# On a Geometric Approach to the Estimation
of Interpolation Projectors

Mikhail Nevskii111 Department of Mathematics, P.G. Demidov Yaroslavl State
University,
Sovetskaya str., 14, Yaroslavl, 150003, Russia,
<EMAIL_ADDRESS>orcid.org/0000-0002-6392-7618 , Alexey Ukhalov222
Department of Mathematics, P.G. Demidov Yaroslavl State University,
Sovetskaya str., 14, Yaroslavl, 150003, Russia,
<EMAIL_ADDRESS>orcid.org/0000-0001-6551-5118

(July 25, 2023)

###### Abstract

Suppose $\Omega$ is a closed bounded subset of ${\mathbb{R}}^{n},$ $S$ is an
$n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=\min\left\\{\sigma\geq
1:\,\Omega\subset\sigma S\right\\}$. Here $\sigma S$ is the result of
homothety of $S$ with respect to the center of gravity with coefficient
$\sigma$. Let $d\geq n+1,$ $\varphi_{1}(x),\ldots,\varphi_{d}(x)$ be linearly
independent monomials in $n$ variables, $\varphi_{1}(x)\equiv 1,$
$\varphi_{2}(x)=x_{1},\ \ldots,\ \varphi_{n+1}(x)=x_{n}.$ Put $\Pi:={\rm
lin}(\varphi_{1},\ldots,\varphi_{d}).$ The interpolation projector
$P:C(\Omega)\to\Pi$ with a set of nodes $x^{(1)},\ldots,x^{(d)}$ $\in\Omega$
is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$
Denote by $\|P\|_{\Omega}$ the norm of $P$ as an operator from $C(\Omega)$ to
$C(\Omega)$ . Consider the mapping $T:{\mathbb{R}}^{n}\to{\mathbb{R}}^{d-1}$
of the form $T(x):=(\varphi_{2}(x),\ldots,\varphi_{d}(x)).$ We have the
following inequalities:
$\frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1$
$\leq\xi(T(\Omega);S)\leq\frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1.$ Here $S$
is the $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ We
discuss this and other relations for polynomial interpolation of functions
continuous on a segment. The results of numerical analysis are presented.

Keywords: polynomial interpolation, projector, norm, absorption coefficient,
estimation

MSC: 41A05, 52B55, 52C07

##  1 Main Definitions and Relations

Let $\Omega$ be a closed bounded subset of ${\mathbb{R}}^{n}$,
$n\in{\mathbb{N}}$. By $C(\Omega)$ we mean the space of continuous functions
$f:\Omega\to{\mathbb{R}}$ with the uniform norm

$\|f\|_{C(\Omega)}:=\max_{x\in\Omega}|f(x)|.$

For a nondegenerate simplex $S\subset{\mathbb{R}}^{n}$, by $\sigma S$ denote
the homothetic copy of $S$ with center of homothety at the center of gravity
of $S$ and ratio $\sigma$. By definition, put
$\xi(\Omega;S):=\min\left\\{\sigma\geq 1:\,\Omega\subset\sigma S\right\\}$.
The inclusion $\Omega\subset S$ is equivalent to the equality
$\xi(\Omega;S)=1.$ We call $\xi(\Omega;S)$ the absorption coefficient of the
set $\Omega$ by the simplex $S$. Further ${\rm ver}(S)$ is a set of vertices
of $S$.

Assume $d\in{\mathbb{N}},$ $d\geq n+1;$ $\varphi_{1}(x),\ldots,\varphi_{d}(x)$
are linearly independent functions, namely monomials in $n$ variables having
the form $x^{\alpha}=x_{1}^{\alpha_{1}}\ldots x_{n}^{\alpha_{n}}.$ Here
$x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$,
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ $\in{\mathbb{Z}}_{+}^{n}$. We suppose
that $\varphi_{1}(x)\equiv 1,$ $\varphi_{2}(x)=x_{1},\ \ldots,\
\varphi_{n+1}(x)=x_{n}.$ By the $d$-dimensional space of polynomials in $n$
variables we mean the set $\Pi:={\rm lin}(\varphi_{1},\ldots,\varphi_{d}).$
Note the important cases: $\Pi=\Pi_{k}\left({\mathbb{R}}^{n}\right)$ — the
space of polynomials of general degree $\leq k$ $(k\in{\mathbb{N}})$ and
$\Pi=\Pi_{\alpha}\left({\mathbb{R}}^{n}\right)$ — the space of polynomials of
degree $\leq\alpha_{i}$ in $x_{i}$ $(\alpha\in{\mathbb{N}}^{n}).$

A collection of points $x^{(1)},\ldots,x^{(d)}\in\Omega$ is called an
admissible set of nodes  for interpolation of functions from $C(\Omega)$ with
the use of polynomials from $\Pi$ iff $\Delta:=\det({\bf A})\neq 0.$ Here $\bf
A$ is $(d\times d)$-matrix

${\bf
A}:=\left(\begin{array}[]{cccc}1&\varphi_{2}\left(x^{(1)}\right)&\ldots&\varphi_{d}\left(x^{(1)}\right)\\\
\vdots&\vdots&\vdots&\vdots\\\
1&\varphi_{2}\left(x^{(d)}\right)&\ldots&\varphi_{d}\left(x^{(d)}\right)\end{array}\right).$

The interpolation projector $P:C(\Omega)\to\Pi$ related to this set of nodes
is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right),\
j=1,\ldots,d.$ An analogue of the Lagrange interpolation formula is the
representation

$Pf(x)=\sum_{j=1}^{d}f\left(x^{(j)}\right)\lambda_{j}(x),\quad\lambda_{j}(x):=\frac{\Delta_{j}(x)}{\Delta},$
(1)

where $\Delta_{j}(x)$ is the determinant which appears from $\Delta$ by
replacing the $j$th row with the row
$\bigl{(}\varphi_{1}(x),\ldots,\varphi_{d}(x)\bigr{)}.$ Polynomials
$\lambda_{j}\in\Pi$ have the property $\lambda_{j}\left(x^{(k)}\right)$ $=$
$\delta_{j}^{k}.$ Their coefficients with respect to the basis $\varphi_{1},$
$\ldots,$ $\varphi_{d}$ form the columns of ${\bf A}^{-1}.$ We call
$\lambda_{j}$ the basic Lagrange polynomials corresponding to the projector
$P$.

In the case $\Pi=\Pi_{1}\left({\bf R}^{n}\right),$ $d=n+1$, and
$\varphi_{j}(x)=x_{j-1}$ $(j=2,\ldots,d)$ polynomials $\lambda_{j}$ are also
called the basic Lagrange polynomials of the simplex $S$ with vertices at the
interpolation nodes. If $\Omega\not\subset S$, then

$\xi(\Omega;S)=(n+1)\max_{1\leq k\leq
n+1}\max_{x\in\Omega}(-\lambda_{k}(x))+1.$ (2)

For convex $\Omega$, equality (2) is proved in [6]; the proof in general case
is just the same. Notice that $\xi({\rm conv}(\Omega);S)=\xi(\Omega;S).$

Further we consider only admissible sets of nodes and those $\Omega$ which
contain such a set.

Denote by $\|P\|_{\Omega}$ the norm of $P$ as an operator from $C(\Omega)$ in
$C(\Omega)$ . From (1), it follows that

$\|P\|_{\Omega}=\max\limits_{x\in\Omega}\sum_{j=1}^{d}|\lambda_{j}(x)|.$ (3)

The equalities $\lambda_{j}(x)=\dfrac{\Delta_{j}(x)}{\Delta}$ are equivalent
to the matrix relation

$\left(\begin{array}[]{cccc}1&1&\ldots&1\\\
\varphi_{2}\left(x^{(1)}\right)&\varphi_{2}\left(x^{(2)}\right)&\ldots&\varphi_{2}\left(x^{(d)}\right)\\\
\vdots&\vdots&&\vdots\\\
\varphi_{d}\left(x^{(1)}\right)&\varphi_{d}\left(x^{(2)}\right)&\ldots&\varphi_{d}\left(x^{(d)}\right)\\\
\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\ \vdots\\\
\lambda_{d}\end{array}\right)=\left(\begin{array}[]{c}1\\\ \varphi_{2}(x)\\\
\vdots\\\ \varphi_{d}(x)\end{array}\right).$ (4)

Let us introduce the mapping $T:{\mathbb{R}}^{n}\to{\mathbb{R}}^{d-1}$ defined
in the way

$y=T(x):=(\varphi_{2}(x),\ldots,\varphi_{d}(x))=(x_{1},\ldots,x_{n},\varphi_{n+1}(x),\ldots,\varphi_{d}(x)).$

We will consider $T$ on the set $\Omega.$ The choice of the first monomials
$\varphi_{j}(x)$ implies the invertibility of $T.$ Denote
$y^{(j)}:=T\left(x^{(j)}\right)$. Relation (4) gives

$\lambda_{1}y^{(1)}+\ldots+\lambda_{d}y^{(d)}=y,\quad\sum_{j=1}^{d}\lambda_{j}=1.$

This means that the numbers $\lambda_{j}(x)$ are the barycentric coordinates
of the point $y=T(x)$ with respect to the $(d-1)$-dimensional simplex with
vertices $y^{(j)}$. Hence,

$\|P\|_{\Omega}=\max_{x\in\Omega}\sum_{j=1}^{d}|\lambda_{j}(x)|=\max\left\\{\sum_{j=1}^{d}|\beta_{j}|:\
\sum_{j=1}^{d}\beta_{j}=1,\ y=\sum_{j=1}^{d}\beta_{j}y^{(j)}\in
T(\Omega)\right\\}.$ (5)

The second equality in (5) expresses the norm of $P$ via barycentric
coordinates of the points of $T(\Omega)$ relatively to the nondegenerate
$(d-1)$-dimensional simplex with vertices $y^{(j)}.$

By $\theta(\Pi;\Omega)$, we denote the minimal norm of a projector
$P:C(\Omega)\to\Pi$:

$\theta(\Pi;\Omega):=\min\limits_{x^{(j)}\in\Omega}\|P\|_{\Omega}.$

A projector whose norm equals $\theta(\Pi;\Omega)$ is called minimal. Also we
consider the following numerical characteristic of $\Omega$ representing the
minimal absorption coefficient of this set by nondegenerate simplices having
vertices in $\Omega$:

$\xi_{n}(\Omega):=\min\left\\{\xi(\Omega;S):\,S\mbox{ --- $n$-dimensional
simplex,}\,{\rm ver}(S)\subset\Omega,\,{\rm vol}(S)\neq 0\right\\}.$

Suppose $x^{(1)},\ldots,x^{(d)}\in\Omega$ is an admissible set of nodes to
interpolate functions from $C(\Omega)$ by polynomials from $\Pi$,
$P:C(\Omega)\to\Pi$ is the corresponding interpolation projector. Then points
$y^{(j)}=T\left(x^{(j)}\right)$ form an admissible set of nodes to interpolate
functions from $C(T(\Omega))$ by polynomials from
$\Pi_{1}\left({\mathbb{R}}^{d-1}\right).$ Consider the interpolation projector
$\overline{P}:C(T(\Omega))\to\Pi_{1}\left({\mathbb{R}}^{d-1}\right)$ with the
nodes $y^{(1)},\ldots,y^{(d)}.$ If $f\in C(\Omega),$ $g\in C(T(\Omega))$ and
$g(y)=f(x)$ while $y=T(x),$ then equalities $Pf\left(x^{(j)}\right)=f_{j}$
lead to equalities
$\overline{P}g\left(y^{(j)}\right)=g_{j}:=g\left(y^{(j)}\right).$ Therefore,
interpolational polynomials $p\in\Pi$ and
$q\in\Pi_{1}\left({\mathbb{R}}^{d-1}\right)$ are also connected by the
relation $p(x)=q(y).$ Let $\lambda_{j}\in\Pi$ be the basic Lagrange
polynomials corresponding to the projector $P,$ $\mu_{j}$ be the basic
Lagrange polynomials corresponding to the projector $\overline{P}$ (or to the
simplex $S={\rm conv}\left(y^{(1)},\ldots,y^{(d)}\right)$ $\subset$
${\mathbb{R}}^{d-1}$), then $\mu(y)=\mu(T(x))=\lambda_{j}(x)$. Hence,

$\|\overline{P}\|_{T(\Omega)}=\|P\|_{\Omega}.$

By this, when estimating the norm $\|P\|_{\Omega}$, it turns out to be
possible to apply geometric inequalities for the norm of projector
$\overline{P}$ under linear interpolation on the $(d-1)$ dimensional set
$T(\Omega)$. This approach was proposed by M. V. Nevskii; his results are
contained in papers [4], [5] and monograph [6]. Let us present the statements
we need.

For a projector $P:C(\Omega)\to\Pi$ with nodes $x^{(j)}$,

$\frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1\leq\xi(T(\Omega);S)\leq\frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1,$
(6)

where $S$ is the $(d-1)$-dimensional simplex with vertices
$T\left(x^{(j)}\right).$ Also we have

$\frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\theta(\Pi;\Omega)-1\right)+1\leq\xi_{d-1}(T(\Omega))\leq\frac{d}{2}\left(\theta(\Pi;\Omega)-1\right)+1.$
(7)

If $\|P\|_{\Omega}\neq 1$, then (6) can be written in the form

$\frac{1}{2}\left(1+\frac{1}{d-1}\right)\leq\frac{\xi(T(\Omega);S)-1}{\|P\|_{\Omega}-1}\leq\frac{d}{2}.$
(8)

For $\theta(\Pi;\Omega)\neq 1$, (7) is equivalent to

$\frac{1}{2}\left(1+\frac{1}{d-1}\right)\leq\frac{\xi_{d-1}(T(\Omega))-1}{\theta(\Pi;\Omega)-1}\leq\frac{d}{2}.$
(9)

We call a point $y=T(x)\in T(\Omega)$ a $1$-point with respect to the simplex
$S={\rm conv}\left(y^{(1)},\ldots,y^{(d)}\right)$, if

$\|P\|_{\Omega}=\sum_{j=1}^{d}|\lambda_{j}(x)|$

and among the numbers $\lambda_{j}(x)$ there is the only one negative. If such
a point exists, then the right-hand inequality in (6) becomes an equality.
This property was proved in [4] in the equivalent form. The notions of a
$1$-vertex of a cube and a $1$-point of an arbitrary set were introduced in
[5] and [8] correspondingly.

If a $1$-point of $T(\Omega)$ exists for a simplex $S={\rm
conv}\left(T\left(x^{(1)}\right),\ldots,T\left(x^{(d)}\right)\right)$ and $S$
satisfies $\xi(T(\Omega);S)=\xi_{d-1}(T(\Omega))$, then the right-hand
relation in (7) turns to equality and the projector $P:C(\Omega)\to\Pi$ with
these nodes $x^{(j))}$ is minimal.

The variant $\Pi=\Pi_{1}\left({\mathbb{R}}^{n}\right)$ was studied in the
authors’ works (see, e. g., [5], [6], [7], [8]). This time $d=$
$\dim\Pi_{1}\left({\mathbb{R}}^{n}\right)$ $=$ $n+1$ and the mapping $T$ is
the identity operator. The most detailed results concerns the cases when
$\Omega$ is an $n$-dimensional cube or an $n$-dimensional Euclidean ball.

In the present paper we discuss relations (6)–(9) for polynomial interpolation
on a segment. Further $n=1$, $\Pi=\Pi_{k}\left({\mathbb{R}}^{1}\right)$,
$k\geq 1$, $d=\dim\Pi_{k}\left({\mathbb{R}}^{1}\right)=k+1$,
$\varphi_{j}(x)=x^{j}$ $(1\leq j\leq{k})$. Let us take $\Omega=[-1,1]$, then

$T(\Omega)=T([-1,1])=\\{(x,\ldots,x^{k})\in{\mathbb{R}}^{k}:\,-1\leq x\leq
1\\}.$

In Sections 2–4, we consider the cases $k=2,3,4$. In Section 5, we present
numerical estimates of the values
$\theta\left(\Pi_{k}\left({\mathbb{R}}^{1}\right);[-1,1]\right)$ and
$\xi_{k}\left(T([-1,1])\right)$, $1\leq k\leq 10$. Finally, Sections 6–7
contain the material related to regular and Chebyshev nodes. We remark all the
cases when there are right-hand equalities in (6)–(9).

The numerical results were obtained by A. Yu. Ukhalov; the detailed data are
given in [10]. The computations were performed using the system Wolfram
Mathematica (see, e. g., [11], [3], [12]). Specially written programs in C++
were also used. To invert matrices and solve extremal problems, functions of
DLIB library [2] were used.

##  2 Quadratic Interpolation on a Segment

The simplest case of nonlinear interpolation is quadratic interpolation on a
segment. An analytical solution of the minimal projector problem with the
indicated geometric approach is given in [4], computer methods were used in
[1]. Let us discuss this case as an illustration.

It is well known (see, e. g., [9]) that the minimal norm of an interpolation
projector is realized for regular nodes and equals $\dfrac{5}{4}$. We claim
that this result can be also obtained with the use of (6)–(7). Additionally,
it turns out that there are infinitely many minimal projectors.

If $\Pi=\Pi_{2}\left({\mathbb{R}}^{1}\right)$, then $d=\dim\Pi=3,$ i. e.,
$k=d-1=2.$ The mapping $T$ has the form $x\longmapsto(x,x^{2})$ and the set
$T(\Omega)=T([-1,1])=\\{(x,x^{2})\in{\mathbb{R}}^{2}:\,-1\leq x\leq 1\\}$ is a
piece of parabola. For the interpolation nodes $-1\leq r<s<t\leq 1$, we have

${\bf A}=\left(\begin{array}[]{ccc}1&r&r^{2}\\\ 1&s&s^{2}\\\
1&t&t^{2}\end{array}\right),$

the simplex $S$ is a triangle with the vertices
$(r,r^{2}),\,(s,s^{2}),\,(t,t^{2})$ lying on this part of parabola. The
absorption of the parabolic sector by this triangle is shown in Figure 1 .

Figure 1: The absorption of the parabolic sector by a triangle

The convexity of the function $\psi(x)=x^{2}$ implies that a $1$-point of the
set $T([-1,1])$ with respect to $S$ exists for any nodes. Hence, there is an
equality in the right-hand part of (6):

$\xi(T(\Omega);S)=\frac{3}{2}\left(\|P\|_{\Omega}-1\right)+1=\frac{3\|P\|_{\Omega}-1}{2}.$
(10)

Since (10) holds true for an arbitrary projector
$P:C[-1,1]\to\Pi_{2}({\mathbb{R}}^{1})$, we have the right-hand equality also
in (7):

$\xi_{2}(T(\Omega))=\frac{3\theta\left(\Pi;\Omega\right)-1}{2}.$

So, in the quadratic case finding the minimum norm of a projector
$\theta\left(\Pi;\Omega\right)$ is equivalent to calculating
$\xi_{2}(T(\Omega))$, i. e., the minimum absorption coefficient of the
specified part of the parabola by a triangle. As nodes of a minimum projector,
one must take the first coordinates of vertices of the result triangle.
Technically, the problem can be reduced to a triangle $S$ with vertices
$(-r,r^{2}),$ $(0,0),$ $(r,r^{2}),$ $0<r\leq 1.$ For this triangle, the
extremal points $y\in T(\Omega)$ can be only $\displaystyle(\pm
1,1),\left(\pm\frac{r}{2},\frac{r^{2}}{4}\right).$ At each of the latter two
points a tangent line to the parabola is parallel to a side of $S.$
Calculations give

$\xi(T(\Omega);S)=\max\left(\frac{11}{8},\frac{3}{r^{2}}-2\right),\quad\|P\|_{\Omega}=\max\left(\frac{5}{4},\frac{2}{r^{2}}-1\right).$

If

$\frac{2\sqrt{2}}{3}=0.942809\ldots\leq r\leq 1,$

then these values do not depend on $r$ and are equal correspondingly to
$\dfrac{11}{8}$ and $\dfrac{5}{4}$ ; these are minimal possible values. Thus,

$\xi_{2}(T(\Omega))=\frac{11}{8},\quad\theta\left(\Pi;\Omega\right)=\frac{5}{4}.$

The minimal is any projector with vertices $-r,$ $0,$ $r$ provided
$r\in\left[\dfrac{2\sqrt{2}}{3},1\right]$ and there are no other minimal
projectors. Note that the relations (7)–(9) look like

$\frac{11}{16}<\frac{11}{8}=\frac{11}{8},\quad\frac{3}{4}<\frac{3}{2}=\frac{3}{2}.$

##  3 Cubic Interpolation on a Segment

In the case $\Pi=\Pi_{3}\left({\mathbb{R}}^{1}\right)$ we have $d=\dim\Pi=4,$
$k=d-1=3.$ Since $T$ has the form $x\longmapsto(x,x^{2},x^{3}),$ then

$T(\Omega)=T([-1,1])=\\{(x,x^{2},x^{3})\in{\mathbb{R}}^{3}:\,-1\leq x\leq
1\\}.$

This is a three-dimensional line with endpoints $(-1,1,-1)$ and $(1,1,1)$
whose projections on the coordinate planes are congruent to the curves
$Y=X^{2},$ $Y=X^{3}$, and $X(t)=t^{2},$ $Y(t)=t^{3};$ the latter one has the
zero angle at the point $X=Y=0.$

For the nodes $-1\leq r<s<t<u\leq 1$,

${\bf A}=\left(\begin{array}[]{cccc}1&r&r^{2}&r^{3}\\\ 1&s&s^{2}&s^{3}\\\
1&t&t^{2}&t^{3}\\\ 1&u&u^{2}&u^{3}\\\ \end{array}\right),$

the simplex $S$ is a tetrahedron with the vertices

$(r,r^{2},r^{3}),\quad(s,s^{2},s^{3}),\quad(t,t^{2},t^{3}),\quad(u,u^{2},u^{3})$
(11)

containing in $T(\Omega)$. The absorption of the set $T(\Omega)$ by this
tetrahedron is shown in Figure 2 .

Figure 2: The absorption of the set $T([-1,1])$ by a tetrahedron

Minimal values of $\|P\|_{\Omega}$ and $\xi(T(\Omega);S)$ were found using a
computer. The minimum of $\|P\|_{\Omega}$ equal to $1.422919...$ is delivered
by symmetrical nodes

$-1\,,\quad-0.417791...\,,\quad 0.417791...\,,\quad 1\,.$ (12)

The minimum of $\xi(T(\Omega);S)$ equal to $1.635778...$ is delivered by a
tetrahedron whose vertices appear after applying the mapping $T$ to the points

$-1,\quad-0.481618...\,,\quad 0.481618...\,,\quad\ 1\,.$

So, the computer calculations give $\theta\,(\Pi;\Omega)=1.422919...\,,$ $\
\xi_{3}(T(\Omega))=1.635778...\,.$ With such values both the inequalities in
(7) are strict and have the form $1.28194...<1.63577...<1.84583...\,.$
Relation (9) may be written as $\frac{2}{3}<1.503307\ldots<2.$

Under quadratic interpolation, a $1$-point of $T(\Omega)$ exists for an
arbitrary projector. But in the case considered this is not so. For example,
let us take

$r=-\frac{\sqrt{2+\sqrt{2}}}{2},\quad s=-\frac{\sqrt{2-\sqrt{2}}}{2},\quad
t=\frac{\sqrt{2-\sqrt{2}}}{2},\quad u=\frac{\sqrt{2+\sqrt{2}}}{2}.$

These are the zeros of the Chebyshev polynomial of degree 4, i. e.,
$8x^{4}-8x^{2}+1$; we call them Chebyshev nodes. Then
$\|P\|_{\Omega}=\sqrt{2+\sqrt{2}}=1.847759\ldots\,,\
\xi(T(\Omega);S)=2.496605\ldots\,$, and inequalities (6) have the form
$1.5651727\ldots<2.496605\ldots$ $<2.695518\ldots\ .$ Since the right-hand
relation in (6) is not an equality, a $1$-point of $T(\Omega)$ with respect to
the simplex with vertices (11) does not exist.

Nethertheless, our calculations show that for some nodes the right-hand
relation in (6) turns an equality (with the accurancy $10^{-11}$). This takes
place for regular nodes and also for the nodes (12) of the minimal projector.

For regular nodes, $\|P\|_{\Omega}=1.63113030\ldots\,,\
\xi(T(\Omega);S)=2.26226061\ldots\,$; relation (6) takes the form
$1.420753\ldots<2.262260\ldots=2.262260\ldots$ . Remark that we maid the
calculations of $\|P\|_{\Omega}$ $\xi(T(\Omega);S)$ with the use of two
various programs. The latter equality holds true with the accurancy not less
than $10^{-18}$.

For the nodes (12), the projector norm is minimal:
$\|P\|_{\Omega}=\theta\,(\Pi;\Omega)=1.422919\ldots,$ the absorption
coefficient is $\xi(T(\Omega);S)=1.845839\ldots$, and relation (6) is written
as $1.281946\ldots<1.845839\ldots=1.845839\ldots$ .

Of course, approximate calculations, even while having good accurancy, do not
garantee the existence of the right-hand equality in (6). The strong proof
gives an approach with the use of the notion of a $1$-point (see Section 1).

For regular nodes

$x^{(1)}=-1,\quad x^{(2)}=-\dfrac{1}{3},\quad x^{(3)}=\dfrac{1}{3},\quad
x^{(4)}=1$

and for the point $x^{*}=-0.699055\ldots$, we have
$\|P\|_{\Omega}=1.631130\ldots=\sum|\lambda_{j}\left(x^{*}\right)|$, while

$\lambda_{1}\left(x^{*}\right)=0.890801\ldots,\quad\lambda_{2}\left(x^{*}\right)=-0.315565\ldots,$
$\lambda_{3}\left(x^{*}\right)=0.360848\ldots,\quad\lambda_{4}\left(x^{*}\right)=0.063915\ldots.$

Consequently, $y^{*}=T\left(x^{*}\right)$ is a $1$-point of $T(\Omega)$ with
respect to a tetrahedron with vertices $T\left(x^{(j)}\right)$. In accordance
with Section 1, the right-hand relation in (6) turns into equality. Note that
$\det({\bf A})=1.053497\ldots.$

For nodes

$x^{(1)}=-1,\quad x^{(2)}=-0.417791\dots,\quad x^{(3)}=0.417791\dots,\quad
x^{(4)}=1$

and for $x^{**}=-0.733172\ldots$, we have
$\|P\|_{\Omega}=1.422919\ldots=\sum|\lambda_{j}\left(x^{**}\right)|$. Since

$\lambda_{1}\left(x^{**}\right)=-0.211459\ldots,\quad\lambda_{2}\left(x^{**}\right)=0.771708\ldots,$
$\lambda_{3}\left(x^{**}\right)=0.381082\ldots,\quad\lambda_{4}\left(x^{**}\right)=0.058668\ldots,$

$y^{**}=T\left(x^{**}\right)$ is a $1$-point of $T(\Omega)$ with respect to a
tetrahedron with vertices $T\left(x^{(j)}\right)$. The existence of a 1-point
means that also in this case we have an equality in the right-hand part of
(6). Here $\det({\bf A})=-1.138679\ldots.$

##  4 Interpolation by Polynomials from $\Pi_{4}\left({\mathbb{R}}^{1}\right)$

If $\Pi=\Pi_{4}\left({\mathbb{R}}^{1}\right)$, then $d=\dim\Pi=5,$ $k=d-1=4.$
The mapping $T$ takes the form $x\longmapsto(x,x^{2},x^{3},x^{4}),$ therefore

$T(\Omega)=T([-1,1])=\\{(x,x^{2},x^{3},x^{4})\in{\mathbb{R}}^{4}:\,-1\leq
x\leq 1\\}.$

For nodes $-1\leq r<s<t<u<v\leq 1$, we have

${\bf A}=\left(\begin{array}[]{ccccc}1&r&r^{2}&r^{3}&r^{4}\\\
1&s&s^{2}&s^{3}&s^{4}\\\ 1&t&t^{2}&t^{3}&t^{4}\\\ 1&u&u^{2}&u^{3}&u^{4}\\\
1&v&v^{2}&v^{3}&v^{4}\\\ \end{array}\right).$

Coordinates of vertices of simplex $S\subset{\mathbb{R}}^{4}$ are written in
the rows of matrix ${\bf A}$ starting from the second column.

The Chebyshev nodes, i. e., the zeroes of the $5$th degree Chebyshev
polynomial $16x^{5}-20x^{3}+5x$, are

$r=-\frac{\sqrt{5+\sqrt{5}}}{2\sqrt{2}},\quad
s=-\frac{\sqrt{5-\sqrt{5}}}{2\sqrt{2}},\quad t=0,\quad
u=\frac{\sqrt{5-\sqrt{5}}}{2\sqrt{2}},\quad
v=\frac{\sqrt{5+\sqrt{5}}}{2\sqrt{2}}.$

In this case, $\|P\|_{\Omega}=\dfrac{1+4\sqrt{5}}{5}=1.988854\ldots\,,$ but
this is not the possible minimum value.

Let us give the results of numerical calculations. The minimal
$\|P\|_{\Omega}$ equal to $1.559490...$ is achieved at symmetrical nodes

$\\-1\,,\quad-0.620911...\,,\quad 0\,,\quad 0.620911...\,,\quad 1\,.$

The minimal $\xi(T(\Omega);S)$ equal to $1.981193...$ is delivered by a
simplex with vertices constructed by operator $T$ from the points

$-1\,,\quad-0.650738...\,,\quad 0\,,\quad 0.650738...\,,\quad 1\,.$

Thus, $\theta\left(\Pi;\Omega\right)=1.559490...\,,\
\xi_{4}(T(\Omega))=1.981193...\,.$ Inequalities in (7) are both strict and
have the form $1.349681...<1.981193...<2.398725...\,.$

##  5 Estimates of
$\theta\left(\Pi_{k}\left({\mathbb{R}}^{1}\right);[-1,1]\right)$ and
$\xi_{k}\left(T([-1,1])\right)$ for $1\leq k\leq 10$

In this section, we present numerical estimates for minimal norms of
interpolation projections acting from $C[-1,1]$ on
$\Pi_{k}\left({\mathbb{R}}^{1}\right)$ and minimum absorption coefficients for
$1\leq k\leq 10$. For brevity, we write
$\theta_{k}:=\theta\left(\Pi_{k}\left({\mathbb{R}}^{1}\right);[-1,1]\right)$,
$\xi_{k}:=\xi_{k}\left(T([-1,1])\right)$.

Table 1: Minimal norms of projectors and corresponding absorption coefficients $k$ | $\theta_{k}\leq$ | $\xi(S)$
---|---|---
1 | 1 | 1
2 | 1.25 | 1.375
3 | 1.422919… | 1.845839…
4 | 1.559490… | 2.224196…
5 | 1.672210… | 2.574785…
6 | 1.768134… | 2.911143…
7 | 1.851599… | 3.239031…
8 | 1.925457… | 3.561425…
9 | 1.991685… | 3.880036…
10 | 2.051705… | 4.195926…
Table 2: Minimal absorption coefficients and corresponding norms of projectors $k$ | $\xi_{k}\leq$ | $||P||$
---|---|---
1 | 1 | 1
2 | 1.375000… | 1.250000…
3 | 1.635778… | 1.604018…
4 | 1.981193… | 1.626067…
5 | 2.210535… | 1.782786…
6 | 2.455130… | 1.858521…
7 | 2.678509… | 1.962845…
8 | 2.907301… | 2.029565…
9 | 3.128316… | 2.108072…
10 | 3.351866… | 2.164915…

Table 1 contains the number $k$ (the degree of an interpolation polynomial),
the upper bound for $\theta_{k}$, and the value $\xi(S):=\xi(T([-1,1]);S)$, i.
e., the absorption coefficient of the set $T([-1,1])$ by the simplex with
vertices $T\left(x^{(j)}\right)$, where $x^{(j)}$ are the nodes of
interpolation projector delivering the estimate of $\theta_{k}$. The values
$\theta_{1}=1$ and $\theta_{2}=1.25$ are precise. The right-hand equality in
(6) takes places for $k=1,2,3$.

In Table 2, we give the upper bounds for minimal absorption coefficients
$\xi_{k}$ and the norms $\|P\|:=\|P\|_{[-1,1]}$ of projectors with the nodes
constructed by vertices of extremal simplices. The values $\xi_{1}=1$ and
$\xi_{2}=1.375$ are precise. The second equality in (6) takes places for
$k=1,2$.

The numerically found nodes of minimal projections and the parametric
coordinates of the vertices of extremal simplices are given in [10]. There one
can also find the estimates for $\theta_{k}$ and $\xi_{k}$ with a large number
of decimal signs. See https://doi.org/10.17632/snh5m99yxr.1.

##  6 Regular Nodes

Computer calculations of absorption coefficients and projector norms for
regular nodes (see Table 3) are complicated by a rapid decrease in the modulus
of determinant of matrix ${\bf A}$. For example, for $k=11$ we have $\det({\bf
A})=3.63581\cdot 10^{-14}$. Further, as $k$ increases, the absolute value of
the determinant rapidly decreases. When computing on systems with standard
floating point representation, it is difficult to guarantee the accuracy of
the results. For this reason, we present the values of the considered
quantities only for $1\leq k\leq 10$. Note that the right-hand equality in (6)
holds for $k=1,2,3$.

Table 3: Absorption coefficients and norms of projectors for regular nodes $k$ | $\xi(S)$ | $||P||$
---|---|---
1 | 1 | 1
2 | 1.375 | 1.25
3 | 2.262260… | 1.631130…
4 | 3.812500… | 2.207824…
5 | 6.167317… | 3.106301…
6 | 9.461457… | 4.549341…
7 | 13.824447… | 6.929739…
8 | 21.876588… | 10.945645…
9 | 41.283675… | 17.848612…
10 | 72.576233… | 29.899955…
Table 4: Absorption coefficients and norms of projectors for Chebyshev nodes $k$ | $\xi(S)$ | $||P||$
---|---|---
1 | 1.414213… | 1.414213…
2 | 2.000000… | 1.666666…
3 | 2.496605… | 1.847759…
4 | 2.962610… | 1.988854…
5 | 3.414213… | 2.104397…
6 | 3.857835… | 2.202214…
7 | 4.296558… | 2.287016…
8 | 4.732050… | 2.361856…
9 | 5.165299… | 2.428829…
10 | 5.596925… | 2.489430…
11 | 6.027339… | 2.544766…
12 | 6.456823… | 2.595678…

##  7 Chebyshev Nodes

The table 4 shows the results of calculating absorption coefficients and
projector norms for Chebyshev nodes (i. e., the nodes coinciding with the
zeros of Chebyshev polynomial of the required degree). The right-hand equality
in (6) holds when $k=1,2$.

As in the case of regular nodes, one can vouch for the accuracy of
calculations of absorption coefficients only for small $k$. We give values for
$1\leq k\leq 12$. Already for $k=12$ we have $\det({\bf A})=3.68529\cdot
10^{-15}$. As $k$ increases, $|\det({\bf A})|$ becomes even smaller, which can
lead to a decrease in precision when using C++ variables of the double type.
Projector norms for Chebyshev nodes were calculated using the exact formula
(see, e. g., [9]), so this remark does not apply to the values of the norms.

Note that the C++ program we use to calculate the norms of projections gives a
satisfactory agreement with the exact formula at least for $k\leq 30$.

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