\phi_g\,d\mathrm{m}(g)\,,
\end{equation*}

and consequently  the first equality follows by the triangle inequality (for Bochner integrals). The second claim is then
a consequence of Corollary <ref>.

Motivated by the practical challenges in verifying the least restrictive definition of accessibility, we now introduce a stronger notion that implies accessibility, hence its name.  Notably, for the applications we will explore later, this stronger notion is the one we will actually be employing.

Let $\big((K,\mu),
(G, \mathrm{m}),\varphi\big)$ be a Rudin triple.
We say that a  $\varphi$-invariant, finite dimensional subspace $S$ of  $C(K)$ is  strongly accessible with respect to $x_0 \in K$, whenever  the following holds:

\begin{equation}\label{point}
\text{$\forall$    $f \in S$ for which $f \circ \varphi_g = f$ for all $g \in G$ with $\varphi_g x_0 = x_0$
\, $\exists$ $\lambda \in \mathbb{C}$  $\colon$ \,   $f = \lambda \,\mathbf{k}_{S}(x_0, \cdot) $.}
\end{equation}

We call $S$  strongly accessible whenever it is strongly accessible with respect to some  $x_0 \in K$.

Now we see how this definition implies accessibility.

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple and let $S$ be a $\varphi$-invariant, finite dimensional subspace of  $C(K)$.
If $S$ is strongly accessible, then it is accessible.

The proof requires  preparation.

Let $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ be a Rudin triple and let $H$ and $S$ be both finite dimensional and $\varphi$-invariant subspaces of $C(K)$.
Then  every operator $T\colon H\to  S$ that commutes with the action of $G$, is a scalar multiple of   $\pi_S$, provided $S$ is strongly accessible.

Moreover, if $H$ is  orthogonal to $S$, and $\mathbf{Q}$ is
a projection from $H \bigoplus S$ onto $S$ that commutes with the action of
on $C(K)$,  then $\mathbf{Q}=\pi_{S}\restrict{H \bigoplus S}.$

Let $S$ be strongly accessible with respect to   $x_0 \in K$. By the assumption on $T$
and Theorem <ref>, (iv)  for  every $g \in G$ with $\varphi_gx_0=x_0$ one has

\[
T \big(\mathbf{k}_H(x_0, \cdot)\big)
\circ \varphi_g
= T \big(\mathbf{k}_H(x_0, \varphi_g (\cdot)) \big)
= T \big(\mathbf{k}_H({\varphi_g^{-1}x_0}, \cdot)  \big) = T \big(\mathbf{k}_H(x_0, \cdot)\big)\,,
\]

and hence by the definition of strong accessability (as in  (<ref>)) there is $\gamma \in \mathbb{C}$ for which

\begin{equation}\label{firststep}
T  \mathbf{k}_H(x_0, \cdot) = \gamma\,  \mathbf{k}_S(x_0, \cdot)\,.
\end{equation}

On the other hand, we conclude from Theorem <ref> (i),(ii) that for all $h \in H$

\[
h= \pi_H h = \int_{K} h(x) \,\mathbf{k}_H(x, \cdot)\,d\mu(x)\,,
\]

where the integral is meant to be  the Bochner integral of the vector-valued function

K \longrightarrow C(K), \quad x \mapsto \big[\zeta \mapsto h(x) \mathbf{k}_H(x, \zeta)\big]\,.

Then by Lemma <ref> (in the form of Remark <ref>) and again Theorem <ref>, (iii) we for all $h \in H$

\begin{align*}
h =
\int_{G} h(\varphi_gx_0)
\mathbf{k}_H(\varphi_gx_0, \cdot)\, d\mathrm{m}(g)
\int_{G} h(\varphi_gx_0)  \mathbf{k}_H(x_0, \cdot)
\circ \varphi_g^{-1}\,d\mathrm{m}( (g)\,,
\end{align*}

so that

\begin{align*}
\int_{G} h(\varphi_gx_0) \, T\big(\mathbf{k}_H(x_0, \cdot) \circ \varphi_g^{-1}\big) d\mathrm{m}( (g)
\\&
\int_{G} h(\varphi_gx_0)\, T\big(\mathbf{k}_H(x_0, \cdot)\big) \circ \varphi_g^{-1}\,d\mathrm{m}( (g)
\\&
\gamma \,
\int_{G} h(\varphi_gx_0) \, \mathbf{k}_S(x_0, \cdot)
\circ \varphi_g^{-1}\,d\mathrm{m}( (g)
\\&
\gamma\, \int_{G} h(\varphi_gx_0) \, \mathbf{k}_S(\varphi_gx_0, \cdot)\,d\mathrm{m}( (g)
\gamma \,
\int_{K} h(x) \,\mathbf{k}_S(x, \cdot)\, d\mu(x)
= \gamma \, \pi_S(h)
\,;
\end{align*}

here we again use that $T$ commutes with the action of $G$ on $C(K)$, Equation (<ref>),
Lemma <ref> (in the form of Remark <ref>), and   Theorem <ref>, (i), (iii (iv).
This proves the first part of the proposition.

To see the second assertion, note  that by  the first part of the lemma  we have $\mathbf{Q}\restrict{H}=\gamma \,\pi_S\restrict {H}$ for some
$\gamma  \in \mathbb{C}$. But since by assumption $H\subset S^\perp$, this implies $\mathbf{Q}\restrict{H}= 0=\pi_{S}\restrict {H}$.
On the other hand,  since $Q$ is a projection onto $S$, we see that  $\mathbf{Q}\restrict{S}=Id_{S}=\pi_S\restrict {S}$, which finishes
the proof.

Let $\mathbf{Q}$  be a projection from $C(K)$ onto $S$, which  commutes with the action of
on $C(K)$.
By  the (density) assumption $(R3)$ from Section <ref>, it suffices to show that for each

\[
\mathbf{Q}\restrict{S_j} = \pi_S\restrict{S_j}\,.
\]

Fixing $\iota$, we define the subspace

\[
H := \big\{f-\pi_Sf\,:\,\,f\in S_j \big\} \subset C(K)\,.
\]

Then $H$ is  $\varphi$-invariant; indeed, by the facts that $\pi_S$ commutes with $G$
and $S_j $ is $\varphi$-invariant, for every   $f\in S_j $
and $g \in G$, we have

\[
(f-\pi_S f)\circ \varphi_g=f\circ \varphi_g -\pi_S f\circ \varphi_g= f\circ \varphi_g -\pi_S(f\circ \varphi_g)\in H\,.
\]

Since $H\perp S$ and $\mathbf{Q}$ commutes with the action of $G$
on $C(K)$,
Lemma <ref> (the second part applied to the restriction  of $\mathbf{Q}$ to $H \oplus S$) shows  that
$\mathbf{Q}\restrict{H \oplus S}=  \pi_S\restrict{H \oplus S}\,,$
so  in particular
\mathbf{Q}\restrict{H}=  \pi_S\restrict{H} = 0\,.
But then
\mathbf{Q}f = \mathbf{Q}(f-\pi_S f)+ \mathbf{Q}(\pi_Sf) =   \pi_Sf
for  $f \in S_j$,
which completes the argument.

Accessability is inherited under finite orthogonal sums and conjugation.

Under the general assumptions on $K$ of  Section <ref>, a finite $L_2(\mu)$-orthogonal sum $S=\bigoplus S_k$ of accessible subspaces $S_k$ is  again accessible,
and $\mathbf{k}_S(x, \cdot) = \sum \mathbf{k}_{S_k}(x, \cdot) $ for all $x \in K$.
In particular, for all $x \in K$

\begin{equation*}\label{integral formula sumA}
\boldsymbol{\lambda}(S_1 \oplus S_2) = \big\|\pi_{S_1} + \pi_{S_2}:C(K) \to S_1 \oplus S_2  \big\|
\int_{K}\big|\mathbf{k}_{S_1}(x, \cdot) +  \mathbf{k}_{S_2}(x, \cdot)\big|\, d\mu
\,.
\end{equation*}

if $S$ is accessible, then its conjugate $\overline{S}$ is accessible, and
$\mathbf{k}_{\overline{S}}(x, \cdot) = \overline{\mathbf{k}_{S}(x, \cdot)}$
for all $x \in K$ .

Fix some $x \in K$.
That $S_1 \oplus S_2$ is  $\varphi$-invariant  is straightforward.
Note that $\pi_{S_1 \oplus S_2}  = \pi_{S_1} + \pi_{S_2}$
is the orthogonal projection from $L_2(\mu)$ onto $S_1 \oplus S_2$. Then by
Theorem <ref>
for all $f\in L_2(\mu)$,  we have

\[
\pi_{S_1 \oplus S_2}f = \pi_{S_1}f + \pi_{S_2}f =
\int_{K}\mathbf{k}_{S_1}(x, \cdot) \, d\mu + \int_{K }\mathbf{k}_{S_2}(x, \cdot) \, d\mu
\int_{K }\big(\mathbf{k}_{S_1}(x, \cdot)\, d\mu +  \mathbf{k}_{S_2}(x, \cdot)\big)\, d\mu\,.
\]

Hence by the uniqueness of reproducing kernels  we get

\[
\mathbf{k}_{S_1+S_2}(x, \cdot)
\,=\,
\mathbf{k}_{S_1}(x, \cdot) +  \mathbf{k}_{S_2}(x, \cdot)\,, \quad\, x\in K\,.
\]

Let us now show that $S_1 \oplus S_2$ is accessible. So let $\mathbf{Q}$ be a projection on
$C(K)$ onto $S_1 \oplus S_2$ which commutes with the action of $G$ on
$C(K)$. We claim that $\mathbf{Q} = \pi_{S_1 \oplus S_2}$. Indeed, consider the two projections

\[
\text{$\mathbf{Q}_{S_1} = \pi_{S_1}\circ   \mathbf{Q} $ \quad and \quad $\mathbf{Q}_{S_2} = \pi_{S_2}\circ   \mathbf{Q} $}
\]

on $C(K)$ onto $S_1$ and ${S_2}$, respectively.
Since $\pi_{S_1}$ and $\pi_{S_2}$ both commute with the action of $G$ on
$C(K)$, also $\mathbf{Q}_{S_1}$ and $\mathbf{Q}_{S_2}$ do. Then by the accessibility of $S_1$ and $S_2$ we see that

\[
\text{$\mathbf{Q}_{S_1} = \pi_{S_1}$ \quad and \quad $\mathbf{Q}_{S_2} = \pi_{S_2}$}\,,
\]

and hence for all $f \in C(K)$ as desired

\[
\mathbf{Q}f = \pi_{S_1}(\mathbf{Q}f)  + \pi_{S_2}(\mathbf{Q}f) = \pi_{S_1}f + \pi_{S_2}f = \pi_{S_1 \oplus S_2} f\,.
\]

To conclude the proof just note that (<ref>) is then a direct consequence of Theorem <ref>.

The proof of the statement regarding the conjugate subspace follows straightforwardly from the
fact that  $\pi_{\overline{S}}(f)=\overline{\pi_S(\overline{f})}$ for $f\in C(K)$.   We leave it to the reader to fill in these details.

§ INTEGRAL REPRESENTATION

We provide integral formulas for the three projection constants of the Banach spaces $\mathcal{H}_{d}(\mathbb{S}^{n-1})$, $\mathcal{P}_{ d}(\mathbb{S}^{n-1})$, and $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$ within the framework of Rudin triples $\big((K,\mu),(G,\mathrm{m}),\varphi\big)$ as explained in Section <ref>. Recall that here all of these three spaces are considered as  finite-dimensional subspaces of  $C(\mathbb{S}^{n-1})$.

To start,
for $K$ we take the sphere $\mathbb{S}^{n-1}$ in the real Hilbert space $\ell^n_2(\mathbb{R}^n)$ endowed with  the normalized surface measure $\sigma_n$.
For  $G$  the orthogonal  group $\mathcal{O}_n$ together with $\mathrm{m}$, its  Haar measure  (compact groups are unimodular), and finally for $\varphi$ we

\[
\varphi: \mathcal{O}_n \longrightarrow \text{Hom}(\mathbb{S}^{n-1})\,, \quad O \mapsto [z \mapsto Oz]\,,
\]

so  $\mathcal{O}_n$ acts on $\mathcal{L}(C(\mathbb{S}^{n-1}))$ through

\begin{equation}\label{Orudin}
\phi: \mathcal{O}_n \longrightarrow \mathcal{L}(C(\mathbb{S}^{n-1})) \,,\quad  O \mapsto [z \mapsto f(Oz) ]\,.
\end{equation}

The triple  $\big( (\mathbb{S}^{n-1},\sigma_n), (\mathcal{O}_n, \mathrm{m}), \varphi\big)$  is a Rudin triple.

It is  well-known that both  properties from $(R1)$ and $(R2)$ as stated in  Definition <ref> are fulfilled.
Let us see, that   the density condition $(R3)$
holds true as well. Indeed,
from Theorem <ref>
we know that all spaces $\mathcal{H}_k ( \mathbb{S}^{n-1})$ form a countable family of
$\mathcal{O}_n$-invariant subspaces with dense union in  $C(\mathbb{S}^{n-1})$.

§.§ Legendre harmonics

Our aim now is to find concrete
descriptions of the reproducing kernels of the three
$\mathcal{H}_{ d}(\mathbb{S}^{n-1})$,
and $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$, which in particular allows
to show that they are accessible.
Although the content of this subsection is essentially familiar to the approximation theory community, we choose to provide full proofs. However, it is important to emphasize that we cannot directly cite the results we need, and in many cases, the proofs have been adapted or require some minor modifications. For this reason, we believe that offering a comprehensive overview is necessary, avoiding the reader's need to constantly refer to other texts to fill in the details.

Given $d \in \mathbb{N}_0$, we demonstrate the existence and uniqueness of a harmonic polynomial that remains invariant under all orthogonal transformations preserving the vector $e_1 \in \mathbb{R}^n$ and which attains  the value 1 at $e_1$; compare with

First, we need to establish some notation. We denote by $\mathcal{O}_n(\xi)$ the set of all orthogonal transformations that fix the vector $\xi \in \mathbb{S}^{n-1}$.

There is a unique  polynomial
$Q \in \mathcal{H}_d(\mathbb{R}^n)$ which fulfills the following two properties:

$Q\circ A = Q$ for all $A\in \mathcal{O}_n(e_1)$ ,

$Q(e_1) = 1$ .

Moreover, this unique polynomial $($known as the Legendre harmonic of degree $d$ in $n$ variables and denoted by $L_{n,d} \colon \mathbb{R}^n \to \mathbb{R})$, is given by

\begin{equation*}
L_{n,d}(x) = \sum_{j=0}^{\lfloor d/2\rfloor} b_j(n,d)  x_1^{d-2j}  \Vert(x_2, \ldots, x_{n})\Vert_2^{2j},
\end{equation*}

where the coefficients $b_j(n,d)$ are determined by:

\begin{equation}\label{unique-harm}
b_j(n,d) = \frac{(-1)^j d!\, \Gamma\Big( \frac{n-1}{2}\Big)}{4^j j!\,(d-2j)!\, \Gamma\big( j+\frac{n-1}{2}\big)}.
\end{equation}

Take $Q$ verifying both properties and choose a representation

\[
Q(x) = \sum_{\substack{\alpha \in \mathbb{N}^n_0\\ |\alpha|=d}} c_\alpha x^\alpha, \quad x \in \mathbb{R}^n\,.
\]

We write $x = (x_1,x') \in \mathbb{R} \times \mathbb{R}^{n-1}$ for  $x \in \mathbb{R}^n$, and note that
every $n\times n$-matrix $A$ belongs to $\mathcal{O}_n(e_1)$ if and only if it has the form

\begin{equation}\label{0a0}
\begin{pmatrix}
1 & 0  \\
0 & A' \\
\end{pmatrix}
\end{equation}

with $A' \in \mathcal{O}_{n-1}$. We fix $x_1 \in \mathbb{R}$,  and consider the polynomial

\[
g(x') := Q(x_1, x'),  \quad x' \in \mathbb{R}^{n-1}.
\]

Then for all $A' \in \mathcal{O}_{n-1}$ and $x' \in \mathbb{R}^{n-1}$

\[
g(A'x')= Q(x_1,A'x' ) = Q(A(x_1,x') ) = Q(x_1,x' )=  g(x')\,,
\]

and in particular $g(x') = g(-x')$. Consequently,

\[
g(t, 0, \ldots, 0) = \sum_{j=0}^{\lfloor d/2\rfloor} a_j t^{2j},  \quad t \in \mathbb{R}\,,
\]

where the complex coefficients $a_j= a_j(x_1)$ are functions in $x_1$.
Choosing for $x' \in \mathbb{R}^{n-1}$ some $A' \in \mathcal{O}_{n-1}$ such that $A' x' = \|x'\|_2 e_1$,
we obtain

\[
\sum_{|\alpha|=d} c_\alpha x_1^{\alpha_1} (x')^{(\alpha_2, \ldots, \alpha_{n-1})} =Q(x_1,x') = g(x') = g(\|x'\|_2 e_1) = \sum_{j=0}^{\lfloor d/2\rfloor} a_j(x_1) \|x'\|_2^{2j}\,.
\]

If we  now compare coefficients,  then for all $x \in \mathbb{R}^n$

\[
Q(x) = \sum_{j=0}^{\lfloor d/2\rfloor} b_j x_1^{d-2j} \|x'\|_2^{2j}\,.
\]

Finally, it remains to verify that  the coefficients $b_j = b_j(n,d)$  are those from (<ref>).  Indeed, using that $Q$ is harmonic,
we see that for all $x \in \mathbb{R}^n$

\[
0 =\triangle Q(x) =
\sum_{j=0}^{\lfloor d/2\rfloor}
\big[(d-2j)(d-2j-1) b_j + (2n-2 +4j)(j+1)b_{j+1}\big]
x_1^{d-2-2j} \|x'\|_2^{2j}\,.
\]

Then for $j = 1, \ldots, \lfloor d/2\rfloor  $

\[
b_j =  -  \frac{(d-2j+2)(d-2j+1)}{2j (2j+n-3) } b_{j-1}\,,
\]

and since $b_0 =1$, we obtain the  desired  claim.

Conversely, let us explain why the polynomial  $L_{n,d}$ has the desired properties. Clearly it is $d$-homogeneous, $L_{n,d}(e_1)=1$,
and the last calculations  prove  its  harmonicity. Moreover, looking at (<ref>) also property $(a)$ is clear.

In order to derive a few important consequences of the preceding proposition, we consider the equality

\begin{equation} \label{legent}
L_{n,d}^{ \diamond}(\langle x, e_1\rangle) =  L_{n,d}(x)  \,, \quad \,x \in \mathbb{S}^{n-1}\,,
\end{equation}

where $L_{n,d}^{ \diamond}\colon [-1,1]\to \mathbb{R} $   is given by

\begin{equation} \label{eq: Legendre polynomial}
L_{n,d}^{ \diamond}(t) :=
\sum_{j=0}^{\lfloor d/2\rfloor} b_j(n,d) \,t^{d-2j}\,    (1-t^2)^{j}\,, \quad\, t\in [-1, 1]\,.
\end{equation}

This  polynomial  is referred to as the 'Legendre polynomial of degree $d$ in $n$ dimensions'. The subsequent consequence of Proposition <ref> is adapted from <cit.>.

Let $f \in \mathcal{H}_d(\mathbb{S}^{n-1})$ and $\xi \in \mathbb{S}^{n-1}$. Then the following are

$f \circ A = f$ for all $A \in \mathcal{O}_n(\xi)$ ,

$f = f(\xi) L_{n,d}^{ \diamond}(\langle\pmb{\cdot},\xi\rangle)$ .

Using the properties of $L_{n,d}$ isolated in Proposition <ref>, the implication $(2)\Rightarrow(1)$ is obvious.
In order to  check that  $(1)\Rightarrow(2)$, we first assume that $\xi = e_1$, and  choose
$g \in \mathcal{H}_d(\mathbb{R}^n)$ such that $g|_{\mathbb{S}^{n-1}} = f$. By assumption for each $A \in \mathcal{O}_n(e_1)$ and $x \in \mathbb{R}^n$

\[
g(Ax) = \|Ax\|^d_2 f \Big(\frac{Ax}{\|Ax\|_2}\Big)
\|x\|^d_2 (f\circ A) \Big(\frac{x}{\|x\|_2}\Big)
=  \|x\|^d_2 f \Big(\frac{x}{\|x\|_2}\Big)
\]

Hence by the 'uniqueness properties' of the Legendre harmonic $L_{n,d}$ from Proposition <ref> there is some constant $c >0$ such that
$g = c L_{n,d}$ on $\mathbb{R}^n$, and inserting here $e_{1}$, we see that $c = g(e_1)=f(e_1)$. Then for all $\eta \in \mathbb{S}^{n-1}$, we get

\begin{equation*}
f(\eta) = g( \eta) = c L_{n,d}(\eta) = f(e_1) L_{n,d}^{ \diamond}(\langle\eta, e_1\rangle)\,.
\end{equation*}

Finally, we take an arbitrary $\xi \in \mathbb{S}^{n-1}$,  and choose  $B \in \mathcal{O}_n$ such that
$Be_1 = \xi$.  Since $B \circ A \circ B^{-1} \in \mathcal{O}_n(\xi)$ for all
$A \in \mathcal{O}_n(e_1)$, by $(1)$ we have $(f \circ B) \circ A \circ B^{-1} = f$ for all
$A \in \mathcal{O}_n(e_1)$, and hence $(f \circ B) \circ A = f \circ B$ for all such $A$. Consequently, $f \circ B = f \circ B(e_1) L_{n,d}^{ \diamond}(\langle\pmb{\cdot},e_1\rangle)$, or equivalently
$f = f(\xi) L_{n,d}^{ \diamond}(\langle\pmb{\cdot},\xi\rangle)$.

As a consequence of the preceding result, we indeed show the accessibility of the Banach spaces of interest.

The space $\mathcal{H}_d(\mathbb{S}^{n-1})$ is strongly  accessible. In particular,
$\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$ are  accessible.

The first statement is an immediate consequence of  Corollary <ref>. Then  the second claim  follows from Lemma <ref>, (<ref>)  combined with Proposition <ref>. Similarly,  for the proof of the third result we  use Proposition <ref> now combined with the first claim and Lemma <ref>, (<ref>) and (<ref>).

The following statement shows an integral representation of the minimum norm projection from $\mathcal{H}_d(\mathbb{S}^{n-1})$ in terms of the Legendre polynomial of degree $d$ in $n$ dimensions;
see also <cit.>.

Let $d,n \in \mathbb{N}$. Then for all $f \in L_2(\sigma_n)$ and  $\xi \in \mathbb{S}^{n-1}$

\begin{equation*}\label{rep-orthoproj}
\big(\pi_{\mathcal{H}_d(\mathbb{S}^{n-1})}f \big)(\xi) =
\int_{\mathbb{S}^{n-1}}\, N_{n,d} \,\,L_{n,d}^{\diamond}(\langle\xi, \eta\rangle) f(\eta)\, d\sigma_n(\eta)\,.
\end{equation*}

We write $S$ for the $\mathcal{O}_n$-invariant  subspace $\mathcal{H}_d(\mathbb{S}^{n-1})$ of $C(\mathbb{S}^{n-1})$.
By Theorem <ref>, (iv) we have
that for all  $A \in \mathcal{O}_n(e_1)$ and $y \in \mathbb{S}^{n-1}$

\begin{equation*}
\mathbf{k}_S(e_1, Ay) = \mathbf{k}_S(A^{-1}e_1, y) = \mathbf{k}_S(e_1, y)\,,
\end{equation*}

so that by Corollary <ref>

\begin{equation*}
\mathbf{k}_S(e_1, \cdot) = \mathbf{k}_S(e_1, e_1)  L_{n,d}^{ \diamond}(\langle\cdot,e_1\rangle)\,.
\end{equation*}

Then Theorem <ref>, (i) and (v) complete the argument.

§.§ Projection constants

As announced our main intention in this article is to collect  information on the projection constants of
$\mathcal{H}_{ d}(\mathbb{S}^{n-1})$,
and $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$.

Let  $n \geq 2$ and $d \geq 1$. Then the following integral formulas hold:

\begin{equation}\label{un-oX}
\boldsymbol{\lambda}\big(\mathcal{H}_{ d}(\mathbb{S}^{n-1})\big) = N_{n,d}\,\int_{\mathbb{S}^{n-1}}\,
\Big|  \,\,L_{n,d}^{ \diamond}(\eta_1)\Big| \,d\sigma_n(\eta)\,.
\end{equation}

\begin{equation}\label{due}
\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\mathbb{S}^{n-1})\big)
\int_{\mathbb{S}^{n-1}}\,
\Big| \sum_{j=0}^{\lfloor d/2\rfloor} N_{n,d-2j} \,\,L_{n,d-2j}^{\diamond}(\eta_1)\Big| \,d\sigma_n(\eta)\,.
\end{equation}

\begin{equation}\label{tres}
\boldsymbol{\lambda}\big(\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})\big)
= \int_{\mathbb{S}^{n-1}}\,
\Big| \sum_{j=0}^{d} N_{n,j} \,\,L_{n,j}^{\diamond}(\eta_1)\Big| \,d\sigma_n(\eta)\,.
\end{equation}

When $d=1$, the outcome of Equation (<ref>) is consistent with Equation (<ref>) for the real Hilbert space $\ell_2^n(\mathbb{R})$; see  the concluding remark in Section <ref>.
As already commented above, the proof  of the preceding theorem follows from our general  program.

Recall that
$\big( (\mathbb{S}^{n-1}, \sigma_n), (\mathcal{O}_n,\mathrm{m}), \varphi)\big)$, where

\begin{equation*}
\phi: \mathcal{O}_n \longrightarrow \mathcal{L}(C(\mathbb{S}^{n-1})) \,,\quad  O \mapsto [z \mapsto f(Oz) ]\,,
\end{equation*}

by Corollary <ref> forms what we call a Rudin triple.
We  write $S$ for  $\mathcal{H}_d(\mathbb{S}^{n-1})$, $\mathcal{P}_{d}(\mathbb{S}^{n-1})$,
or $\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})$.
Then the  $\mathcal{O}_n$-invariant finite dimensional subspace $S$ of $C(\mathbb{S}^{n-1})$ by Corollary <ref> is  accessible. From Theorem <ref> we deduce that

\begin{equation}\label{wellinger}
\boldsymbol{\lambda}(S) =  \big\|\pi_S\colon C(\mathbb{S}^{n-1}) \to S\big\|=\int_{\mathbb{S}^{n-1} }|\mathbf{k}_S(e_1, \eta) | d\sigma_n(\eta)\,.
\end{equation}

But by Corollary <ref> we know that for all $\eta \in \mathbb{S}^{n-1} $

\begin{equation*}
\mathbf{k}_{\mathcal{H}_d(\mathbb{S}^{n-1})}(e_1, \eta) =  N_{n,d} \,\,L_{n,d}^{\diamond}(\langle e_1, \eta\rangle)\,,
\end{equation*}

which completes the proof of   (<ref>). As a consequence,  (<ref>) and  (<ref>)
are immediate from Lemma <ref>  and  Proposition <ref>.

§ CONCRETE FORMULAS

We need a few basic preliminaries on Jacobi polynomials.
constitute an important  rather wide class
of orthogonal polynomials, from which Chebyshev, Legendre, Laguerre and
Gegenbauer polynomials follow as special cases. The main aim here is to relate them
with the Legendre polynomials $L_{n,d}^{ \diamond}$ defined above in  (<ref>).

We start with a simple observation that will be useful for our purposes. Integrating in spherical coordinates together with the  variable change  $t= \cos \theta$ and Equation
(<ref>), it is not difficult to see that
for every $f \in C[-1,1]$ we have

\begin{align}\label{umrechnen}
\begin{split}
\int_{\mathbb{S}^{n-1}}
f(\eta_1) d\sigma_n(\eta)
\frac{1}{\pmb{\omega}_n}
\,\,
\int_{\mathbb{S}^{n-1}}
f(\eta_1) ds_n(\eta)
\\&
\frac{\pmb{\omega}_{n-1}}{\pmb{\omega}_n}
\,\, \int_{-1}^1 f(t) \; (1-t^2)^{\frac{n-3}{2}} dt
\\&
\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(\frac{n-1}{2})}
\,\, \int_{-1}^1 f(t) \; (1-t^2)^{\frac{n-3}{2}} dt \,.
\end{split}
\end{align}

Jacobi polynomials. Let $\alpha, \beta  > -1$.  Then we denote the sequence of $d$-homogeneous  Jacobi  polynomials on the interval $[-1,1]$ by   $\big(P_d^{(\alpha, \beta)}\big)_{d \in \mathbb{N}_0}$. They are determined by being
$d$-homogeneous,   orthogonal with respect to the inner product

\[
\langle P,Q\rangle = \int_{-1}^{1} P(t)Q(t)\, (1-t)^\alpha  (1+t)^\beta dt\,,
\]

and normalized by

\[
P_d^{(\alpha, \beta)}(1) = \binom{d+\alpha}{d}\,.
\]

There are various concrete  ways of writing such polynomials – for example by the so-called Rodrigues formula

\begin{equation}\label{rodjac}
P_d^{(\alpha, \beta)}(t) = \frac{(-1)^d}{2^d d!}
(1-t)^{-\alpha} (1+t)^{-\beta}
\Big(\frac{\partial}{\partial t}\Big)^d \big[(1-t)^{\alpha+d} (1+t)^{\beta+d}\big]\,.
\end{equation}

Gegenbauer  polynomials.
Fixing $\lambda>-1/2$, Gegenbauer polynomials are orthogonal on the interval $[-1,1]$ with respect to the weight function $(1-t^2)^{\lambda-\frac{1}{2}}$.
As usual, we take the following Rodrigues type formula as  definition:

\begin{equation}\label{RodriguezGegenbauer}
C_d^{(\lambda)}(t)=\frac{(-1)^d}{2^d \;\; d!} \frac{\Gamma(\lambda + \frac{1}{2}) \;\; \Gamma(d+2\lambda)}{\Gamma(2\lambda) \;\; \Gamma(\lambda+n+\frac{1}{2})} (1-t^2)^{-\lambda+1/2} \Big(\frac{\partial}{\partial t}\Big)^d [(1-t^2)^{d+\lambda-\frac{1}{2}}]\,.
\end{equation}

So, up to a multiplicative factor,  Gegenbauer polynomials $C_d^{(\lambda)}$ are the Jacobi polynomials $ P_d^{(\lambda- \frac{1}{2},\lambda- \frac{1}{2})}$,
more precisely,

\begin{align}\label{gegjac}
C_d^{(\lambda)}(t) = \frac{\Gamma(2\lambda +d)\Gamma(\lambda +\frac{1}{2})}{\Gamma(2\lambda)\Gamma(\lambda +\frac{1}{2}+d)}
\,P_d^{(\lambda- \frac{1}{2},\lambda- \frac{1}{2})}(t)\,.
\end{align}

We are going to need the following orthogonality relation:

\begin{equation}\label{Gegenbauerpoly}
\int_{-1}^1 C_{d}^{(\alpha)}(t) \;\; C_{k}^{(\alpha)}(t) \;\;(1-t^2)^{\lambda-\frac{1}{2}} \;\;dt = \frac{\pi\;\; 2^{1-\alpha} \;\;\Gamma(d+2\alpha)}{d!\;\;(d+\alpha)\;\;[\Gamma(\alpha)]^2} \, \delta_{d,k}\,.
\end{equation}

Legendre  polynomials of dimension $\pmb{n}$ and homogeneity $\pmb{d}$. Recall the definition of the polynomials $L_{n,d}^{ \diamond}$
from Theorem <ref>.  Again they may be formulated in 'Rodrigues form':
For $n \ge 2$ and  $d\ge 2$ we have

\begin{equation}\label{rodriguez}
L_{n,d}^{ \diamond} (t)= (-1)^d \frac{\Gamma(\frac{n-1}{2})}{2^d \Gamma(d+\frac{n-1}{2})} \,\,
(1-t^2)^{-\frac{n-3}{2}} \Big(\frac{\partial}{\partial t}\Big)^d(1-t^2)^{d+\frac{n-3}{2}}\,, \quad\ t\in [-1, 1]
\end{equation}

(see, e.g., <cit.>). By inspecting  (<ref>),  (<ref>),(<ref>), and (<ref>), we may relate the polynomials $L_{n,d}^{ \diamond}$ with  the Jacobi polynomials $P_d^{(\frac{n-3}{2}, \frac{n-3}{2})}$ as well as the
Gegenbauer polynomials $C_d^{(\frac{n}{2}-1)}$.

Given $n>2$ and $d \in \mathbb{N}$, for all $t \in [-1,1]$

\begin{align*}
L_{n,d}^{ \diamond}(t) & = \frac{d! \; (n-3)!}{(d+n-3)!} C^{(\frac{n}{2}-1)}_d(t)
\\
& = \frac{d! \; (n-3)!}{(d+n-3)!} \,
\frac{\sqrt{\pi} 2^{-\frac{n-3}{2}} \sqrt{(d+n-3)!}}{\sqrt{d! \; (d+\frac{n}{2} - 1) } \; \Gamma(\frac{n}{2}-1)}
\widehat{C}^{(\frac{n}{2}-1)}_d(t)
\\
& = \frac{d! \; (n-3)!}{(d+n-3)!} \,
\frac{\Gamma(n-2 +d)\Gamma(\frac{n-1}{2})}{\Gamma(n-2)\Gamma(\frac{n-1}{2}+d)}\,
P_d^{(\frac{n-3}{2}, \frac{n-3}{2})}(t);
\end{align*}

here $\widehat{C}^{(\frac{n}{2}-1)}_{d} \colon [-1,1] \to \mathbb{R}$ stands for the normalized Gegenbauer polynomial, that is, its $L_2$-norm on the interval $[-1,1]$ with respect to the weight function $(1 - t^2)^{(\frac{n}{2}-1)–\frac{1}{2}} = (1 - t^2)^{\frac{n-3}{2}}$ is one.

We emphasize some particular cases of importance that are known within the literature (see also <cit.>): For $n=2$ we  get the $d$-th Chebychev polynomial

\begin{equation}\label{d=2}
L_{2,d}^{ \diamond}(t)   = \cos(d \arccos t)\,,
\end{equation}

for $n=3$ the (standard) $d$-th Legendre polynomial

\begin{equation}\label{d=3}
L_{3,d}^{ \diamond}(t) =  \frac{1}{2^d d! } \,\,
\Big(\frac{\partial}{\partial t}\Big)^d (t^2-1)^{d}\,,
\end{equation}

and for $n=4$ the $d$-th  Chebychev polynomial of second kind

\begin{equation}\label{d=4}
L_{4,d}^{ \diamond}(t) =\frac{1}{d+1}\frac{\sin ((d+1) \arccos t)}{\sin (\arccos t)}\,, \quad\, t\in [-1, 1]\,.
\end{equation}

§.§ Harmonics

We start with the calculation of the projection constant of $\mathcal{H}_{ d}(\mathbb{S}^{1})$,
so the dimension $n$ equals $2$.

For all $d \in \mathbb{N}$

\[
\boldsymbol{\lambda}\big(\mathcal{H}_{ d}(\mathbb{S}^{1})\big) = \frac{4}{\pi}\,.
\]

Note that the $2$-dimensional space  $\mathcal{H}_{d}(\mathbb{S}^{1})$ (see again  Equation (<ref>)) consists of all linear combinations of the polynomials $\text{Re}(x_1+ix_2)^d$ and $\text{Im}(x_1+ix_2)^d$, so in particular  $(x_1+ix_2)^d \in \mathcal{H}_{ d}(\mathbb{S}^{1}) $.

Denote  by $\lambda_2$  the (non-normalized) Lebesgue surface measure on $\mathbb{S}^{1}$, and deduce from  Theorem <ref> that

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{H}_{ d}(\mathbb{S}^{1})\big)
= 2 \int_{\mathbb{S}^{1}} |L_{2,d}^{ \diamond}(t)  (\langle e_1, \eta \rangle)| d\sigma_2(\eta)
\\&
= \frac{2}{2\pi}  \int_{\mathbb{S}^{1}} |L_{2,d}^{ \diamond}(t)  (\langle e_1, \eta \rangle)|\, d\lambda_2(\eta)
\\&
= \frac{1}{\pi}  \int_{0}^{2\pi} |L_{2,d}^{ \diamond}(t)  (\cos t)| dt
\\&
=\frac{1}{\pi}  \Big(\int_{0}^{\pi} |L_{2,d}^{ \diamond}(t)  (\cos t)| dt + \int_{\pi}^{2\pi} |L_{2,d}^{ \diamond}(t)  (\cos t)| dt\Big)
\\&
=\frac{1}{\pi}  \Big(\int_{0}^{\pi} |L_{2,d}^{ \diamond}(t)  (\cos t)| dt + \int_{\pi}^{2\pi} |L_{2,d}^{ \diamond}(t)  (-\cos (t-\pi))| dt\Big)\,.
\end{align*}

Then by (<ref>)

\begin{align*}
\int_{0}^{\pi} |L_{2,d}^{ \diamond}(t)  (\cos t)| dt
\int_{0}^{\pi} \big|\cos \big(d \arccos (\cos t)\big)\big| dt
\\&
=\int_{0}^{\pi} |\cos (d t )| dt
\frac{1}{d} \int_{0}^{d\pi} |\cos t| dt
=  \int_{0}^{\pi} |\cos t| dt   =  2\,,
\end{align*}

and, using this and the symmetry  of $\arccos$,

\begin{align*}
\int_{\pi}^{2\pi} |L_{2,d}^{ \diamond}(t)  (-\cos (t-\pi))| dt
=\int_{\pi}^{2\pi} |\cos \big(d \arccos(-\cos (t-\pi))\big)| dt
\\&
=\int_{\pi}^{2\pi} \big|\cos \big(d (\pi -\arccos(\cos (t-\pi))\big)\big| dt
\\&
= \int_{\pi}^{2\pi} |\cos (d (2\pi - t))| dt
= \int_{0}^{\pi} |\cos (d t )| dt = 2\,.
\end{align*}

So, together, we get as desired
\boldsymbol{\lambda}\big(\mathcal{H}_{ d}(\mathbb{S}^{1})\big)
= \frac{1}{\pi} (2+2)
\frac{4}{\pi}$ .

Let us turn to the case $n>2$.

For all $n>2$ and $d\in \mathbb{N}$, we have

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big) & =
\frac{\Gamma(\frac{n-1}{2})}{\sqrt{\pi} \,\Gamma(n-1)} \,
\frac{2d+n-2}{n-2}
\,
\frac{\Gamma(n-2+d) \Gamma(\frac{n-1}{2})}{\Gamma(n-2) \Gamma(\frac{n-1}{2} +d)}
\,
\int_{-1}^{1} \big| P_d^{(\frac{n-3}{2},\frac{n-3}{2})}(t)\big| (1-t^2)^{\frac{n-3}{2}} dt
\,.
\end{align*}

Alternatively, we have in terms of Gegenbauer polynomials that

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big) & = \frac{2d+n-2}{n-2}  \int_{\mathbb{S}^{n-1}} \big |C^{(\frac{n}{2}-1)}_d(\eta_1) \big|\,d\sigma_n(\eta) \\
& = \frac{\sqrt{2d+n-2}}{n-2}  \;\; \frac{\sqrt{\pi} 2^{-\frac{n-4}{2}} \sqrt{(d+n-3)!}}{\sqrt{d! } \; \Gamma(\frac{n}{2}-1)} \int_{\mathbb{S}^{n-1}} \big | \widehat{C}^{(\frac{n}{2}-1)}_d(\eta_1) \big |\,d\sigma_n(\eta) \,.
\end{align*}

This follows by combining Theorem <ref> with Equation (<ref>) and Lemma <ref>, and performing the corresponding numerical simplifications.

For fixed dimension $n$, we now show the asymptotic order of $\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)$ as the homogeneity degree $d$ goes to infinity. As the following theorem shows, this projection constant behaves as the square root of the dimension of $\mathcal{H}_{d}(\mathbb{S}^{n-1})$, meeting the Kadet-Snobar estimate. Recall that after fixing $n$ we for large $d$ have

\[
\dim \mathcal{H}_{d}(\mathbb{S}^{n-1}) \,\sim_{c(n)} \, d^{n-2}\,.
\]

More precisely,

\begin{equation} \label{asymp dimension ratio}
\lim_{d \to +\infty} \frac{\dim \mathcal{H}_{d}(\mathbb{S}^{n-1}) }{d^{n-2}} = \frac{2}{(n-2)!}\,.
\end{equation}

Indeed, the previous limit can be deduced from the fact that by Equation (<ref>) the dimension is exactly

$$\dim \mathcal{H}_{d}(\mathbb{S}^{n-1}) = \frac{(2d+n-2) \Gamma(d+n-2)}{\Gamma(d+1) (n-2)!},$$

combined with the following well-known formula for the asymptotic of ratios of Gamma functions: Given fixed numbers $a,b >0$,

\begin{align}\label{mainasym}
\lim_{x\to +\infty} \frac{\Gamma(x + a)}{ \Gamma(x + b) x^{a-b}} =1.
\end{align}

For $n>2$

\begin{equation} \label{Eq: limit projection harmonic A}
\lim_{d \to +\infty} \frac{\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}{\sqrt{\dim \mathcal{H}_{d}(\mathbb{S}^{n-1})}} \,\, = \,\,\frac{  \Gamma(\frac{n}{2})}{ 2^{\frac{2n-15}{2}} \sqrt{(n-3)!} \;\;  \Gamma(\frac{n-1}{2}) \; \; \sqrt{\pi} \;\; (n-2)^{3/2} \;\; \Gamma(\frac{n-2}{4})^2}.
\end{equation}

In particular,

\begin{equation} \label{Eq: limit projection harmonic B}
\lim_{d \to +\infty} \frac{\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}{d^{\frac{n-2}{2}}} \,\, = \,\,\frac{  \Gamma(\frac{n}{2})}{ 2^{\frac{2n-17}{2}} [(n-3)!]^{3/2} \;\;  \Gamma(\frac{n-1}{2}) \; \; \sqrt{\pi} \;\; (n-2)^{5/2} \;\; \Gamma(\frac{n-2}{4})^2}.
\end{equation}

By Theorem <ref> we have

\begin{align*}
\frac{\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}{\sqrt{\mbox{dim}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}}
\\
= \sqrt{\frac{d!(n-2)!}{(2d+n-2)(d+n-3)!}} \,\frac{\sqrt{2d+n-2}}{(n-2)}  \;\; \frac{\sqrt{\pi} 2^{-\frac{n-4}{2}} \sqrt{(d+n-3)!}}{\sqrt{d! } \; \Gamma(\frac{n}{2}-1)} \int_{\mathbb{S}^{n-1}} \big | \widehat{C}^{(\frac{n}{2}-1)}_d(\eta_1) \big|\; d\sigma_n(\eta)
\\
& = \frac{\sqrt{\pi}  \;\;2^{-\frac{n-4}{2}} \;\; (n-3)!}{ \sqrt{(n-2)!} \;\; \Gamma(\frac{n}{2}-1)} \int_{\mathbb{S}^{n-1}}
\big|\widehat{C}^{(\frac{n}{2}-1)}_d(\eta_1)\big|\,d\sigma_n(\eta)\,,
\end{align*}

and  consequently it follows from Equation (<ref>) that

\begin{align*}
\frac{\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}{\sqrt{\mbox{dim}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}} & =  \frac{  2^{-\frac{n-4}{2}} \;\: (n-3)! \; \; \Gamma(\frac{n}{2})}{ \sqrt{(n-2)!} \;\; \Gamma(\frac{n}{2}-1) \; \; \Gamma(\frac{n-1}{2})} \int_{-1}^1
\big|\widehat{C}^{(\frac{n}{2}-1)}_d(t)\big| \; (1-t^2)^{\frac{n-3}{2}} dt\,.
\end{align*}

From  <cit.> we know that

\[
\lim_{d\to \infty} \; \; \int_{-1}^1
|\hat{C}^{(\frac{n}{2}-1)}_d(t)| \; (1-t^2)^{\frac{n-3}{2}} dt
=\left(\frac{2}{\pi}\right)^{3/2} \int_0^\pi (1-\cos^2 \theta)^{\frac{n-2}{4}} \;d\theta \,.\]

By the change $t=\cos \theta$ of variables, together with (<ref>) ($d=k=0$ and $\alpha=\frac{n-2}{4}$) this integral equals

\begin{align*} \left(\frac{2}{\pi}\right)^{3/2} \int_{-1}^1 (1-t^2 )^{\frac{n-2}{4} - \frac{1}{2}} \;d\theta & = \left(\frac{2}{\pi}\right)^{3/2} \frac{\pi 2^{1 - 2(\frac{n-2}{4})} \Gamma(\frac{n}{2} - 1)}{ \frac{n-2}{4} \;\; \Gamma(\frac{n-2}{4})^2}\, = \, \frac{2^{- \frac{(n-11)}{2}} \;\; \Gamma(\frac{n}{2} - 1)}{ \sqrt{\pi} \;\; (n-2) \;\; \Gamma(\frac{n-2}{4})^2}\,.
\end{align*}

This shows that

\begin{align*}
\lim_{d \to +\infty} \frac{\boldsymbol{\lambda}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}{\sqrt{\mbox{dim}\big(\mathcal{H}_{d}(\mathbb{S}^{n-1})\big)}}  & = \frac{  2^{-\frac{n-4}{2}} \;\: (n-3)! \; \; \Gamma(\frac{n}{2})}{ \sqrt{(n-2)!} \;\; \Gamma(\frac{n}{2}-1) \; \; \Gamma(\frac{n-1}{2})} \;\; \frac{2^{- \frac{(n-11)}{2}} \;\; \Gamma(\frac{n}{2} - 1)}{ \sqrt{\pi} \;\; (n-2) \;\; \Gamma(\frac{n-2}{4})^2} \\
& = \frac{  2^{-\frac{2n-15}{2}} \;\: (n-3)! \; \; \Gamma(\frac{n}{2})}{ \sqrt{(n-2)!} \;\; \Gamma(\frac{n}{2}-1) \; \; \Gamma(\frac{n-1}{2})} \;\; \frac{ \Gamma(\frac{n}{2} - 1)}{ \sqrt{\pi} \;\; (n-2) \;\; \Gamma(\frac{n-2}{4})^2} \\
& = \,\,\frac{  2^{-\frac{2n-15}{2}}  \; \; \Gamma(\frac{n}{2})}{ \sqrt{(n-3)!} \;\;  \Gamma(\frac{n-1}{2}) \; \; \sqrt{\pi} \;\; (n-2)^{3/2} \;\; \Gamma(\frac{n-2}{4})^2},
\end{align*}

which is exactly (<ref>). On the other hand, (<ref>) follows from the previous limit and  (<ref>).

§.§ Homogeneous polynomials

As before in Proposition <ref> we start with the $2$-dimensional case.

For all $d \in \mathbb{N}$

\[
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{1})\big) =
\frac{1}{2\pi} \int_{0}^{2\pi}\bigg| \frac{\sin\big((\frac{d+1}{2})t \big)}{\sin\big(\frac{t}{2} \big)}\bigg|\,dt\,.
\]

In particular,

\[
\lim_{d \to \infty}  \frac{\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{1})\big)}{\log d} \,=\, \frac{4}{\pi^2}\,.
\]

It should be pointed out that in the case of the complex Hilbert space $\ell_2^2(\mathbb{C})$, the projection constant $\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^2(\mathbb{C})\big)$ is bounded by $2$ for any degree $d$ (see again (<ref>) and also (<ref>)). However, the previous result shows that for the real case we have a significant difference. In fact, the projection constant of $\mathcal{P}_{d}(\ell_2^2(\mathbb{R}))=\mathcal{P}_{d}(\mathbb{S}^{1})$ has a logarithmic increase with respect to $d$.

We follow the ideas of the proof of  Propositions <ref>.
Suppose first that $d$ is even.
Note that $N_{2,0}=1$ and that, on the other hand, by Equation (<ref>) we have $N_{2,\ell}=2$ for every $\ell\geq 1$.
Then we conclude from Theorem <ref> that

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{1})\big)
=  \int_{\mathbb{S}^{1}} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\langle e_1, \eta \rangle)\big| d\sigma_2(\eta)
\\&
=\frac{1}{2\pi}\int_{\mathbb{S}^{1}} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\langle e_1, \eta \rangle)\big| d\lambda_2(\eta)
\\&
=\frac{1}{2\pi}\int_{0}^{2\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\cos t)\big| dt
\\&
=\frac{1}{2\pi}\Big(\int_{0}^{\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\cos t)\big| dt
\int_{\pi}^{2\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\cos t)\big| dt\Big)\,.
\end{align*}

By (<ref>) we know that $L_{2,2\ell}^{ \diamond}$ is the  $d$-th Chebychev polynomial, that is,

\begin{equation*}
L_{2,2\ell}^{ \diamond}(t)   = \cos(2\ell \arccos t)\,,
\end{equation*}

and so we may rewrite the preceding two integrals as follows:

\begin{align*}
\int_{0}^{\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\cos t)\big|\,dt
\int_{0}^{\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} \cos (2\ell t)\big|\,dt
\\&
= \frac{1}{2}  \int_{0}^{2\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} \cos (\ell s)\big|\,ds =  \frac{1}{2} \int_{0}^{2 \pi}  \bigg|\frac{\sin\big(( \frac{d+1}{2})s \big)}{\sin\big(\frac{s}{2} \big)}\bigg| \,ds
\end{align*}

\begin{align*}
\int_{\pi}^{2\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} L_{2,2\ell}^{\diamond}(\cos t)\big|\,dt
\int_{\pi}^{2\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} \cos \big(2\ell (2\pi - t)\big)\big|\,dt
\\
& =
\frac{1}{2} \int_{0}^{2\pi} \big|1+ 2 \sum_{\ell=1}^{\frac{d}{2}} \cos \big(\ell s)\big)\big|\,ds
= \frac{1}{2} \int_{0}^{2\pi}  \bigg|\frac{\sin\big(( \frac{d+1}{2})s \big)}{\sin\big(\frac{s}{2} \big)}\bigg|\,ds
\,.
\end{align*}

Combining,  we obtain the required formula for the even case.

Suppose now that $d$ is odd. In this case  Theorem <ref> gives that

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{1})\big)
=  \int_{\mathbb{S}^{1}} \big| 2 \sum_{\ell=0}^{\floor{\frac{d}{2}}} L_{2,2\ell+1}^{\diamond}(\langle e_1, \eta \rangle)\big| d\sigma_2(\eta)
\\&
=\frac{1}{2\pi}\Big(\int_{0}^{\pi} \big| 2 \sum_{\ell=0}^{\floor{\frac{d}{2}}} L_{2,2\ell+1}^{\diamond}(\cos t)\big| dt
\int_{\pi}^{2\pi} \big| 2 \sum_{\ell=0}^{\floor{\frac{d}{2}}} L_{2,2\ell+1}^{\diamond}(\cos t)\big| dt\Big)\,.
\\&
=\frac{1}{2\pi}\Big(\int_{0}^{\pi} \big| 2 \sum_{\ell=0}^{\floor{\frac{d}{2}}} \cos((2\ell +1) t\big| dt
\int_{\pi}^{2\pi} \big| 2 \sum_{\ell=0}^{\floor{\frac{d}{2}}} \cos((2\ell +1)(2\pi- t)\big| dt\Big)\,.
\end{align*}

Now note that

$$ \sum_{\ell=0}^{\floor{\frac{d}{2}}} \cos\big((2\ell+1)\frac{s}{2}\big) = \textbf{Re} \left( \sum_{\ell=0}^{\floor{\frac{d}{2}}} e^{i (2 \ell +1) \frac{s}{2}} \right) = \frac{1}{2} \frac{\sin\big((\floor{\frac{d}{2}}+1\big)s)}{\sin(\frac{s}{2})}= \frac{1}{2} \frac{\sin\big((\frac{d+1}{2})s\big)}{\sin(\frac{s}{2})}\,,$$

and then the  proof follows the same steps as before, with the substitution $t=\frac{s}{2}$ as previously employed.

Finally, the limiting formula follows from the well-known properties of the Dirichlet kernel.

We go on with the case $n>2$.
The following result is an analog of Theorem <ref>.
Recall that the floor function $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. Conversely, the ceil function $\lceil x \rceil$ represents the smallest integer greater than or equal to $x$. It is worth noting that both functions are involved in the following formula.

Given $n>2$ and $d\in \mathbb{N}$, define

\[
c(n,d) :=
\,\,
\frac{\Gamma(\lceil\frac{d}{2}\rceil+\frac{n}{2})}{\Gamma(\lceil\frac{d}{2}\rceil+\frac{1}{2})}
\,\,
\frac{ \Gamma(\lfloor\frac{d}{2}\rfloor+\frac{1}{2}+ \frac{n}{2})}{\Gamma(\lfloor\frac{d}{2}\rfloor+1)}
\,\,
\frac{ \Gamma(d+1)}{\Gamma(d+ \frac{n}{2} +\frac{1}{2})}\,.
\]

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\mathbb{S}^{n-1})\big) \,\, = \,\,
\frac{2^{n-2}\Gamma(\frac{n}{2})}{\sqrt{\pi} \,\Gamma(n-1)}
\, c(n,d)\, \int_{-1}^{1} \big| P_d^{(\frac{n-1}{2},\frac{n-1}{2})}(t)\big| (1-t^2)^{\frac{n-3}{2}} dt\,.
\end{align*}

When $d=1$, the preceding outcome aligns with Rutovitz's findings for the real Hilbert space from Equation (<ref>). To delve deeper into this interpretation, see  the concluding remark in Section <ref>.

We have to distinguish the even degrees $d$ form the   odd ones, and start with the even case.
By Theorem <ref> one has

\begin{equation*}
\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^n(\mathbb{R}))\big)
\int_{\mathbb{S}^{n-1}}\,
\big| \mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )\big| \,d\sigma_n(\eta)
\end{equation*}

$$\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
=  \sum_{j=0}^{\frac{d}{2}} N_{n,j} \,\,L_{n,2j}^{\diamond}(\eta_1)\,.

\alpha = \frac{n-3}{2}\,.
Then by Lemma <ref>  and Equation (<ref>) we get  that

\begin{align*}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\sum_{j=0}^{\frac{d}{2}} \frac{(4j+n-2)(2j+n-3)!}{(2j)!(n-2)!}
\,\,
\frac{(2j)! \;\; (n-3)!}{(2j+n-3)!}
\,\,
\frac{\Gamma(n-2 +2j)\Gamma(\frac{n-1}{2})}{\Gamma(n-2)\Gamma(\frac{n-1}{2}+2j)}
\,\,
P_{2j}^{(\alpha, \alpha)}(\eta_1)
\,,
\end{align*}

and hence

\begin{align} \label{kern}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,\, \sum_{j=0}^{\frac{d}{2}}\, \frac{2(2j +\alpha + \frac{1}{2})\Gamma(2(j+\alpha +\frac{1}{2}))}{\Gamma(2j+\alpha+1)}\,\,
P_{2j}^{(\alpha, \alpha)}(\eta_1)\,.
\end{align}

By <cit.>, we have

\begin{align}\label{szegoe-even}
P_{2j}^{(\alpha, \alpha)}(\eta_1) = \frac{\Gamma(2j+\alpha+1)j!}{\Gamma(j+\alpha+1)(2j)!} \,\,
P_{j}^{(\alpha, -\frac{1}{2})}(2\eta_1^2-1)\,,
\end{align}

\begin{align}\label{szegoe-evenB}
\sum_{j=0}^{\frac{d}{2}}\,
\frac{(2j + \alpha -\frac{1}{2}+1)\Gamma(j+\alpha-\frac{1}{2}+1)}{ \Gamma(j-\frac{1}{2}+1)}
\,
P_{j}^{(\alpha, -\frac{1}{2})}(2\eta_1^2-1)
\frac{\Gamma(\frac{d}{2} + \alpha -\frac{1}{2} +2)}{\Gamma(\frac{d}{2}  -\frac{1}{2} +1)}
P_{\frac{d}{2}}^{(\alpha+1, -\frac{1}{2})}(2\eta_1^2-1)\,.
\end{align}

Then, applying (<ref>) and (<ref>) to (<ref>), we get

\begin{align} \label{rearranging-even}
\begin{split}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}
\,
\sum_{j=0}^{\frac{d}{2}}\,
\,
\frac{(2j + \alpha -\frac{1}{2}+1)\Gamma(j+\alpha-\frac{1}{2}+1)}{ \Gamma(j-\frac{1}{2}+1)}
\,
P_{j}^{(\alpha, -\frac{1}{2})}(2\eta_1^2-1)\,,
\end{split}
\end{align}

\begin{align*}
\frac{2(2j +\alpha + \frac{1}{2})\Gamma(2(j+\alpha +\frac{1}{2}))}{\Gamma(2j+\alpha+1)}
\,\,
\frac{\Gamma(2j+\alpha+1)j!}{\Gamma(j+\alpha+1)(2j)!}
\,\,
\frac{ \Gamma(j-\frac{1}{2}+1)}{(2j + \alpha -\frac{1}{2}+1)\Gamma(j+\alpha-\frac{1}{2}+1)}
\\&
\,\,
\frac{\Gamma(2(j+\alpha +\frac{1}{2}))j!}{\Gamma((j+\alpha+\frac{1}{2})+\frac{1}{2})(2j)!}
\,\,
\frac{ \Gamma(j+\frac{1}{2})}{\Gamma(j+\alpha+\frac{1}{2})}\,.
\end{align*}

We claim that
C(n,j) = 2^{n-2}\,.
Indeed, we use the formula

$$\Gamma\Big(j+\frac{1}{2}\Big) = \frac{(2j)!\sqrt{\pi}}{4^j j!}$$

and for $x = j + \alpha + \frac{1}{2}$  the
Legendre duplication formula

\begin{equation} \label{Legendre duplication formula}
\Gamma(x)\, \Gamma\Big(x+ \frac{1}{2}\Big) = 2^{1-2x}\, \sqrt{\pi}\Gamma(2x)\,,
\end{equation}

to see that

\[
C(n,j) = 2 \frac{\Gamma(2x)}{\Gamma(x + \frac{1}{2})\Gamma(x)} \frac{j!}{(2j)!}
\Gamma\Big(j+\frac{1}{2}\Big) = 2^{n-2}.
\]

Summarizing, we get

\begin{align*}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}
\,
\frac{\Gamma(\frac{d}{2}+\frac{n}{2})}{\Gamma(\frac{d}{2}+\frac{1}{2})}
\,
P_{\frac{d}{2}}^{(\alpha+1, -\frac{1}{2})}(2\eta_1^2-1)\,.
\end{align*}

Finally, we once again use <cit.> to obtain

\begin{align*}
P_{\frac{d}{2}}^{(\alpha+1, -\frac{1}{2})}(2\eta_1^2-1)
\frac{\Gamma(\frac{d}{2} + \frac{n-1}{2} +1) \Gamma(d+1)}{\Gamma(d + \frac{n-1}{2} +1)\Gamma(\frac{d}{2}+1)}
\end{align*}

and conclude that

\begin{align*}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}
\,
\end{align*}

\[
\frac{\Gamma(\frac{d}{2}+\frac{n}{2})}{\Gamma(\frac{d}{2}+\frac{1}{2})}
\,
\frac{\Gamma(\frac{d}{2} + \frac{n-1}{2} +1) \Gamma(d+1)}{\Gamma(d + \frac{n-1}{2} +1)\Gamma(\frac{d}{2}+1)}\,.
\]

Consequently, by Equation (<ref>) we have

\begin{align} \label{fin}
\begin{split}
\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\ell_2^n(\mathbb{R}))\big)
\\[1ex]&
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}
\,
\,            \int_{\mathbb{S}^{n-1}}\, \big| P_{d}^{(\frac{n-1}{2},\frac{n-1}{2})}(\eta_1)\big|  \,d\sigma_n(\eta)
\\[1ex]&
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}
\,
\, \frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(\frac{n-1}{2})}\int_{-1}^1\, \big| P_{d}^{(\frac{n-1}{2},\frac{n-1}{2})}(t)\big|  (1-t^2)^{\frac{n-3}{2}} dt\,,
\end{split}
\end{align}

and this finishes  the even case of our claim.

The proof of the  odd case is similar - but with some subtle differences.
If $d$ is odd, then
we are going to replace (<ref>)  by

\begin{align}\label{szegoe-odd}
P_{2j+1}^{(\alpha, \alpha)}(\eta_1) = \frac{\Gamma(2j+\alpha+2)j!}{\Gamma(j+\alpha+1)(2j+1)!} \,\,
\eta_1 P_{j}^{(\alpha, \frac{1}{2})}(2\eta_1^2-1)\,,
\end{align}

and (<ref>) by

\begin{align}\label{szegoe-oddB}
\sum_{j=0}^{\lfloor\frac{d}{2}\rfloor}\,
\,
\frac{(2j + \alpha +\frac{1}{2}+1)\Gamma(j+\alpha+\frac{1}{2}+1)}{\Gamma(j+\frac{1}{2}+1)}
\,
\eta_1 P_{j}^{(\alpha, \frac{1}{2})}(2\eta_1^2-1)
\frac{\Gamma(\lfloor\frac{d}{2}\rfloor + \alpha +\frac{1}{2} +2)}{\Gamma(\lfloor\frac{d}{2}\rfloor  +\frac{1}{2} +1)}
\,
\eta_1 P_{\lfloor\frac{d}{2}\rfloor}^{(\alpha, \frac{1}{2})}(2\eta_1^2-1)\,,
\end{align}

where again $\alpha = \frac{n-3}{2}$ (see  <cit.>).
Starting as above,
we have

\begin{align}\label{kero-odd}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
= \sum_{j=0}^{\lfloor\frac{d}{2}\rfloor} N_{n,2j+1} \,\,L_{n,2j+1}^{ \diamond}(\eta_1)\,,
\end{align}

and then  (as for (<ref>))  we obtain

\begin{align} \label{kern-odd}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,\, \sum_{j=0}^{\lfloor\frac{d}{2}\rfloor}\,
\frac{2((2j+1) +\alpha + \frac{1}{2})
\Gamma(2(j+\alpha +1)) }{\Gamma((2j+1)+\alpha +1)}\,\,
P_{2j+1}^{(\alpha, \alpha)}(\eta_1)\,.
\end{align}

Then as in (<ref>)

\begin{align} \label{rearranging-odd}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2}))}{\Gamma(n-1)}
\,
\sum_{j=0}^{\lfloor\frac{d}{2}\rfloor}\,
\,
\frac{(2j + \alpha +\frac{1}{2}+1)\Gamma(j+\alpha+\frac{1}{2}+1)}{ \Gamma(j+\frac{1}{2}+1)}
\,
\eta_1 P_{j}^{(\alpha, \frac{1}{2})}(2\eta_1^2-1)\,,
\end{align}

\begin{align*} \label{C-odd}
\begin{split}
\\&
\frac{2((2j+1) +\alpha + \frac{1}{2})
\Gamma(2(j+\alpha +1)) }{\Gamma((2j+1)+\alpha +1)}\,
\frac{\Gamma(2j+\alpha+2)j!}{\Gamma(j+\alpha+1)(2j+1)!} \,
\frac{\Gamma(j+\frac{1}{2}+1)}{(2j + \alpha +\frac{1}{2}+1)\Gamma(j+\alpha+\frac{1}{2}+1)}
\\&
\frac{\Gamma(2(j+\alpha +1)) j!}{\Gamma(j+\alpha+1)(2j+1)!} \,
\frac{\Gamma(j+\frac{1}{2}+1)}{\Gamma(j+\alpha+\frac{1}{2}+1)}
\,.
\end{split}
\end{align*}

Using that

$$\Gamma((j+1)+\frac{1}{2}) = \frac{\sqrt{\pi}}{4^{j+1}} \frac{(2(j+1))!}{(j+1)!}= \frac{1}{2}\frac{\sqrt{\pi}}{4^j }\frac{(2j+1)!}{j!}$$

and again the Legendre duplication formula  from
(now for $x=j+\alpha +1$), we as above get

\[
C(n,j) =  \frac{\Gamma(2x)}{\Gamma(x)\Gamma(x+\frac{1}{2})} \frac{\sqrt{\pi}}{4^j}
\]

Then we derive from (<ref>) and (<ref>) that

\begin{align} \label{rearranging-oddBB}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)} 2^{n-2}
\,
\frac{\Gamma(\lfloor\frac{d}{2}\rfloor + \alpha +\frac{1}{2} +2)}{\Gamma(\lfloor\frac{d}{2}\rfloor  +\frac{1}{2} +1)}
\,
\eta_1 P_{\lfloor\frac{d}{2}\rfloor}^{(\alpha, \frac{1}{2})}(2\eta_1^2-1)\,,
\end{align}

Reversing the process, using  <cit.>,  we conclude from

\begin{align*}
\eta_1P_{\lfloor\frac{d}{2}\rfloor}^{(\alpha+1, \frac{1}{2})}(2\eta_1^2-1)
\frac{\Gamma(\lfloor\frac{d}{2}\rfloor + \frac{n-1}{2} +1) \Gamma(2\lfloor\frac{d}{2}\rfloor+2)}
{\Gamma(2\lfloor\frac{d}{2}\rfloor + \frac{n-1}{2} +2)\Gamma(\lfloor\frac{d}{2}\rfloor+1)}
\end{align*}

\begin{align*}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)} 2^{n-2}
\,
\frac{\Gamma(\lfloor\frac{d}{2}\rfloor + \frac{n}{2}+1)}{\Gamma(\lfloor\frac{d}{2}\rfloor  +\frac{1}{2} +1)}
\,
\frac{\Gamma(\lfloor\frac{d}{2}\rfloor + \frac{n}{2}+ \frac{1}{2} ) \Gamma(2\lfloor\frac{d}{2}\rfloor+2)}
{\Gamma(2\lfloor\frac{d}{2}\rfloor + \frac{n}{2} +\frac{3}{2})\Gamma(\lfloor\frac{d}{2}\rfloor+1)}
\end{align*}

But $2\lfloor\frac{d}{2}\rfloor+1 = d$    and $\lfloor\frac{d}{2}\rfloor +1 = \lceil\frac{d}{2}\rceil$, hence finally

\begin{align*}
\mathbf{k}_{\mathcal{P}_{d}(\mathbb{S}^{n-1})}(e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)} 2^{n-2}
\,
\frac{\Gamma(\lceil\frac{d}{2}\rceil+\frac{n}{2})}{\Gamma(\lceil\frac{d}{2}\rceil+\frac{1}{2})}
\,\,
\frac{ \Gamma(\lfloor\frac{d}{2}\rfloor+\frac{1}{2}+ \frac{n}{2})}{\Gamma(\lfloor\frac{d}{2}\rfloor+1)}
\,\,
\frac{ \Gamma(d+1)}{\Gamma(d+ \frac{n}{2} +\frac{1}{2})}\,.
\end{align*}

The proof finishes as in (<ref>).

The following consequence in particular shows that  here $\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\mathbb{S}^{n-1})\big)$
does not meet the Kadets-Snobar theorem since, fixing $n$, we for large $d$ have (see again the Equations (<ref>),
(<ref>),and (<ref>))

\[
\dim \mathcal{P}_{d}(\mathbb{S}^{n-1}) \,\sim_{c(n)} \, d^{n-1}\,.
\]

For each integer $n >2$

\[
\lim_{d \to \infty}
\frac{\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{n-1})\big)}{d^\frac{n-2}{2}}
\,= \, \frac{2^{\frac{n}{2}+1}}{\pi^{\frac{3}{2}}}   \,\,
\frac{\Gamma(\frac{n}{2})\Gamma(\frac{n}{4}-\frac{1}{2})}{\Gamma(n-1)\Gamma(\frac{n}{4})}
\,.
\]

Using again the formula (<ref>) for the asymptotic of ratios of Gamma functions
and distinguishing odd $d$'s from even ones, we have for $c(n,d)$, as defined in Theorem <ref>, that

\[
\lim_{d \to \infty} \,\, \frac{c(n,d)}{d^\frac{n-1}{2}}\,\, = \,\,
\frac{1}{2^{n-2}}\,.
\]

The following limit  from <cit.> is crucial for our purposes: For $\alpha, \beta, \lambda, \mu > -1$
such that $2 \lambda > \alpha - \frac{3}{2}$

\begin{align} \label{doro-eq}
\begin{split}
\lim_{d \to \infty} \,\, \sqrt{d}\,\,
\int_{0}^{1}
(1-x)^\lambda (1+x)^\mu &|P^{(\alpha, \beta)}_d(x)| dx
\\&
\,= \, \frac{2^{\lambda+\mu +1}}{ \sqrt{\pi}} \frac{2}{\pi}
\int_{0}^{\frac{\pi}{2}}  \big(\sin (\theta/2) \big)^{2\lambda-\alpha +\frac{1}{2}}
\big(\cos (\theta/2) \big)^{2\mu-\beta +\frac{1}{2}} d \theta\,,
\end{split}
\end{align}

and we are going to use this equation for $\alpha = \beta = \frac{n-1}{2}$
and $\lambda = \mu = \frac{n-3}{2}$. Moreover, we need to know that by
Jacobi polynomials satisfy

\begin{align*} \label{SZ1}
P_d^{(\alpha, \beta)} (t) = (-1)^d P_d^{(\beta,\alpha)} (-t), \quad\, t\in [-1, 1]\,,
\end{align*}

\[
\int_{-1}^{1} \big| P_d^{(\frac{n-1}{2},\frac{n-1}{2})}(t)\big| (1-t^2)^{\frac{n-3}{2}} dt
= 2
\int_{0}^{1} \big| P_d^{(\frac{n-1}{2},\frac{n-1}{2})}(t)\big| (1-t^2)^{\frac{n-3}{2}} dt\,.
\]

Combining all this with Theorem <ref>, gives

\begin{align*}
\lim_{d \to \infty}
\frac{\boldsymbol{\lambda}\big(\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})\big)}{d^\frac{n-2}{2}}
\\&
\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(n-1)}
\,
\lim_{d \to \infty} \,\, \frac{c(n,d)}{d^\frac{n-1}{2}}
\,
\lim_{d \to \infty} \,\, \sqrt{d}\,
\int_{0}^{1} \big| P_d^{(\frac{n-1}{2},\frac{n-1}{2})}(t)\big| (1-t^2)^{\frac{n-3}{2}} dt
\\&
\frac{2^{n-2}\Gamma(\frac{n}{2})}{\sqrt{\pi} \,\Gamma(n-1)}
\,
\frac{1}{2^{n-2}}
\,
\frac{2^{n-2}}{\sqrt{\pi}} \frac{2}{\pi}
\int_{0}^{\frac{\pi}{2}}  \big(\sin (\theta/2) \big)^{\frac{n}{2}-2}
\big(\cos (\theta/2) \big)^{\frac{n}{2}-2} d \theta \\
&=  \frac{2^n \Gamma(\frac{n}{2})}{\pi^2 \Gamma(n-1)} \,\, \int_{0}^{\frac{\pi}{2}}  \big(\sin (\theta/2) \big)^{\frac{n}{2}-2}
\big(\cos (\theta/2) \big)^{\frac{n}{2}-2} d \theta
\,.
\end{align*}

It remains to check that

\begin{equation*}
\text{\bf I}
:= \int_{0}^{\frac{\pi}{2}}  \big(\sin (\theta/2) \big)^{\frac{n}{2}-2}
\big(\cos (\theta/2) \big)^{\frac{n}{2}-2} d \theta =
\frac{\sqrt{\pi}}{2^{\frac{n}{2}-1}}
\frac{\Gamma(\frac{n}{4}-\frac{1}{2})}{\Gamma(\frac{n}{4})}\,.
\end{equation*}

Indeed, using  the identity  $ 2\sin(\theta/2) \cos (\theta/2) = \sin \theta$ and the substitution
$t = \sin^2 \theta$ a simple calculation leads to

\[
\text{\bf I} = \frac{1}{2^{\frac{n}{2}-1}}\,\,  \text{\bf B}\Big(\frac{n}{4}-\frac{1}{2},\frac{1}{2}\Big)
= \frac{1}{2^{\frac{n}{2}-1}}\,\, \frac{\Gamma(\frac{n}{4}-\frac{1}{2})\Gamma(\frac{1}{2})}{\Gamma(\frac{n}{4})}\,,
\]

where $\text{\bf B}$ as usual stands for the Beta-function.

Summarizing we see that projection constants of spaces of homogeneous
polynomials in the real and complex case behave substantially different. Indeed, we have
for fixed $n$ and  large $d$ that

\[
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{n-1})\big)
\,\sim_{c(n)} d^{\frac{n-2}{2}}
\quad \text{and} \quad
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}_{\mathbb{C}}^{n-1})\big)
\,\sim_{c(n)} \,1,
\]

whereas for  fixed $d$ and large $n$

\[
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{n-1})\big)
\,\sim_{c(d)} n^{\frac{d}{2}}
\quad \text{and} \quad
\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}_{\mathbb{C}}^{n-1})\big)
\,\sim_{c(d)} n^{\frac{d}{2}}\,.
\]

Here the first result is a repetition of Corollary <ref>, the third one
was proved in Example 6.2 of [10],
whereas the second and fourth statements  are simple consequences of the
Ryll-Wojtaszczyk formula (<ref>).

§.§ Finite degree polynomials

Starting we apply  our general result to show a formula  for the projection constant $\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{1})\big)$ for the Banach space $\mathcal{P}_{ \leq d}(\mathbb{S}^{1})$ of all complex-valued degree-$d$ polynomials on the one dimensional real euclidean sphere. It is worth noting that, as a by-product, we recover the well-known Lozinski-Kharshiladze formula for the projection constant of the Banach space  $\textbf{Trig}_{\leq d}(\mathbb{T})$ of all  complex-valued degree-$d$ polynomials

\[
P(z) = \sum_{|k|\leq d} c_k z^n, \quad\, z \in \mathbb{T}
\]

defined on the  circle group $\mathbb{T}$,
equipped with the uniform norm over $\mathbb{T}$. This formula states that $\boldsymbol{\lambda}\big(\textbf{Trig}_{\leq d}(\mathbb{T})\big)$ equals the Lebesgue constant of the Dirichlet kernel (see <cit.> and  [22]).

For all $d \in \mathbb{N}$, we have

\[
\boldsymbol{\lambda}\big(\textbf{Trig}_{\leq d}(\mathbb{T})\big) = \boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{1})\big) =
\frac{1}{2\pi} \int_{0}^{2\pi}\bigg| \frac{\sin\big((d+ \frac{1}{2})t \big)}{\sin\big(\frac{t}{2} \big)}\bigg|\,dt\,.
\]

In particular,

\[
\lim_{d \to \infty}  \frac{\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{1})\big)}{\log d} \,=\, \frac{4}{\pi^2}\,.
\]

We claim that the linear mapping
$\Phi \colon \textbf{\bf Trig}_{\leq d}(\mathbb T) \to \mathcal{P}_{\leq d}(\mathbb{S}^{1})$,  which for  $P(z) = \sum_{|k|\leq d} c_k z^k$ is given by the formula:

\[
\Phi(P)(x, y):=
\sum_{k=1}^ d c_{-k} (x-iy)^k +c_0+
\sum_{k=1}^ d c_k (x+iy)^k, \quad\, (x, y) \in \mathbb{S}^{1}\,,
\]

is an isometric isomorphism. Being obviously isometric, it is injective.
On the other hand, we deduce from  Proposition <ref> that

\[
\mathcal{P}_{\leq d}(\mathbb{S}^{1}) = \mathcal H_{\leq d} (\mathbb{S}^{1})= \bigoplus_{k \leq d}\mathcal{H}_k(\mathbb{S}^{1})\,.
\]

Hence by Equation (<ref>) we see  that  $\text{dim} \,\mathcal{P}_{\leq d}(\mathbb{S}^{1}) = 2d +1=\textbf{\bf Trig}_{\leq d}(\mathbb{T})$. Combining,
$\Phi$ is as desired an isometric isomorphism,  and  as a consequence

\[
\boldsymbol{\lambda}\big(\textbf{Trig}_{\leq d}(\mathbb{T})\big) = \boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{1})\big)\,.
\]

Now we apply Theorem <ref> and  Equation (<ref>) to get

\begin{equation*}
\mathbf{k}_{\mathcal P_{\leq d} (\mathbb S^1)} (e_1,\eta) =
1+ \sum_{\ell=1}^{d} 2 L_{2,\ell}^{\diamond}(\langle e_1, \eta \rangle), \quad \eta \in
\mathbb S^1\,.
\end{equation*}

Then similar to the proofs of   Propositions <ref> and <ref> we have

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{1})\big)
=  \frac{1}{2\pi}\bigg(\int_{0}^{\pi} \Big|1+ 2 \sum_{\ell=1}^{d} \cos (\ell t)\Big|\,dt + \int_{\pi}^{2\pi} \Big|1+ 2 \sum_{\ell=1}^{d} \cos \big(\ell (2\pi - t)\big)\Big|\,dt\bigg),
\end{align*}

and the argument follows as before.

For $n >2$ we get the following formula - complementing a result of Ragozin <cit.>.

For  $n >2$ and $d \in \mathbb{N}$

\[
\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{n-1})\big) =
\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(n-1)}\,
\frac{\Gamma(d+n-1)}{\Gamma(d+ \frac{n-1}{2})}
\,
\int_{-1}^{1} \Big| P_d^{(\frac{n-1}{2},\frac{n-3}{2})}(t)\Big| (1-t^2)^{\frac{n-3}{2}} dt\,.
\]

The proof follows the  lines of the proof of Theorem <ref>. Again we start with Theorem <ref> which shows that

\[
\mathbf{k}_{\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})}( e_1, \eta )
= \sum_{j=0}^{d} N_{n,j} \,\,L_{n,j}^{\diamond}(\eta_1)
\]

\begin{equation*}
\boldsymbol{\lambda}\big(\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})\big)
\int_{\mathbb{S}^{n-1}}\,
\big| \mathbf{k}_{\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})}(\langle e_1, \eta \rangle)\big| \,d\sigma_n(\eta)\,.
\end{equation*}

Then, as before,
by Lemma <ref>  and Equation (<ref>) we  have

\begin{align} \label{kern-leqd}
\mathbf{k}_{\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})}( e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,\, \sum_{j=0}^{d}\,(2j +n-2) \, \frac{\Gamma(n+j-2)}{\Gamma(\frac{n-1}{2}+j)}\,
P_{j}^{(\alpha, \alpha)}(\eta_1)\,,
\end{align}

where $\alpha = \frac{n-3}{2}$. Now we first rewrite this expression getting

\begin{align*}
\mathbf{k}_{\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})}( e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,
\sum_{j=0}^{d}
\frac{(2j +\alpha +\alpha +1)\Gamma(j +\alpha +\alpha +1)}{\Gamma(j +\alpha +1)} \,
P_{j}^{(\alpha, \alpha)}(\eta_1)
\,,
\end{align*}

But by
we have

\begin{equation*}
\sum_{j=0}^{d}
\frac{(2j +\alpha +\alpha +1)\Gamma(j +\alpha +\alpha +1)}{\Gamma(j +\alpha +1)} \,
P_{j}^{(\alpha, \alpha)}(\eta_1)
\,=\,
\frac{\Gamma(d +\alpha +\alpha +2)}{\Gamma(d +\alpha +1)}
P_{d}^{(\alpha+1, \alpha)}(\eta_1)\,,
\end{equation*}

which shows

\begin{align*}
\mathbf{k}_{\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})}( e_1, \eta )
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,
\frac{\Gamma(d +\alpha +\alpha +2)}{\Gamma(d +\alpha +1)}
P_{d}^{(\alpha+1, \alpha)}(\eta_1)
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,
\frac{\Gamma(d +n-1)}{\Gamma(d +\frac{n-1}{2})}
P_{d}^{(\frac{n-1}{2}, \frac{n-3}{2})}(\eta_1)
\,.
\end{align*}

Consequently, by (<ref>) we have

\begin{align*}
\boldsymbol{\lambda}\big(\mathcal{P}_{\leq d}(\mathbb{S}^{n-1})\big)
\frac{\Gamma(\frac{n-1}{2})}{\Gamma(n-1)}\,
\frac{\Gamma(d +n-1)}{\Gamma(d +\frac{n-1}{2})}\,
\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(\frac{n-1}{2})}
\,
\int_{-1}^1\, \big| P_{d}^{(\frac{n-1}{2}, \frac{n-3}{2})}(t)\big|  (1-t^2)^{\frac{n-3}{2}} dt\,,
\end{align*}

which completes the proof.

Note that for each $n$  and  large  $d$

$$\dim \mathcal{P}_{ \leq d}(\mathbb{S}^{n-1}) \,\sim_{c(n)}\,  d^{n-1}.$$

Indeed, this easily follows by combining the Equations (<ref>), (<ref>)
and (<ref>).
Hence, as in the case of Corollary <ref> (see also  Remark <ref>), this again shows that here the Kadets-Snobar theorem leads to a weak upper estimate
- in contrast to  the result for homogeneous spherical harmonics from
Theorem <ref>.

For each integer $n > 2$

\[
\lim_{d \to \infty} \frac{\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{2})\big)}{d^{\frac{n-2}{2}}}
\,=\,
\frac{2^{\frac{n}{2}+1}}{\pi^{\frac{3}{2}}}\,\,
\frac{\Gamma(\frac{n}{2})\Gamma(\frac{n}{2}-1)}{ \Gamma(n-1)\Gamma(\frac{n}{4}+\frac{1}{4})}
\,.
\]

The proof is similar to that of Corollary <ref>. We use that by <cit.> (see also the main result of [24])
for $\alpha > -\frac{1}{2}$
and $\beta > -1$

\begin{align*}
\lim_{d \to \infty} \,\, \sqrt{d}\,\,
\int_{0}^{1}
(1-x)^\alpha (1+x)^\beta \big|P^{(\alpha+1, \beta)}_d(x)\big| dx
\,=\,
\frac{2^{\alpha + \beta +2}}{\pi^{\frac{3}{2}}}\,
\frac{\Gamma(\frac{\alpha}{2}+\frac{1}{4})\Gamma(\frac{\beta}{2}+\frac{3}{4}) }
{\Gamma(\frac{\alpha + \beta}{2}+1)}\,.
\end{align*}

In [31]  this equality is  in fact  proved as an application of (<ref>),
and in the following we apply it to $\alpha = \beta = \frac{n-3}{2}$. Additionally,  using (<ref>), we have

\[
\lim_{d \to \infty}  \frac{1}{d^{\frac{n-1}{2}}}
\,
\frac{\Gamma(d+n-1)}{\Gamma(d+ \frac{n-1}{2})} \, =\, 1\,.
\]

Combining with Theorem <ref>, we get

\begin{align*}
\lim_{d \to \infty}
\frac{\boldsymbol{\lambda}\big(\mathcal{P}_{ d}(\mathbb{S}^{n-1})\big)}{d^\frac{n-2}{2}}
\\&
\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(n-1)}\,
\,
\lim_{d \to \infty}  \frac{1}{d^{\frac{n-1}{2}}}
\,
\frac{\Gamma(d+n-1)}{\Gamma(d+ \frac{n-1}{2})}
\,
\lim_{d \to \infty} \,\, \sqrt{d}\,
\int_{-1}^{1} \big| P_d^{(\frac{n-1}{2},\frac{n-1}{3})}(t)\big| (1-t^2)^{\frac{n-3}{2}} dt
\\&
\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\, \Gamma(n-1)}\,
\,          \,
\frac{2^{n-1}}{\pi^{\frac{3}{2}}}\,
\frac{\Gamma(\frac{n}{4}-\frac{1}{2})\Gamma(\frac{n}{4})}{ \Gamma(\frac{n}{4}+\frac{1}{4})}
=\frac{2^{n-1}\Gamma(\frac{n}{2})}{\pi^2\, \Gamma(n-1)}\,
\frac{\Gamma(\frac{n}{4}-\frac{1}{2})\Gamma(\frac{n}{4})}{ \Gamma(\frac{n}{4}+\frac{1}{4})}
\,.                         \end{align*}

Then the Legendre duplication formula
(<ref>) gives  for $x = \frac{n}{4} -\frac{1}{2} $ that

\[
\Gamma\Big( \frac{n}{4} -\frac{1}{2} \Big) \Gamma\Big( \frac{n}{4}\Big) \,= \,\frac{\sqrt{\pi}}{2^{\frac{n}{2}-2}} \,
\Gamma\Big( \frac{n}{2} -1 \Big)\,,
\]

and this concludes the argument.

We finish with the following  special case $n=3$ of Theorem <ref> and its  Corollary <ref>.
Note that according to Theorem <ref> we here may replace the projection constant of  $\mathcal{P}_{ \leq d}(\mathbb{S}^{2})$ by the norm of the orthogonal projection $\pi_{\mathcal{P}_{ \leq d}(\mathbb{S}^{2})} \colon C(\mathbb{S}^{2}) \to \mathcal{P}_{ \leq d}(\mathbb{S}^{2})$.
In this form the result is due to Gronwall <cit.>
(see also  <cit.>).

For  every  $d \in \mathbb{N}$

\[
\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{2})\big)
= \frac{d+1}{2} \int_{-1}^{1}
|P_d^{(1,0)}(t)| dt\,.
\]

\begin{align*}
\lim_{d \to \infty} \frac{\boldsymbol{\lambda}\big(\mathcal{P}_{ \leq d}(\mathbb{S}^{2})\big)}{\sqrt{d}}
\,=\,
2 \sqrt{\frac{2}{\pi}}\,.
\end{align*}

§ TRANSITION TO REAL COEFFICIENTS

Throughout the article, our function spaces under study had always the real unit sphere $\mathbb{S}^{n-1}$ as their domain and the complex field $\mathbb{C}$ as their codomain. In particular, the polynomials belonging to these spaces had complex coefficients in their monomial expansion.

It begs the question, what happens with their counterparts when dealing just with real numbers?
That is, the underlying polynomials have only real coefficients in their power series expansion.

It is not difficult to see that the analogous spaces have the same dimension as $\mathbb{R}$-vector spaces. Moreover, the same proofs (following line by line) we provided hold. In fact note that the kernels appearing in the theorems are polynomials with real values. Thus, the exact same conclusions can be obtained in this context of functions reaching the real field $\mathbb{R}$.

To illustrate this, recall from (<ref>) that $\mathcal{P}_{d}(\mathbb{S}^{n-1})=\mathcal{P}_{d}(\ell_2^n(\mathbb{R}))$, and note that a simple calculation shows that for $d=1$ then Theorem <ref> for
$\boldsymbol{\lambda}\big(\mathcal{P}_{d}(\mathbb{S}^{n-1})\big)$ is in fact consistent with Rutovitz's result for the real Hilbert space (<ref>).

§ ACKNOWLEDGMENTS

The second author would like to express gratitude to Professor Mariano Suarez Alvarez for bringing reference [2] to our attention. He would also like to thank Professor Luciano Abadias for elucidating a calculation in his article [1], which has been instrumental for our purposes.
The fourth author would like to thank Professor Dorothee Haroske for her assistance in obtaining and providing us the book  [31].

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