# Highly localized kernels on space of homogeneous type

Yuan Xu  Department of Mathematics, University of Oregon, Eugene, OR
97403–1222, USA<EMAIL_ADDRESS>
###### Abstract.

Highly localized kernels based on orthogonal polynomials have been studied and
utilized over several regular domains. Much of the results deduced via these
kernels can be treated uniformly in the framework of localizable spaces of
homogeneous type. We clarify the concept of localness and provide a list of
such localizable spaces.

###### 2010 Mathematics Subject Classification:

41A10, 41A63, 42C10, 42C40

The author was partially supported by Simons Foundation Grant #849676.

For the proceedings of the Constructive Theory of Functions, Lozenets,
Bulgaria, 2023.

##  1\. Introduction

Highly localized kernels based on orthogonal polynomials have been
instrumental in studying various problems in approximation theory and
computational analysis over regular domains in the past two decades. Such
kernels have near-exponential decay off the diagonal and also serve as kernels
for integral operators that are polynomials of near-best approximation on the
domain, which makes them ideal tools for studying several problems in
approximation theory and harmonic analysis. They have been established for
weighted function spaces on the interval, the unit sphere, the unit ball, and
the simplex in the Euclidean spaces [3, 13, 14, 9, 10, 18, 19, 20, 21] as well
as on ${\mathbb{R}}_{+}^{d}$ and ${\mathbb{R}}^{d}$ [12, 22], and more
recently, on the conic domains [28, 30].

In the literature, much of the analysis based on highly localized kernels is
carried out domain by domain, relying on the geometry of each domain.
Recently, in [28], a framework was developed that provides a unified treatment
for all compact domains that are equipped with highly localized kernels. In
this expository paper, we aim to clarify the concept of localization and
explain the essence of the framework.

The setup is on the space of homogeneous type, which is a measure space
$(\Omega,\mu,{\mathsf{d}})$ on the set $\Omega$ with a positive measure $\mu$
and a metric ${\mathsf{d}}$ such that all open balls

$B(x,r)=\\{y\in\Omega:{\mathsf{d}}(x,y)<r\\}$

are measurable and $\mu$ is a regular measure satisfying the doubling property

$\mu(B(x,2r))\leq c\mu(B(x,r)),\qquad\forall x\in\Omega,\quad\forall r>0,$

where $c$ is independent of $x$ and $r$. Such a measure $\mu$ is called a
doubling measure. In our setup, the domain $\Omega$ will be either an
algebraic surface or a domain with a non-empty interior, and
$\mathrm{d}\mu={\mathsf{w}}(x)\mathrm{d}x$ with ${\mathsf{w}}$ being a
nonnegative weight function defined on $\Omega$ that defines a well-
established inner product

(1.1)
${\langle}f,g{\rangle}_{{\mathsf{w}}}=\int_{\Omega}f(x)g(x){\mathsf{w}}(x)\mathrm{d}x$

on the space of polynomials restricted to $\Omega$. We shall write the
homogeneous space as $(\Omega,{\mathsf{w}},{\mathsf{d}})$ with ${\mathsf{w}}$
being a doubling weight, and we write
${\mathsf{w}}(E)=\int_{E}\mathrm{d}{\mathsf{w}}$ for the subset
$E\subset\Omega$.

We need kernels defined in terms of orthogonal polynomials. Let
${\mathcal{V}}_{n}^{d}({\mathsf{w}})$ be the space of orthogonal polynomials
of degree $n$ with respect to this inner product. Let
$P_{n}({\mathsf{w}};\cdot,\cdot)$ be the reproducing kernel of the space
${\mathcal{V}}_{n}^{d}({\mathsf{w}})$. The orthogonal projection operator
$\operatorname{proj}_{n}:L^{2}(\Omega,{\mathsf{w}})\mapsto{\mathcal{V}}_{n}^{d}({\mathsf{w}})$
can be written as

(1.2)
$(\operatorname{proj}_{n}f)(x)=\int_{\Omega}P_{n}({\mathsf{w}};x,y)f(y){\mathsf{w}}(y)\mathrm{d}y,\qquad
f\in L^{2}(\Omega,{\mathsf{w}}).$

Let $\widehat{a}$ be a cut-off function, defined as a compactly supported
nonnegative function in $C^{\infty}({\mathbb{R}}_{+})$ such that
$\widehat{a}(t)=1$ for $0\leq t\leq 1$ and $\widehat{a}(t)=0$ for $t\geq 2$.
Then our highly localized kernels are defined by

(1.3)
$L_{n}({\mathsf{w}};x,y):=\sum_{j=0}^{\infty}\widehat{a}\Big{(}\frac{j}{n}\Big{)}P_{j}({\mathsf{w}};x,y).$

The kernel is regarded as highly localized if it satisfies an upper-bound
estimate that contains a factor $(1+n{\mathsf{d}}(x,y))^{-{\kappa}}$ for any
given ${\kappa}>0$, which means that the kernel decays faster than any
polynomial power away from the diagonal $x=y$; see the next section for a
precise definition.

We call $(\Omega,{\mathsf{w}},{\mathsf{d}})$ a localizable space of
homogeneous type if it possesses highly localized kernels. For such a space,
the framework in [28] allows us to establish several results in approximation
theory and harmonic analysis in weighted function spaces, such as
$L^{p}(\Omega,{\mathsf{w}})$, that include a characterization of best
approximation by polynomials and localized tight frames, among other results.
In other words, several results can be regarded as properties of localizable
space of homogeneous type. The definition of highly localized kernels in [28]
requires three assertions that need to hold. As we shall show in the next
section, one of the assertions is redundant as it holds for every doubling
weight.

At the moment, the localizable spaces of homogenous type are known only for a
few regular domains. This is not surprising given the complexity of
multivariate orthogonal polynomials and, even more so, the kernel
$P_{n}({\mathsf{w}};\cdot,\cdot)$. Most highly localized kernels are
established when a closed-form formula for the reproducing kernel
$P_{n}({\mathsf{w}};\cdot,\cdot)$ is known, which we call an addition formula,
taking a cure from the addition formula for the spherical harmonics. The
existence of such a formula is rare and reflects something fundamental about
the localizable space of homogeneous type. We shall provide a list of examples
of localizable spaces of homogeneous type.

The paper is organized as follows. In the next section, we provide a precise
definition of highly localized kernels and, as a consequence, localizable
space of homogeneous type, and a sketch of some results that can be
established with the help of such kernels. In the third section, we provide a
list of examples of known localizable spaces.

##  2\. Localized space of homogeneous type

###  2.1. Orthogonal structure and localized kernels

Let $(\Omega,{\mathsf{w}},{\mathsf{d}})$ be a space of homogeneous type
associated with the inner product ${\langle}\cdot,\cdot{\rangle}_{\mathsf{w}}$
defined in (1.1). The dimension of the space
${\mathcal{V}}_{n}^{d}({\mathsf{w}})$ of orthogonal polynomials depends on the
geometry of $\Omega\subset{\mathbb{R}}^{d}$. For our purpose, we only consider
two cases. The first case is when the interior of $\Omega$ is an open set,
such as the unit ball or the simplex in ${\mathbb{R}}^{d}$, for which

(2.1) $\dim{\mathcal{V}}_{n}^{d}(\Omega,{\mathsf{w}})=\binom{n+d-1}{n},\quad
n=0,1,2,\ldots.$

The second case is when the domain $\Omega$ is a quadratic algebraic surface
of the form $\\{x\in{\mathbb{R}}^{d}:\phi(x)=0\\}$, where $\phi$ is a
quadratic polynomial, such as the unit sphere ${\mathbb{S}^{d-1}}$ with
$\phi(x)=1-\|x\|^{2}$, for which

(2.2)
$\dim{\mathcal{V}}_{n}^{d}(\Omega,{\mathsf{w}})=\binom{n+d-2}{n}+\binom{n+d-3}{n-1},\quad
n=1,2,3,\ldots.$

We assume that ${\mathsf{w}}$ is regular so that the orthogonal decomposition

$L^{2}(\Omega,\varpi)=\bigoplus_{n=0}^{\infty}{\mathcal{V}}_{n}^{d}(\Omega,{\mathsf{w}}):\qquad
f=\sum_{n=0}^{\infty}\operatorname{proj}_{n}f$

holds, where $\operatorname{proj}_{n}$ is the orthogonal projection operator.
This gives the Fourier orthogonal decomposition of $f$ since, if $P_{\nu,n}$
is an orthonormal basis of ${\mathcal{V}}_{n}^{d}(\varpi)$, then

$\operatorname{proj}_{n}f=\sum_{1\leq\nu\leq\dim{\mathcal{V}}_{n}^{d}(\Omega,{\mathsf{w}})}\widehat{f}_{\nu,n}P_{\nu,n},\qquad\widehat{f}_{\nu,n}:={\langle}f,P_{\nu,n}{\rangle}_{\mathsf{w}}.$

The reproducing kernel $P_{n}({\mathsf{w}};\cdot,\cdot)$ of
${\mathcal{V}}_{n}^{d}(\Omega,{\mathsf{w}})$ is the kernel of the operator
$\operatorname{proj}_{n}$, as shown in (1.2), which can be written as

$P_{n}({\mathsf{w}};x,y)=\sum_{1\leq\nu\leq\dim{\mathcal{V}}_{n}^{d}(\Omega,{\mathsf{w}})}P_{\nu,n}(x)P_{\nu,n}(y)$

in terms of any orthonormal basis of ${\mathcal{V}}_{n}^{d}(\varpi)$. In the
multidimensional setting, orthogonal polynomials are complicated and described
by multiple indices even when they can be written down, and the kernel
$P_{n}({\mathsf{w}};\cdot,\cdot)$ is a sum of multiple layers of products of
polynomials. Such a sum can hardly be useful for obtaining estimates of
kernels like $L_{n}({\mathsf{w}})$ defined in (1.3).

In all cases when highly localized kernels are known, their desired upper
bound is established with the help of an addition formula for the reproducing
kernel. In most of the cases, the addition formula is of the form

(2.3)
$P_{n}({\mathsf{w}};x,y)=\int_{{\mathbb{R}}^{m}}Z_{n}(\xi(x,y;u))\mathrm{d}\mu(u)$

where $Z_{n}$ is a polynomial, often orthogonal polynomial, of one variable,
$\xi:(x,y;z)\mapsto{\mathbb{R}}$ for $x,y\in{\mathbb{R}}^{d}$, and
$\mathrm{d}\mu$ is a measure on ${\mathbb{R}}^{m}$, which can degenerate to
point evaluations. The simplest example is the unit sphere with orthogonal
polynomials being spherical harmonics with respect to the surface measure, for
which the closed-form formula is given by

(2.4)
$P_{n}({\mathsf{w}};x,y)=\frac{n+{\lambda}}{{\lambda}}C_{n}^{\lambda}({\langle}\xi,\eta{\rangle})\quad\hbox{with}\quad{\lambda}=\frac{d-2}{2},$

where $C_{n}^{\lambda}$ is the Gegenbauer polynomial of degree $n$ (see
Subsection 3.2 below). This formula is called the addition formula for
spherical harmonics, a name that we adopt for closed-formula of reproducing
kernels in all other cases. If an addition formula of the form (2.3) exists,
then the kernel $L_{n}({\mathsf{w}};\cdot,\cdot)$ defined in (1.3) can be
written as

$L_{n}({\mathsf{w}};x,y)=\int_{{\mathbb{R}}^{m}}L_{n}\big{(}\xi(x,y;\mu)\big{)}\mathrm{d}\mu(u),$

where $L_{n}$ on the right-hand side is a function of one variable given by

$L_{n}(t)=\sum_{j=0}^{\infty}\widehat{a}\Big{(}\frac{j}{n}\Big{)}Z_{j}(t),$

which allows us to deduce an upper bound of the kernel
$L_{n}({\mathsf{w}};\cdot,\cdot)$ from that of $L_{n}$.

We should emphasize that the addition formula is something special. If it is
of the form (2.3), then it indicates that the reproducing kernel, hence the
projection operator $\operatorname{proj}_{n}f$, has an implicit one-
dimensional structure, which allows us to relate the study of the Fourier
orthogonal series to that of the Fourier orthogonal series in one variable.
For example, it is well-known that a substantial portion of the Fourier series
on the unit sphere is reduced to that of the Fourier-Gegenbauer series. The
same holds for other regular domains and for any space of homogeneous type
that possesses an addition formula (cf. [28, 29, 30]).

###  2.2. Highly localized kernels

Let $(\Omega,{\mathsf{w}},{\mathsf{d}})$ be a space of homogeneous type. The
existing examples of the highly localized kernel suggests the following
definition.

######  Definition 2.1.

The kernels $L_{n}({\mathsf{w}};\cdot,\cdot)$, $n=1,2,\ldots$, are called
highly localized if they satisfy the following assertions:

  1. Assertion 1. For ${\kappa}>0$ and $x,y\in\Omega$,

$|L_{n}({\mathsf{w}};x,y)|\leq
c_{\kappa}\frac{1}{\sqrt{{\mathsf{w}}\\!\left(B(x,n^{-1})\right)}\sqrt{{\mathsf{w}}\\!\left(B(y,n^{-1})\right)}\left(1+n{\mathsf{d}}(x,y)\right)^{\kappa}}.$

  2. Assertion 2. For $0<\delta\leq\delta_{0}$ with some $\delta_{0}<1$ and $x_{1}\in B(x_{2},\frac{\delta}{n})$,

$\left|L_{n}({\mathsf{w}};x_{1},y)-L_{n}({\mathsf{w}};x_{2},y)\right|\leq
c_{\kappa}\frac{n{\mathsf{d}}(x_{1},x_{2})}{\sqrt{{\mathsf{w}}\\!\left(B(x_{1},n^{-1})\right)}\sqrt{{\mathsf{w}}\\!\left(B(x_{2},n^{-1})\right)}\left(1+n{\mathsf{d}}(x_{2},y)\right)^{\kappa}}.$

This definition contains one less assertion in comparison with the definition
given in [28]. The third assertion contained in [28] implies that the first
two assertions are sharp. For example, it implies that

(2.5)
$\int_{\Omega}\left|L_{n}({\mathsf{w}};x,y)\right|^{p}{\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y)\leq
c\left[{\mathsf{w}}\\!\left(B\\!\left(x,n^{-1}\right)\right)\right]^{1-p},$

where $\mathrm{d}{\mathsf{m}}$ is the Lebesgue measure on $\Omega$, which is
frequently needed when we utilize the highly localized kernels. The third
assertion imposes a condition on the weight function ${\mathsf{w}}$:

Assertion 3. For sufficient large ${\kappa}>0$, there is a constant
$c_{\kappa}>0$ such that

$\displaystyle\int_{\Omega}\frac{{\mathsf{w}}(y)}{{\mathsf{w}}\\!\left(B(y,n^{-1})\right)\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\kappa}}}\mathrm{d}{\mathsf{m}}(y)\leq
c_{\kappa}.$

This assertion is stated in [28] as part of the definition for
$L_{n}({\mathsf{w}})$ being a highly localized kernel. It is, however,
redundant since it holds for all doubling weight.

######  Proposition 2.2.

If $(\Omega,{\mathsf{w}},{\mathsf{d}})$ is a space of homogeneous type, then
Assertion 3 holds.

###### Proof.

It is known that the doubling weight ${\mathsf{w}}$ satisfies

(2.6)
$\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{-{\alpha}({\mathsf{w}})}\leq\frac{{\mathsf{w}}\\!\left(B\left(y,n^{-1}\right)\right)}{{\mathsf{w}}\\!\left(B\left(x,n^{-1}\right)\right)}\leq\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\alpha}({\mathsf{w}})},$

where ${\alpha}({\mathsf{w}})>0$ is a positive constant. Fix $n$ and define
$\Omega_{0,n}(x)=B(x,n^{-1})$ and

$\Omega_{m,n}(x)=\left\\{y:2^{m-1}n^{-1}\leq{\mathsf{d}}(x,y)\leq
2^{m}n^{-1}\right\\},\qquad m=1,2,\ldots.$

Then $\Omega=\cup_{m=0}^{\infty}\Omega_{m,n}(x)$. Let $J_{n}$ denote the left-
hand side of the inequality in Assertion 3. Using (2.6), we obtain

$\displaystyle J_{n}\,$
$\displaystyle=\sum_{m=0}^{\infty}\int_{\Omega_{m,n}(x)}\frac{{\mathsf{w}}(y)}{{\mathsf{w}}\\!\left(B(y,n^{-1})\right)\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\kappa}}}\mathrm{d}{\mathsf{m}}(y)$
$\displaystyle\leq\sum_{m=0}^{\infty}\frac{1}{{\mathsf{w}}\\!\left(B(x,n^{-1})\right)}\int_{\Omega_{m,n}(x)}\frac{{\mathsf{w}}(y)}{\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\kappa}-{\alpha}(w)}}\mathrm{d}{\mathsf{m}}(y).$

Since ${\mathsf{d}}(x,y)\geq 2^{m-1}n^{-1}$ and
$\Omega_{m,n}(x)=B(x,2^{m}n^{-1})\setminus B(x,2^{m-1}n^{-1})$, it follows
that

$\displaystyle
J_{n}\leq\sum_{m=0}^{\infty}\frac{1}{{\mathsf{w}}\\!\left(B(x,n^{-1})\right)}\frac{{\mathsf{w}}\\!\left(B(x,2^{m}n^{-1})\right)}{(2^{m-1})^{{\kappa}-{\alpha}({\mathsf{w}})}}\leq\sum_{m=0}^{\infty}\frac{(c_{\mathsf{w}})^{m}}{(1+2^{m-1})^{{\kappa}-{\alpha}({\mathsf{w}})}}\leq\frac{1}{2^{{\kappa}-{\alpha}({\mathsf{w}})}-c_{\mathsf{w}}}$

where we have used the doubling condition of ${\mathsf{w}}$, with
$c_{\mathsf{w}}$ denoting the smallest constant for the doubling condition, in
the second inequality. Choosing ${\kappa}$ large shows that $J_{n}$ is
bounded. ∎

######  Definition 2.3.

A space $(\Omega,{\mathsf{w}},{\mathsf{d}})$ of homogeneous type is called
localizable if it possesses highly localized kernels for one doubling weight
$\varpi$ on $\Omega$.

This definition is given in [28], where several problems are studied in a
localizable space of homogeneous type, with few additional assumptions, and it
provides a uniform treatment of analysis on several regular domains that are
derived domain by domain in the literature. As a consequence, as soon as we
can show a space of homogeneous type is localizable, a variety of results will
follow right away. This is the motivation for the study in [28], where the
framework is developed and applied to new spaces over conic domains.

It is worthwhile to emphasize that we need highly localized kernel only for
one doubling weight $\varpi$ on $\Omega$ in order for
$(\Omega,{\mathsf{w}},{\mathsf{d}})$ being localizable. Indeed, most of the
results in [28, 30] hold for all doubling weight in a localizable space.

###  2.3. Analysis in localizable space of homogeneous type

The highly localized kernels have been an important tool for several problems
in approximation theory and harmonic analysis. Here we briefly outline a few
of them to illustrate their usefulness.

####  2.3.1. Best approximation by polynomials

Let $L_{n}({\mathsf{w}};\cdot,\cdot)$ be the operator defined in (1.3) via a
cut-off function $\widehat{a}$, which is a resampling of the kernel
$P_{2n}({\mathsf{w}};\cdot,\cdot)$. We can define an integral operator that
has $L_{n}\big{(}\varpi;\cdot,\cdot)$ as its kernel. For convenience, we
denote the operator by $L_{n}(\varpi)*f$ in the notation of a pseudo
convolution and define it as

(2.7)
$L_{n}({\mathsf{w}})*f(x)=\int_{\Omega}f(y)L_{n}\big{(}{\mathsf{w}};x,y){\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y).$

The definition of the cut-off function $\widehat{a}$ shows that $L_{n}*f=f$,
whenever $f$ is a polynomial of degree at most $n$, and $L_{n}*f$ is a
polynomial of degree at most $2n$. If $L_{n}({\mathsf{w}})$ were highly
localized, then $L_{n}({\mathsf{w}})*f$ should be a good approximation to the
function $f$. This is indeed the case. Let $E_{n}(f)_{p,{\mathsf{w}}}$ denote
the error of the best polynomial approximation of degree at most $n$ to $f$,

$E_{n}(f)_{p,{\mathsf{w}}}:=\int_{P\in\Pi_{n}(\Omega)}\|f-P\|_{p,{\mathsf{w}}},\qquad
1\leq p\leq\infty,$

where $\|\cdot\|_{p,{\mathsf{w}}}$ denotes the norma of
$L^{p}(\Omega,{\mathsf{w}})$ and $\Pi_{n}(\Omega)$ denote the space of
polynomials restricted on $\Omega$. Then $L_{n}({\mathsf{w}})*f$ is a near-
best polynomial of best approximation, as specified in the following theorem.

######  Theorem 2.4.

Let $(\Omega,{\mathsf{w}},{\mathsf{d}})$ be a localizable space of homogeneous
type. Let $f\in L^{p}(\Omega,{\mathsf{w}})$, $1\leq p<\infty$ and $f\in
C(\Omega)$ for $p=\infty$. Then the operator $L_{n}({\mathsf{w}})*f$ is
bounded in $L^{p}(\Omega,{\mathsf{w}})$ and

$\|f-L_{n}*f\|_{p,{\mathsf{w}}}\leq c_{p}E_{n}(f)_{p,{\mathsf{w}}},\quad 1\leq
p\leq\infty.$

If $Z_{n}$ in the addition formula (2.3) is a normalized Jacobi polynomial
given by

$Z_{n}^{({\alpha},{\beta})}(t)=\frac{P_{n}^{({\alpha},{\beta})}(1)P_{n}^{({\alpha},{\beta})}(t)}{h_{n}^{({\alpha},{\beta})}},$

where $h_{n}^{({\alpha},{\beta})}$ is the $L^{2}$ norm of
$P_{n}^{({\alpha},{\beta})}$, which holds for several domains as will be seen
in the next section, then we can define a translation operator
$S_{{\theta},{\mathsf{w}}}$ by

(2.8)
$\operatorname{proj}_{n}({\mathsf{w}};S_{{\theta},{\mathsf{w}}}f)=\frac{P_{n}^{({\alpha},{\beta})}(\cos{\theta})}{P_{n}^{({\alpha},{\beta})}(1)}\operatorname{proj}_{n}({\mathsf{w}};f),\quad
n=0,1,2,\ldots.$

This is a bounded operator and satisfies, in particular, that

$\|S_{{\theta},\varpi}f\|_{p,\varpi}\leq\|f\|_{p,\varpi}\quad\hbox{and}\quad\lim_{{\theta}\to
0}\|S_{{\theta},\varpi}f-f\|_{\varpi,p}=0.$

for $f\in L^{p}(\varpi;\Omega)$ if $1\leq p\leq\infty$, or $f\in C(\Omega)$ if
$p=\infty$. These properties allow us to define a modulus of smoothness in the
localizable space of homogeneous type. For $r>0$, we define the $r$-th
difference operator

$\triangle_{{\theta},\varpi}^{r}=\left(I-S_{{\theta},\varpi}\right)^{r/2}=\sum_{n=0}^{\infty}(-1)^{n}\binom{r/2}{n}(S_{{\theta},\varpi})^{n},$

where $I$ denotes the identity operator, in the distribution sense. Then the
modulus smoothness is defined by

(2.9) $\omega_{r}(f,t)_{p,\varpi}:=\sup_{0<\theta\leq
t}\left\|\triangle_{{\theta},\varpi}^{r}f\right\|_{p,\varpi},\quad 0<t<\pi,$

which satisfies the usual properties of the modulus of smoothness.
Furthermore, it can be used to prove both direct and inverse theorems in a
characterization of the best approximation $E_{n}(f)_{p,{\mathsf{w}}}$ by
polynomials. We refer to [28] for details.

####  2.3.2. Positive cubature rules and Christoffel functions

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. A cubature rule of degree
$n$ for ${\mathsf{w}}$ is a finite linear combination of point evaluations
that satisfies

$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{k=1}^{N}{\lambda}_{k}f(x_{k}),\qquad\forall
f\in\Pi_{n}(\Omega),$

where ${\lambda}_{k}\in{\mathbb{R}}$ and $x_{k}\in\Omega$. The cubature rule
is called positive if all ${\lambda}_{k}$ are positive. The positivity is
important for numerical stability and for many applications. Positive cubature
rules exist if the set of its nodes is well separated as defined below.

######  Definition 2.5.

Let $\Xi$ be a discrete set in $\Omega$.

  1. (a)

A finite collection of subsets $\\{S_{z}:z\in\Xi\\}$ is called a partition of
$\Omega$ if $S_{z}^{\circ}\cap S_{y}^{\circ}=\emptyset$ when $z\neq y$ and
$\Omega=\bigcup_{z\in\Xi}S_{z}$.

  2. (b)

Let ${\varepsilon}>0$. A discrete subset $\Xi$ of $\Omega$ is called
${\varepsilon}$-separated if ${\mathsf{d}}(x,y)\geq{\varepsilon}$ for every
two distinct points $x,y\in\Xi$.

  3. (c)

$\Xi$ is called maximal if there is a constant $c_{d}>1$ such that

$1\leq\sum_{z\in\Xi}\chi_{B(z,{\varepsilon})}(x)\leq c_{d},\qquad\forall
x\in\Omega,$

where $\chi_{E}$ denotes the characteristic function of the set $E$.

The item (c) implies $\Omega=\bigcup_{z\in\Xi}B(z,{\varepsilon})$. The
positive cubature rules have been studied by many authors and in various
domains. It takes the following form.

######  Theorem 2.6.

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. Let $\Xi$ be a maximum
$\frac{\delta}{n}$-separated subset of $\Omega$. There is a $\delta_{0}>0$
such that for $0<\delta<\delta_{0}$ there exist positive numbers
${\lambda}_{z}$, $z\in\Xi$, so that

(2.10)
$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{z\in\Xi}{\lambda}_{z}f(z),\qquad\forall
f\in\Pi_{n}(\Omega).$

Moreover, the weight ${\lambda}_{z}$ satisfies ${\lambda}_{z}\geq
c_{1}{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)$.

This theorem is first proved for the unit sphere [17, 18]. The proof is
standard by now and can be easily adapted to any space of homogeneous type.
The existence of positive ${\lambda}_{z}$ in (2.10) is based on the Farkas
lemma and uses the Marcinkiewicz-Zygmund inequality. The proof latter
inequality does not rely on highly localized kernels, nor is the proof of the
above theorem. Thus, these results are properties for the space of homogeneous
type. With such generality, they have become a useful tool for the
discretization of $L^{p}(\Omega,{\mathsf{w}})$ for fairly general $\Omega$ and
${\mathsf{w}}$.

In the case that the space is localizable, we can utilize the highly localized
kernels to provide a matching upper bound for the coefficients ${\lambda}_{z}$
via the Christoffel function, which plays an important role in studying the
tight polynomial frame in the next subsection. Let ${\mathsf{w}}$ be a
doubling weight on $\Omega$. The Christoffel function
${\lambda}_{n}({\mathsf{w}};\cdot)$ is defined by

(2.11)
$\displaystyle{\lambda}_{n}({\mathsf{w}};x):=\inf_{\begin{subarray}{c}g(x)=1\\\
g\in\Pi_{n}(\Omega)\end{subarray}}\int_{\Omega}|g(x)|^{2}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x).$

The function is of interest in its own right and has various applications.
Recall that $P_{n}({\mathsf{w}};\cdot,\cdot)$ denote the reproducing kernel of
${\mathcal{V}}_{n}(\Omega,{\mathsf{w}})$. it is known [5, Theorem 3.6.6.]
that, for $n=0,1,2,\ldots$,

(2.12)
$\displaystyle{\lambda}_{n}({\mathsf{w}};x)=\frac{1}{K_{n}({\mathsf{w}};x,x)},\qquad
K_{n}({\mathsf{w}};x,y):=\sum_{k=0}^{n}P_{k}({\mathsf{w}};x,y).$

We need to obtain lower and upper bound for ${\lambda}_{n}({\mathsf{w}};x)$.
If $(\Omega,{\mathsf{w}},{\mathsf{d}})$ is localizable, then the lower bound
follows immediately from the highly localized kernel $L_{n}({\mathsf{w}})$,
since

$K_{n}({\mathsf{w}};x,x)\leq L_{n}({\mathsf{w}};x,x)\leq
c\,{\mathsf{w}}\big{(}B(x,n^{-1})\big{)}$

where the second inequality follows from setting $y=x$ in Assertion 1, which
is equivalent to ${\lambda}_{n}({\mathsf{w}};x)\geq
c{\mathsf{w}}\big{(}B(x,n^{-1})\big{)}$ by (2.12). For the upper bound, we can
use (2.11) by selecting a particular function $g(y)$ that is localized at $x$,
which we choose as a fast decaying polynomial. In one variable, fast decaying
polynomials are studied in [11], see also [23], which leads to fast decaying
polynomials on the unit sphere by the addition formula. For the unit ball,
such polynomials are constructed in [21]. In the space of homogeneous type, we
assume that such polynomials exist by making the following assertion.

Assertion 4. Let $\Omega$ be compact. For each $x\in\Omega$, there is a
nonnegative polynomial $T_{x}$ of degree at most $n$ that satisfies

  1. (1)

$T_{x}(x)=1$, $T_{x}(y)\geq\delta>0$ for $y\in B(x,\frac{1}{n})$ for some
$\delta$ independent of $n$, and, for each ${\gamma}>1$,

$0\leq T_{x}(y)\leq c_{\gamma}(1+n{\mathsf{d}}(x,y))^{-{\gamma}},\qquad
y\in\Omega;$

  2. (2)

there is a polynomial $q_{n}$ such that $q_{n}(x)T_{x}(y)$ is a polynomial of
degree at most $rn$, for some positive integer $r$, in $x$-variable and
$c_{1}\leq q_{n}(x)\leq c_{2}$ for $x\in\Omega$ for some positive numbers
$c_{1}$ and $c_{2}$.

This was formulated in [28] and used to establish the following [28, Lemma
2.1.5]:

######  Lemma 2.7.

Assume Assertion 4. Let ${\alpha}>0$ be a positive number and let $x\in\Omega$
be fixed. For a doubling weight ${\mathsf{w}}$ on $\Omega$, there is a
polynomial $Q_{x}$ of degree $n$ such that, for all $y\in\Omega$,

(2.13)
$c_{1}(1+n{\mathsf{d}}(x,y))^{{\alpha}}{\mathsf{w}}(B(y,\tfrac{1}{n}))\leq
Q_{x}(y)\leq
c_{2}(1+n{\mathsf{d}}(x,y))^{{\alpha}}{\mathsf{w}}(B(y,\tfrac{1}{n})),$

where $c_{1}$ and $c_{2}$ are positive constant independent of $n$, $x$ and
$y$.

The inequality (2.13) shows that $Q_{x}$ is highly localized and decays near-
exponentially away from $y=x$. Using this lemma, we obtain an upper bound of
${\lambda}_{n}({\mathsf{w}})$ and, in particular, a matching upper bound for
${\lambda}_{z}$ in Theorem 2.6.

######  Theorem 2.8.

Let $(\Omega,{\mathsf{w}},{\mathsf{d}})$ be a localizable space of homogeneous
type and assume Assertion 4. Then

${\lambda}_{n}({\mathsf{w}};x)\sim{\mathsf{w}}\big{(}B(y,\tfrac{1}{n})\big{)}.$

Moreover, the coefficients ${\lambda}_{z}$ in the cubature rule (2.10)
satisfies
${\lambda}_{z}\sim{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)$.

It is shown in [28, 30] that this theorem holds for conic domains.

####  2.3.3. Localized polynomial tight frames

The study of such tight frames starts with the unit sphere in [18], where they
are called needlelits, and then studied for several regular domains, such as
(cf. [3, 4, 12, 9, 20, 21, 22]), and they have been applied in both
computational analysis (cf. [1, 2, 7, 8, 15, 24]) and in decomposition of
functions spaces (cf. [13, 14, 10, 19, 22]). We follow the framework in [28].

Let $(\Omega,\varpi,{\mathsf{d}})$ be a localizable homogeneous space. We
consider a highly localized kernel $L_{n}({\mathsf{w}};x,y)$ with its cut-off
function $\widehat{a}$ satisfies

$\displaystyle\begin{split}\widehat{a}(t)\geq\rho>0,\qquad&\mbox{if
$t\in[3/5,5/3]$},\\\
[\widehat{a}(t)]^{2}+[\widehat{a}(2t)]^{2}=1,\qquad&\mbox{if
$t\in[1/2,1]$.}\end{split}$

Such a cut-off function can be easily constructed. The last assumption on
$\widehat{a}$ implies

(2.14)
$\sum_{j=0}^{\infty}\left[\widehat{a}\left(\frac{t}{2^{j}}\right)\right]^{2}=1,\quad
t\in[1,\infty).$

In terms of the kernel $L_{n}({\mathsf{w}};\cdot,\cdot)$ defined via the above
cut-off function $\widehat{a}$, we define

$F_{0}(x,z):=1,\quad\hbox{and}\quad
F_{j}(x,z):=L_{2^{j-1}}({\mathsf{w}};x,z),\quad j=1,2,3,\ldots.$

and define, accordingly, $F_{j}*f=L_{2^{j-1}}({\mathsf{w}})*f$, that is,

$F_{j}*f(x):=\int_{\Omega}f(y)F_{j}(x,y){\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y),\qquad
j=0,1,2,\ldots.$

The $*$ operation is associative. By the orthogonality of the reproducing
kernel, it follows readily that

$\displaystyle
F_{j}*F_{j}*f=\sum_{k=1}^{2^{j}}\left|\widehat{a}\left(\frac{k}{2^{j-1}}\right)\right|^{2}\operatorname{proj}_{k}({\mathsf{w}};f).$

Hence, the following semi-discrete Calderón type decomposition follows from
(2.14),

(2.15)
$f=\sum_{k=0}^{\infty}\operatorname{proj}_{k}({\mathsf{w}};f)=\sum_{j=0}^{\infty}F_{j}*F_{j}*f,\qquad
f\in L^{2}(\Omega,{\mathsf{w}}).$

Since $F_{j}(x,\cdot)F_{k}(\cdot,y)$ is a polynomial of degree $2^{j}$,
discretization by a cubature rule of degree $2^{j}$ puts $F_{j}*F_{k}$ as a
sum, which leads to a tight frame if the cubature rule has positive
coefficients. We use the positive cubature rule in Theorem 2.6. For
$j=0,1,\ldots,$ let ${\varepsilon}_{j}=\frac{\delta}{2^{j}}$ and let $\Xi_{j}$
be a maximal ${\varepsilon}_{j}$-separated subset in $\Omega$, where $\delta$
is chosen so that the cubature rule (2.10) holds. Thus, there are
${\lambda}_{z,j}>0$ for $z\in\Xi_{j}$ such that

$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{z\in\Xi_{j}}{\lambda}_{z,j}f(z),\qquad
f\in\Pi_{2^{j}}(\Omega);$

moreover, ${\lambda}_{z,j}\sim{\mathsf{w}}(B(z,2^{-j}))$ since we assume that
Assertion 4 holds for ${\mathsf{w}}$. We use this cubature to discretize
$F_{j}*F_{k}$, which leads to a tight frame.

For $z\in\Xi_{j}$ and $j=1,2,\ldots$, we define our frame elements by

$\psi_{z,j}(x):=\sqrt{{\lambda}_{z,j}}F_{j}(x,z).$

Then $\Psi:=\big{\\{}\psi_{z,j}:z\in\Xi_{j},\quad 1\leq j\leq\infty\big{\\}}$
is a frame system [28, Theorem 2.20]:

######  Theorem 2.9.

Assume that $\Omega$ admits a localizable homogenous space. Let ${\mathsf{w}}$
be a doubling weight satisfying Assertion 4. If $f\in
L^{2}(\Omega,{\mathsf{w}})$, then

(2.16) $f=\sum_{j=0}^{\infty}\sum_{z\in\Xi_{j}}\langle
f,\psi_{z,j}\rangle_{\mathsf{w}}\psi_{z,j}\qquad\mbox{in
$L^{2}(\Omega,{\mathsf{w}})$}$

and

(2.17)
$\|f\|_{2,{\mathsf{w}}}=\Big{(}\sum_{j=0}^{\infty}\sum_{z\in\Xi_{j}}|\langle
f,\psi_{z,j}\rangle_{\mathsf{w}}|^{2}\Big{)}^{1/2}.$

The frame element $\psi_{z,j}$ has a near exponential rate of decay away from
its center with respect to the distance ${\mathsf{d}}(\cdot,\cdot)$ on
$\Omega$.

######  Theorem 2.10.

Let $(\Omega,\varpi,{\mathsf{d}})$ be a localizable space of homogeneous type.
There, there is a constant $c_{\kappa}>0$ depending only on ${\kappa}$, $d$,
$\varpi$ and $\widehat{a}$ such that for $z\in\Xi_{j}$, $j=0,1,\dots$,

(2.18) $|\psi_{z,j}(x)|\leq
c_{\kappa}\frac{1}{\sqrt{\varpi(B(z,2^{-j}))}(1+2^{j}{\mathsf{d}}(x,z))^{\kappa}},\quad
x\in\Omega.$

Besides the regular domains mentioned before, these localized tight frames
hold on conic domains as shown in [28].

##  3\. Examples of localizable space of homogeneous type

We state several examples of $(\Omega,{\mathsf{w}},\mathrm{d})$ that are
localizable spaces of homogeneous type. In each case, we also include the
addition formula utilized to prove Assertions 1 and 2.

###  3.1. The interval $[-1,1]$ with the Jacobi weight

For ${\alpha},{\beta}>-1$, the Jacobi weight function is defined by

$w_{{\alpha},{\beta}}(t):=(1-t)^{\alpha}(1+t)^{\beta},\qquad-1<x<1.$

The intrinsic distance of the interval $[-1,1]$ is defined by

${\mathsf{d}}_{[-1,1]}(t,s)=\arccos\left(ts+\sqrt{1-t^{2}}\sqrt{1-s^{2}}\right),$

which is the projection of the distance $|{\theta}-\phi|$ on the upper half of
the unit circle if we set $t=\cos{\theta}$ and $s=\cos\phi$. The space
$([-1,1],w_{{\alpha},{\beta}},{\mathsf{d}}_{[-1,1]})$ is a localizable
homogeneous space if ${\alpha},{\beta}\geq-\frac{1}{2}$. Let
$P_{n}^{({\alpha},{\beta})}$ be the usual Jacobi polynomials. Then the kernel
$L_{n}$, denoted by $L_{n}^{({\alpha},{\beta})}$, is of the form

(3.1)
$L_{n}^{({\alpha},{\beta})}(x,y)=\sum_{j=0}^{\infty}\widehat{a}\Big{(}\frac{j}{n}\Big{)}\frac{P_{j}^{({\alpha},{\beta})}(x)P_{j}^{({\alpha},{\beta})}(y)}{h_{j}^{({\alpha},{\beta})}},$

where $\widehat{a}$ is an admissible cutoff function and
$h_{j}^{({\alpha},{\beta})}$ is the norm square of $P_{j}^{(\\\ a,{\beta})}$
in the $L^{2}([-1,1],w_{{\alpha},{\beta}})$. Let the ball $B(x,r)$ be defined
with respect to this distance. Then the Jacobi weight is a doubling weight and
satisfies,

$w_{{\alpha},{\beta}}(B(x,n^{-1}))\sim
n^{-1}(1-t+n^{-2})^{{\alpha}+\frac{1}{2}}(1+t+n^{-2})^{{\beta}+\frac{1}{2}}=:w_{{\alpha},{\beta}}(n;t).$

The first Assertion of the highly localized kernel is established in [20],
which states: for any ${\kappa}>0$ and $n\geq 1$,

$|L_{n}^{({\alpha},{\beta})}(t,s)|\leq
c_{\kappa}\frac{1}{\sqrt{w_{{\alpha},{\beta}}(n;t)}\sqrt{w_{{\alpha},{\beta}}(n;s)}}\left(1+n{\mathsf{d}}_{[-1,1]}(t,s)\right)^{-{\kappa}},$

where $\alpha,\beta\geq-1/2$ and $0<{\varepsilon}\leq 1$. The second assertion
is stated in [13, Theorem 2.2].

In this one-dimensional case, there is no addition formula but a product
formula that writes the product of two Jacobi polynomials of the same
parameters as an integral of one Jacobi polynomial of the same parameters,
which allows us to write $L_{n}^{({\alpha},{\beta})}(x,y)$ as an integral of
$L_{n}^{({\alpha},{\beta})}(z,1)$.

###  3.2. Unit sphere

For the unit sphere ${\mathbb{S}^{d-1}}$ of ${\mathbb{R}}^{d}$, we consider
the space $({\mathbb{S}^{d-1}},\mathrm{d}\sigma,{\mathsf{d}}_{\SS})$, where
$\mathrm{d}\sigma$ is the surface measure and ${\mathsf{d}}_{\SS}$ is the
geodesic distance defined by

${\mathsf{d}}_{\SS}(\xi,\eta)=\arccos{\langle}\xi,\eta{\rangle},\qquad\xi,\eta\in{\mathbb{S}^{d-1}}.$

The orthogonal polynomials are the classical spherical harmonics, which are
homogeneous harmonic polynomials restricted to the unit sphere. The space
${\mathcal{V}}_{n}^{d}({\mathbb{S}^{d-1}},\mathrm{d}\sigma)$ is usually denote
by ${\mathcal{H}}_{n}^{d}$ and its dimension is given by (2.2). Let
$\\{Y_{\nu,n}:1\leq\nu\leq\dim{\mathcal{H}}_{n}^{d}\\}$ be an orthonormal
basis of ${\mathcal{H}}_{n}^{d}$. The classical addition formula for the
spherical harmonics gives a closed-form formula for the reproducing kernel of
${\mathcal{H}}_{n}^{d}$,

${\mathsf{P}}_{n}(\xi,\eta):=\sum_{1\leq\nu\leq\dim{\mathcal{H}}_{n}^{d}}Y_{\nu,n}(\xi)Y_{\nu,n}(\eta)=Z_{n}^{\frac{d-2}{2}}({\langle}\xi,\eta{\rangle}),\quad\forall\xi,\eta\in{\mathbb{S}^{d-1}},$

where we use $Z_{n}^{\lambda}$ to denote a multiple of the Gagenbauer
polynomials $C_{n}^{\lambda}$,

$Z_{n}^{\lambda}(t):=\frac{n+{\lambda}}{{\lambda}}C_{n}^{\lambda}(t),\qquad{\lambda}\geq
0,$

where the case ${\lambda}=0$ holds under the limit so that
$Z_{n}^{0}(t)=T_{n}(t)$, the Chebyshev polynomial of the first kind. The
addition formula reduces the estimate of the highly localized kernel to the
kernel of the Jacobi polynomials $P_{n}^{(\mu,\mu)}(1,z)$ with
$\mu=\frac{d-3}{2}$. In this case, the Assertion 1 is of a particularly simple
form

$\left|L_{n}(\xi,\eta)\right|\leq
c_{\kappa}\frac{n^{d-1}}{(1+n{\mathsf{d}}_{\SS}(\xi,\eta))^{{\kappa}}},$

which is also the first highly localized kernel known in the literature
([18]).

The space ${\mathcal{H}}_{n}^{d}$ has another important property worth
mentioning: it is the eigenspace of the Laplace-Beltrami operator
$\Delta_{0}$, which is the restriction of the Laplace operator $\Delta$ on the
unit sphere; see, for example, [4, Section 1.4] for its explicit expression.
More precisely, it is known [4, (1.4.9)] that

(3.2) $\Delta_{0}Y=-n(n+d-2)Y,\qquad Y\in{\mathcal{H}}_{n}^{d}.$

We shall call $\Delta_{0}$ the spectral operator. The two properties, the
addition formula and the spectral operator, are fundamental for approximation
and harmonic analysis on the unit sphere.

###  3.3. Unit ball

For the unit ball ${\mathbb{B}}^{d}$ of ${\mathbb{R}}^{d}$, we consider the
space $({\mathbb{B}}^{d},{\mathsf{w}}_{\mu},{\mathsf{d}}_{\mathbb{B}})$, where
${\mathsf{w}}_{\mu}$ is the weight function defined by

${\mathsf{w}}_{\mu}(x)=(1-\|x\|^{2})^{\mu-\frac{1}{2}},\qquad\mu>-\frac{1}{2},\quad
x\in{\mathbb{B}}^{d},$

and ${\mathsf{d}}_{\mathbb{B}}$ is the intrinsic distance on
${\mathbb{B}}^{d}$ defined by

${\mathsf{d}}_{\mathbb{B}}(x,y)=\arccos\Big{(}{\langle}x,y{\rangle}+u\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\Big{)},\quad
x,y\in{\mathbb{B}}^{d}.$

Several orthogonal basis of the space
${\mathcal{V}}_{n}^{d}({\mathbb{B}}^{d},{\mathsf{w}}_{\mu})$ can be given
explicitly. The reproducing kernel $P_{n}({\mathsf{w}}_{\mu};\cdot,\cdot)$
satisfies an addition formula [25]: for $\mu\geq 0$ and
$x,y\in{\mathbb{B}}^{d}$,

$\displaystyle
P_{n}({\mathsf{w}}_{\mu};x,y)=c_{\mu-\frac{1}{2}}\int_{-1}^{1}Z_{n}^{\mu+\frac{d-1}{2}}$
$\displaystyle\Big{(}{\langle}x,y{\rangle}+u\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\Big{)}(1-u^{2})^{\mu-1}\mathrm{d}u,$

where $\mu>0$ and it holds for $\mu=0$ under the limit

(3.3) $\lim_{\mu\to
0+}c_{\mu-\frac{1}{2}}\int_{-1}^{1}f(t)(1-t^{2})^{\mu-1}\mathrm{d}u=\frac{f(1)+f(-1)}{2}.$

It is known that ${\mathsf{w}}_{\mu}$ is a doubling weight on the unit ball
and

${\mathsf{w}}_{\mu}\big{(}B(x,n^{-1})\big{)}\sim
n^{-d}\left(\sqrt{1-\|x\|^{2}}+n^{-1}\right)^{2\mu}=:{\mathsf{w}}_{\mu}(n;x).$

In this case, the Assertion 1 takes the form

$\left|L_{n}({\mathsf{w}}_{\mu};x,y)\right|\leq
c_{k}\frac{1}{\sqrt{{\mathsf{w}}_{\mu}(n;x)}\sqrt{{\mathsf{w}}_{\mu}(n,y)}(1+n{\mathsf{d}}_{\mathbb{B}}(x,u))^{\kappa}}.$

This inequality and Assertion 2 are established in [21]. Consequently, the
space $({\mathbb{B}}^{d},{\mathsf{w}},{\mathsf{d}})$ is a localizable space of
homogeneous type.

###  3.4. Simplex

For the simplex $\triangle^{d}$ of ${\mathbb{R}}^{d}$ defined by

$\triangle^{d}:=\left\\{x:x_{1},\geq 0,\ldots,x_{d}\geq 0,\,|x|\leq
1\right\\},\qquad|x|:=x_{1}+\ldots+x_{d},$

we consider the space
$(\triangle,{\mathsf{w}}_{\gamma},{\mathsf{d}}_{\triangle})$, where
${\mathsf{w}}_{\gamma}$ is the classical Jacobi weight function

${\mathsf{w}}_{\gamma}(x)=x_{1}^{{\gamma}_{1}}\cdots
x_{d}^{{\gamma}_{d}}(1-|x|)^{{\gamma}_{d+1}},\qquad{\gamma}_{i}>-1,\quad
x\in\triangle^{d},$

and ${\mathsf{d}}_{\triangle}$ is the intrinsic distance on $\triangle^{d}$
defined by

${\mathsf{d}}_{\triangle}(x,y)=\arccos\Big{(}\sqrt{x_{1}y_{1}}+\cdots+\sqrt{x_{d}y_{d}}+\sqrt{1-|x|}\sqrt{1-|y|}\Big{)}.$

Like the case of the unit ball, several orthogonal bases of the space
${\mathcal{V}}_{n}^{d}(\triangle^{d},{\mathsf{w}}_{\gamma})$ can be given
explicitly. The reproducing kernel
${\mathsf{P}}_{n}({\mathsf{w}}_{\gamma};\cdot,\cdot)$ also satisfies an
addition formula [5, Theorem 5.2.4]: for ${\alpha}_{i}\geq-\frac{1}{2}$ and
$x,y\in\triangle^{d}$,

(3.4)
$P_{n}({\mathsf{w}}_{\gamma};x,y)=c_{\alpha}\int_{-1}^{1}Z_{n}^{(|{\gamma}|+d-\frac{1}{2},-\frac{1}{2})}\left(2\xi(x,y;t)^{2}-1\right)\prod_{i=1}^{d+1}(1-t_{i}^{2})^{{\gamma}_{i}-\frac{1}{2}}\mathrm{d}t,$

where $Z_{n}^{({\alpha},{\beta})}$ is a constant multiple of the Jacobi
polynomial

$Z_{n}^{({\alpha},{\beta})}(t)=\frac{P_{n}^{({\alpha},{\beta})}(1)P_{n}^{({\alpha},{\beta})}(t)}{h_{n}^{({\alpha},{\beta})}}$

and $\xi(x,y,t)$ is a function defined by

$\xi(x,y;t)=\sqrt{x_{1}y_{1}}t_{1}+\cdots+\sqrt{x_{d}y_{d}}t_{d}+\sqrt{x_{d+1}y_{d+1}}t_{d+1}.$

In this case, ${\mathsf{w}}_{\gamma}$ is known to be a doubling weight and it
satisfies, for ${\gamma}_{i}\geq 0$, $1\leq i\leq d+1$,

${\mathsf{w}}_{\gamma}(B(x,n^{-1}))\sim
n^{-d}\prod_{i=1}^{d+1}(\sqrt{x_{i}}+n^{-1})^{2{\gamma}_{i}+1},\quad
x\in\triangle^{d},$

which follows, for example, from [4, Lemma 11.3.6] by making a change of
variables $x_{j}\mapsto\sqrt{x_{j}}$. In this case, Assertion 1 was
established in [9] and Assertion 2 appeared more recently in [6] for
${\mathsf{w}}_{\gamma}$. Thus, the space
$(\triangle^{d},{\mathsf{w}},{\mathsf{d}}_{\triangle})$ is a localizable space
of homogeneous type.

###  3.5. Conic surface

For the conic surface ${\mathbb{V}}_{0}^{d+1}$ of ${\mathbb{R}}^{d+1}$ defined
by

${\mathbb{V}}_{0}^{d+1}:=\left\\{(x,t):\|x\|=t,\,x\in{\mathbb{R}}^{d}\,0\leq
t\leq 1\right\\}$

we consider the space
$({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{-1,{\gamma}},{\mathsf{d}}_{{\mathbb{V}}_{0}})$,
where the weight function ${\mathsf{w}}_{-1,{\gamma}}$ is the Jacobi weight
function with one parameter equal to $-1$,

${\mathsf{w}}_{-1,{\gamma}}(t)=t^{-1}(1-t)^{{\gamma}},\qquad{\gamma}>-1,$

and ${\mathsf{d}}_{{\mathbb{V}}_{0}}$ in the intrinsic distance function on
the conic surface [28, Definition 4.1],

${\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s)):=\arccos\left(\sqrt{\frac{{\langle}x,y{\rangle}+ts}{2}}+\sqrt{1-t}\sqrt{1-s}\right).$

Orthogonal polynomials in this setting were first studied in [26], where it is
shown, in particular, that the reproducing kernel
$P_{n}({\mathsf{w}}_{-1,{\gamma}};\cdot,\cdot)$ for the space
${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},{\mathsf{w}}_{-1,{\gamma}})$ satisfies
the addition formula

(3.5) $\displaystyle
P_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))=c_{{\gamma}}\int_{[-1,1]^{2}}$
$\displaystyle
Z_{n}^{({\gamma}+d-\frac{3}{2},-\frac{1}{2})}\big{(}2\zeta(x,t,y,s;v)^{2}-1\big{)}$
$\displaystyle\times(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v$

for ${\gamma}>-\frac{1}{2}$ and under the limit when ${\gamma}=-\frac{1}{2}$.
The weight ${\mathsf{w}}_{-1,{\gamma}}$ is an integrable doubling weight with
respect to ${\mathsf{d}}_{{\mathbb{V}}_{0}}$ and it satisfies [28, (4.4)]

$\displaystyle{\mathsf{w}}_{{\beta},{\gamma}}\big{(}{\mathsf{c}}((x,t),n^{-1})\big{)}\sim
r^{d}(t+n^{-2})^{\frac{d-2}{2}}(1-t+n^{-2})^{{\gamma}+\frac{1}{2}}.$

Both Assertions 1 and 2 are established in [28] with the help of the addition
formula (3.5) for ${\mathsf{w}}_{-1,{\gamma}}$, where the work of carrying out
the estimates is fairly involved. Consequently, the space
$({\mathbb{V}}_{0}^{d+1},{\mathsf{w}},{\mathsf{d}}_{{\mathbb{V}}_{0}})$ is a
localizable space of homogeneous type.

###  3.6. Solid cone

For the solid cone ${\mathbb{V}}^{d+1}$ of ${\mathbb{R}}^{d+1}$ bounded by
${\mathbb{V}}_{0}^{d+1}$ and the hyperplane $t=1$,

${\mathbb{V}}^{d+1}:=\left\\{(x,t):\|x\|\leq t,\,x\in{\mathbb{R}}^{d}\,0\leq
t\leq 1\right\\}$

we consider the weight function ${\mathsf{w}}_{{\gamma},\mu}$ defined by

${\mathsf{w}}_{{\gamma},\mu}(x,t)=(t^{2}-\|x\|^{2})^{\mu-\frac{1}{2}}(1-t)^{{\gamma}},\qquad{\beta}>-\frac{1}{2},\quad{\gamma}>-1,$

and ${\mathsf{d}}_{{\mathbb{V}}}$ is the intrinsic distance function on the
conic surface [28, Definition 4.1],

${\mathsf{d}}_{{\mathbb{V}}}((x,t),(y,s)):=\arccos\left(\sqrt{\frac{{\langle}x,y{\rangle}+\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}+ts}{2}}+\sqrt{1-t}\sqrt{1-s}\right).$

Orthogonal polynomials in this setting were first studied in [26], where it is
shown, in particular, that the reproducing kernel
$P_{n}({\mathsf{w}}_{{\gamma},\mu};\cdot,\cdot)$ for the space
${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},{\mathsf{w}}_{{\gamma},\mu})$ satisfies
the addition formula

(3.6) $\displaystyle
P_{n}\left({\mathsf{w}}_{{\gamma},\mu};(x,t),(y,s)\right)=c$
$\displaystyle\int_{[-1,1]^{3}}Z_{n}^{2\mu+{\gamma}+d}\big{(}2\zeta(x,t,y,s;u,v)\big{)}$
$\displaystyle\times(1-u^{2})^{\mu-1}(1-v_{1}^{2})^{\mu+\frac{d-3}{2}}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}u\mathrm{d}v.$

where $\zeta(x,t,y,s;u,v)\in[-1,1]$ is defined by

$\zeta(x,t,y,s;t)=v_{1}\sqrt{\frac{1}{2}\left(ts+{\langle}x,y{\rangle}+\sqrt{t^{2}-\|x\|^{2}}\sqrt{s^{2}-\|y\|^{2}}u\right)}+v_{1}\sqrt{1-t}\sqrt{1-s}.$

The ${\mathsf{w}}_{{\gamma},\mu}$ is an integrable doubling weight with
respect to ${\mathsf{d}}_{{\mathbb{V}}}$ and it satisfies [28, (4.4)]

$\displaystyle{\mathsf{w}}_{{\gamma},\mu}\big{(}B((x,t),n^{-1})\big{)}\sim
n^{-d+1}(t+n^{-2})^{\frac{d-1}{2}}(1-t+n^{-2})^{{\gamma}+\frac{1}{2}}(t^{2}-\|x\|^{2}+n^{-2})^{\mu}.$

Both Assertions 1 and 2 are established in [28], so that the space
$({\mathbb{V}}^{d+1},{\mathsf{w}},{\mathsf{d}}_{{\mathbb{V}}})$ is a
localizable space of homogeneous type.

###  3.7. Double conic and hyperbolic surface

For $\varrho\geq 0$, we consider the quadratic surface defined by

${\mathbb{X}}_{0}^{d+1}=\left\\{(x,t):\|x\|^{2}=t^{2}-\varrho^{2},\,x\in{\mathbb{R}}^{d},\,\varrho\leq|t|\leq\sqrt{\varrho^{2}+1}\right\\},$

which is a hyperbolic surface if $\varrho>0$ and a double conic surface if
$\varrho=0$. We consider the space
$({\mathbb{X}}_{0},{\mathsf{w}}_{{\beta},{\gamma}},{\mathsf{d}}_{{\mathbb{X}}_{0}})$,
where ${\mathsf{w}}_{{\beta},{\gamma}}$ is the weight function defined by, for
$d\geq 2$, ${\beta}>-\frac{1}{2}$ and ${\gamma}>-\frac{1}{2}$,

${\mathsf{w}}_{{\beta},{\gamma}}(t)=|t|(t^{2}-\varrho^{2})^{{\beta}-\frac{1}{2}}(\varrho^{2}+1-t^{2})^{{\gamma}-\frac{1}{2}},\qquad\varrho\leq|t|\leq\sqrt{\varrho^{2}+1},$

and ${\mathsf{d}}_{{\mathbb{X}}_{0}}$ is the distance function on
${\mathbb{X}}_{0}^{d+1}$ given by

$\displaystyle{\mathsf{d}}_{{\mathbb{X}}_{0}}((x,t),(y,s))=\arccos\left({\langle}x,y{\rangle}+\sqrt{1+\varrho^{2}-t^{2}}\sqrt{1+\varrho^{2}-s^{2}}\right).$

When $\varrho=0$, or on the double conic surface, the weight function becomes

${\mathsf{w}}_{{\beta},{\gamma}}(t)=|t|^{2{\beta}}(1-t^{2})^{{\gamma}-\frac{1}{2}}.$

The weight function ${\mathsf{w}}_{{\beta},{\gamma}}$ on the hyperbolic
surface is a doubling weight and satisfies

${\mathsf{w}}_{{\beta},{\gamma}}\big{(}B((x,t),n^{-1})\big{)}=n^{-d}\big{(}t^{2}-\varrho^{2}+n^{-2}\big{)}^{{\beta}}\big{(}1-t^{2}+\varrho^{2}+n^{-2}\big{)}^{{\gamma}}.$

The orthogonal polynomials in this setting are first studied in [27]. The
space of orthogonal polynomials
${\mathcal{V}}_{n}({\mathbb{X}}_{0}^{d+1},{\mathsf{w}})$ of degree $n$ has the
same dimension as the space of spherical harmonics ${\mathcal{H}}_{n}^{d+1}$.
If the weight function ${\mathsf{w}}$ is even in the $t$ variable, this space
can be factored as

${\mathcal{V}}_{n}\left({\mathbb{X}}_{0}^{d+1},{\mathsf{w}}\right)={\mathcal{V}}_{n}^{\mathsf{E}}\big{(}{\mathbb{X}}_{0}^{d+1},{\mathsf{w}}\big{)}\bigoplus{\mathcal{V}}_{n}^{\mathsf{O}}\big{(}{\mathbb{X}}_{0}^{d+1},{\mathsf{w}}\big{)},$

where the subspace
${\mathcal{V}}_{n}^{\mathsf{E}}({\mathbb{X}}_{0}^{d+1},{\mathsf{w}})$ consists
of orthogonal polynomials that are even in the $t$ variable, and the subspace
${\mathcal{V}}_{n}^{\mathsf{O}}({\mathbb{X}}_{0}^{d+1},{\mathsf{w}})$ consists
of orthogonal polynomials that are odd in the $t$ variable. Then

$\dim{\mathcal{V}}_{n}^{\mathsf{E}}\big{(}{\mathbb{X}}_{0}^{d+1},{\mathsf{w}}\big{)}=\binom{n+d-1}{n}\quad\hbox{and}\quad\dim{\mathcal{V}}_{n}^{\mathsf{O}}\big{(}{\mathbb{X}}_{0}^{d+1},{\mathsf{w}}\big{)}=\binom{n+d-2}{n-1}.$

It turns out that a closed-form formula holds for
$P_{n}^{\mathsf{E}}({\mathsf{w}}_{{\beta},{\gamma}})$ and it has the simplest
form when ${\beta}=0$, in which case the addition formula is given by [27,
(5.11)]

(3.7) $\displaystyle
P_{n}^{\mathsf{E}}\big{(}{\mathsf{w}}_{0,{\gamma}};(x,t),(y,s)\big{)}$
$\displaystyle=c_{{\gamma}-\frac{1}{2}}\int_{-1}^{1}(1-v^{2})^{{\gamma}-1}$
$\displaystyle\times
Z_{n}^{{\gamma}+\frac{d-1}{2}}\bigg{(}{\langle}x,y{\rangle}+v\sqrt{1+\varrho^{2}-s^{2}}\sqrt{1+\varrho^{2}-t^{2}}\bigg{)}\mathrm{d}v.$

The kernel $P_{n}^{\mathsf{O}}({\mathsf{w}}_{{\beta},{\gamma}})$ is related to
$P_{n}^{\mathsf{E}}({\mathsf{w}}_{\beta},{\gamma})$ by a simple relation [27,
(5.3)]

(3.8) $\displaystyle
P_{n}^{\mathsf{O}}\big{(}{\mathsf{w}}_{{\beta},{\gamma}};(x,t),(y,s)\big{)}=\frac{{\beta}+{\gamma}+\frac{d+1}{2}}{{\beta}+\frac{d+1}{2}}st\,P_{n-1}^{\mathsf{E}}\big{(}{\mathsf{w}}_{{\beta}+1,{\gamma}};(x,t),(y,s)\big{)}$

for ${\beta},{\gamma}>-\frac{1}{2}$ and ${\beta}\geq-\frac{1}{2}$ under the
limit when $\varrho=0$, which shows that
$P_{n}^{\mathsf{O}}\big{(}{\mathsf{w}}_{-1,{\gamma}};\cdot,\cdot\big{)}$ has a
simple closed formula in the form of (3.7) when $\varrho=0$ but with a
different weigh function ${\mathsf{w}}_{-1,{\gamma}}$ in place of
${\mathsf{w}}_{0,{\gamma}}$.

In order to use the addition formula, we need to consider the subspace of
homogeneous type $({\mathbb{X}}_{0}^{d+1},{\mathsf{w}},{\mathsf{d}})$ defined
according to the parity. Thus, instead of the kernel $L_{n}({\mathsf{w}})$, we
consider

$L_{n}^{\mathsf{E}}\big{(}{\mathsf{w}};(x,t),(y,s)\big{)}=\sum_{j=0}^{\infty}\widehat{a}\left(\frac{j}{n}\right)P_{j}^{\mathsf{E}}\big{(}{\mathsf{w}};(x,t),(y,s)\big{)}$

and the kernel $L_{n}^{\mathsf{O}}({\mathsf{w}})$ defined similarly, which
uses only polynomials that are even (odd respectively) in the $t$ variable. In
this case, we can decompose the surface ${\mathbb{X}}_{0}^{d+1}$ as an upper
part and a lower part,

${\mathbb{X}}_{0}^{d+1}={\mathbb{X}}_{0,+}^{d+1}\cup{\mathbb{X}}_{0,-}^{d+1}=\\{(x,t)\in{\mathbb{X}}_{0}^{d+1}:t\geq
0\\}\cup\\{(x,t)\in{\mathbb{X}}_{0}^{d+1}:t\leq 0\\}.$

If a function $f(x,t)$ is even in the $t$ variable, then we only need to
consider its restriction on ${\mathbb{X}}_{0,+}^{d+1}$. In particular, if we
consider, for example, approximation and tight polynomial frames for functions
that are even in the $t$ variable, then the framework established in [28]
applies.

The above setup can also be regarded as studying a function $f$ defined on
${\mathbb{X}}_{0,+}^{d+1}={\mathbb{V}}_{0}^{d+1}$ and extends its definition
to ${\mathbb{X}}_{0}^{d+1}$ by defining $f(x,t)=f(x,-t)$. It should be noted,
however, that the polynomial space $\Pi_{n}({\mathbb{V}}_{0}^{d+1})$ of
polynomials on ${\mathbb{V}}_{0}^{d+1}$ and the
$\Pi_{n}({\mathbb{X}}_{0,+})^{d+1}$ are not the same and, in fact, have
different dimensions.

Using $L_{n}^{\mathsf{E}}({\mathsf{w}})$, both Assertions 1 and 2 are
established in [30]. Thus, if ${\mathsf{w}}$ is a doubling weight even in the
$t$ variable, then
$({\mathbb{X}}_{0,+}^{d+1},{\mathsf{w}},{\mathsf{d}}_{{\mathbb{X}}_{0}})$ is a
localizable space of homogeneous type.

###  3.8. Double cone and hyperboloid

For $\varrho\geq 0$, we consider the domain defined by

${\mathbb{X}}^{d+1}=\left\\{(x,t):\|x\|^{2}\leq
t^{2}-\varrho^{2},\,x\in{\mathbb{R}}^{d},\,\varrho\leq|t|\leq\sqrt{\varrho^{2}+1}\right\\},$

which is the hyperboloid, bounded by the hyperbolic surface
${\mathbb{X}}^{d+1}$ and two hyperplanes $t=\pm 1$, if $\varrho>0$ and a solid
double cone if $\varrho=0$. We consider the space
$({\mathbb{X}}^{d+1},{\mathsf{w}}_{{\beta},{\gamma},\mu},{\mathsf{d}}_{{\mathbb{X}}})$,
where ${\mathsf{w}}_{{\beta},{\gamma},\mu}$ is the weight function defined by

${\mathsf{w}}_{{\beta},{\gamma},\mu}(x,t)=|t|(t^{2}-\varrho^{2})^{{\beta}-\frac{1}{2}}(1+\varrho^{2}-t^{2})^{{\gamma}-\frac{1}{2}}(t^{2}-\varrho^{2}-\|x\|^{2})^{\mu-\frac{1}{2}}$

and ${\mathsf{d}}_{{\mathbb{X}}}$ is the distance function on
${\mathbb{X}}^{d+1}$ given by

$\displaystyle{\mathsf{d}}_{{\mathbb{X}}}((x,t),(y,s))=\arccos\left({\langle}x,y{\rangle}+\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}+\sqrt{1+\varrho^{2}-t^{2}}\sqrt{1+\varrho^{2}-s^{2}}\right).$

The weight function ${\mathsf{w}}_{{\beta},{\gamma},\mu}$ on the hyperboloid
is a doubling weight and satisfies

${\mathsf{w}}_{{\beta},{\gamma},\mu}\big{(}B((x,t),n^{-1})\big{)}=n^{-d-1}\big{(}t^{2}-\varrho^{2}+n^{-2})^{\beta}(1+\varrho^{2}-t^{2}+n^{-2}\big{)}^{{\gamma}}\big{(}t^{2}-\varrho^{2}-\|x\|^{2}+n^{-2}\big{)}^{\mu}.$

As in the case of hyperbolic surface, if the weight function ${\mathsf{w}}$ is
even in the $t$ variable, then the space of orthogonal polynomials
${\mathcal{V}}_{n}({\mathbb{X}}^{d+1},{\mathsf{w}})$ can be factored as

${\mathcal{V}}_{n}\left({\mathbb{X}}^{d+1},{\mathsf{w}}\right)={\mathcal{V}}_{n}^{\mathsf{E}}\big{(}{\mathbb{X}}^{d+1},{\mathsf{w}}\big{)}\bigoplus{\mathcal{V}}_{n}^{\mathsf{O}}\big{(}{\mathbb{X}}^{d+1},{\mathsf{w}}\big{)},$

where the subspace
${\mathcal{V}}_{n}^{\mathsf{E}}({\mathbb{X}}^{d+1},{\mathsf{w}})$ consists of
orthogonal polynomials that are even in the $t$ variable, and the subspace
${\mathcal{V}}_{n}^{\mathsf{O}}({\mathbb{X}}^{d+1},{\mathsf{w}})$ consists of
orthogonal polynomials that are odd in the $t$ variable. A closed-form formula
holds for $P_{n}^{\mathsf{E}}({\mathsf{w}}_{{\beta},{\gamma},\mu})$, which
takes the simplest form if ${\beta}=\frac{1}{2}$ [27, Corollary 5.6],

$\displaystyle
P_{n}^{\mathsf{E}}\big{(}{\mathsf{w}}_{\frac{1}{2},{\gamma},\mu};(x,t),(y,s)\big{)}$
$\displaystyle=c\int_{-1}^{1}\int_{-1}^{1}Z_{n}^{{\gamma}+\mu+\frac{d}{2}}\big{(}\zeta(x,t,y,s;u,v)\big{)}(1-v^{2})^{{\gamma}-1}(1-u^{2})^{\mu-1}\mathrm{d}u\mathrm{d}v,$

where

$\displaystyle\zeta(x,t,y,s;u,v)=\Big{(}{\langle}x,y{\rangle}\,$
$\displaystyle+u\sqrt{t^{2}-\varrho^{2}-\|x\|^{2}}\sqrt{s^{2}-\varrho^{2}-\|y\|^{2}}\Big{)}\mathrm{sign}(st)$
$\displaystyle+v\sqrt{1+\varrho^{2}-s^{2}}\sqrt{1+\varrho^{2}-t^{2}}.$

Moreover, the kernel $P_{n}^{\mathsf{O}}({\mathsf{w}}_{{\beta},{\gamma},\mu})$
is related to $P_{n}^{\mathsf{E}}({\mathsf{w}}_{{\beta}+1,{\gamma},\mu})$ by a
formula that is similar to (3.8). Notice again the shift in the subscript
${\beta}$ by $1$, which shows that the addition formula does not apply to
$P_{n}^{\mathsf{O}}\big{(}{\mathsf{w}}_{\frac{1}{2},{\gamma},\mu})$.

Defining the kernel $L_{n}^{\mathsf{E}}({\mathsf{w}})$ that uses only
$P_{j}^{\mathsf{E}}({\mathsf{w}})$ as in the case of hyperbolic surface, which
uses only polynomials that are even in the $t$ variable. Then both Assertions
1 and 2 are established in [30]. If we decompose the the domain
${\mathbb{X}}^{d+1}$ as an upper part and a lower part,

${\mathbb{X}}^{d+1}={\mathbb{X}}_{+}^{d+1}\cup{\mathbb{X}}_{-}^{d+1}=\\{(x,t)\in{\mathbb{X}}^{d+1}:t\geq
0\\}\cup\\{(x,t)\in{\mathbb{X}}^{d+1}:t\leq 0\\},$

then $({\mathbb{X}}_{+}^{d+1},{\mathsf{w}},{\mathsf{d}}_{{\mathbb{X}}})$ is a
localizable space of homogeneous type if ${\mathsf{w}}$ is even in the $t$
variable.

###  3.9. Other double domains of conic and hyperbolic type

Orthogonal structure on several solid domains, beyond the domains bounded by
quadratic surfaces, is preserved under an appropriate structure transformation
on the hyperboloid ${\mathbb{X}}^{d+1}$, as shown in [31]. This includes the
addition formula and highly localized kernels so that we obtain localized
space of homogeneous type on such domains. We state one example to give a
flavor.

For $0<\mathfrak{b}<1$, we consider the domain inside the unit ball
${\mathbb{B}}^{d+1}$ but outside the ellipsoid
$\\{(x,t):\mathfrak{b}(1-\|x\|^{2})\leq t^{2}\\}\subset{\mathbb{B}}^{d+1}$,
which is defined by

$\mathbb{X}^{d+1}=\left\\{(x,t):\sqrt{1-\tfrac{t^{2}}{\mathfrak{b}}}\leq\|x\|\leq\sqrt{1-t^{2}},\,\,|t|\leq
1\right\\}.$

We consider the space
$({\mathbb{X}}^{d+1},{\mathsf{w}}_{{\beta},{\gamma},\mu},{\mathsf{d}}_{{\mathbb{X}}})$,
where the weigh function is defined by

$\displaystyle{\mathsf{w}}_{{\beta},{\gamma}}(x,t)=\big{(}1-\|x\|^{2}-t^{2}\big{)}^{\beta}\big{(}t^{2}-\mathfrak{b}(1-\|x\|^{2})\big{)}^{\gamma}\big{(}t^{2}-\mathfrak{b}+\|x\|^{2}\big{)}^{\frac{1}{2}}$

and the distance function ${\mathsf{d}}_{X}X$ is defined by

$\displaystyle{\mathsf{d}}_{{\mathbb{X}}}((x,t),(y,s))=\arccos$
$\displaystyle\left({\langle}x,y{\rangle}+\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\right.$
$\displaystyle\qquad\left.+\sqrt{\frac{1-t^{2}-\|x\|^{2}}{1-\mathfrak{b}}}\sqrt{\frac{1-s^{2}-\|y\|^{2}}{1-\mathfrak{b}}}\right).$

Then the corresponding kernel
$L_{n}^{\mathsf{E}}({\mathsf{w}}_{{\beta},{\gamma}})$ is highly localized with
respect to the distance ${\mathsf{d}}_{\mathbb{X}}$. Consequently,
$({\mathbb{X}}_{+}^{d+1},{\mathsf{w}},{\mathsf{d}}_{{\mathbb{X}}})$ is a
localizable space of homogeneous type if ${\mathsf{w}}$ is even in the $t$
variable.

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