Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball

TL;DR

This paper investigates geometric estimates for the norm of interpolation projectors in the space of continuous functions on a Euclidean ball, relating these estimates to the volume of simplices used in interpolation.

Contribution

It introduces a method to estimate the operator norm of interpolation projectors from below using the volume of the simplex within the Euclidean ball.

Findings

The norm of the interpolation projector can be bounded below by the volume of the simplex.

The approach provides a geometric perspective on interpolation error estimates.

The method applies to nondegenerate simplices in the Euclidean unit ball.

Abstract

Let $B_n$ be the Euclidean unit ball in ${\mathbb R}^n$ given by the inequality $\|x\|\leq 1$, $\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}$. By $C(B_n)$ we mean the space of continuous functions $f:B_n\to{\mathbb R}$ with the norm $\|f\|_{C(B_n)} := \max\limits_{x\in B_n}|f(x)|$. The symbol $\Pi_1\left({\mathbb R}^n\right)$ denotes the set of polynomials in $n$ variables of degree $\leq 1$, i.e., the set of linear functions upon ${\mathbb R}^n$. Assume $x^{(1)}, \ldots, x^{(n+1)}$ are the vertices of an $n$-dimensional nondegenerate simplex $S\subset B_n$. The interpolation projector $P:C(B_n)\to \Pi_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right) = f\left(x^{(j)}\right).$ Denote by $\|P\|_{B_n}$ the norm of $P$ as an operator from $C(B_n)$ onto $C(B_n)$. We describe the approach in which $\|P\|_{B_n}$ can be estimated from below via the volume of $S$.

Equations

$$\Vert x\Vert :={\left(\sum _{i=1}^n{x}_i^2\right)}^{\frac{1}{2}},B_n:=\{x\in {\mathbb{R}}^n:\Vert x\Vert \le 1\},Q_n:=[0,1{]}^n.$$

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

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

$$Pf(x)=\sum _{j=1}^{n+1}f\left(x^{(j)}\right){\lambda }_j(x).$$

$$\Vert P{\Vert }_{\Omega }=\underset{x\in \Omega }{\max }\sum _{j=1}^{n+1}|{\lambda }_j(x)|.$$

$$\Vert P{\Vert }_{\Omega }=\underset{x\in \mathrm{ver}(\Omega )}{\max }\sum _{j=1}^{n+1}|{\lambda }_j(x)|,$$

$${\chi }_n(t):=\frac{1}{2^{n}n!}{\left[(t^2-1{)}^n\right]}^{(n)}$$

$$E_{n,\gamma}:=\Bigl{\{}x\in{\mathbb{R}}^{n}:\sum_{j=1}^{n}\left|x_{j}\right|+\Bigl{|}1-\sum_{j=1}^{n}x_{j}\Bigr{|}\leq\gamma\Bigr{\}}.$$

$$\mathrm{me}{\mathrm{s}}_{\mathrm{n}}(E_{n,\gamma })=\frac{{\chi }_n(\gamma )}{n!}.$$

$${\theta }_n(Q_n)\ge {\chi }_n^{-1}\left(\frac{1}{{\nu }_n}\right)={\chi }_n^{-1}\left(\frac{n!}{h_n}\right).$$

$$\Vert P{\Vert }_{B_n}\ge {\chi }_n^{-1}\left(\frac{{\varkappa }_n}{\mathrm{vol}(S)}\right)={\chi }_n^{-1}\left(\frac{n!{\varkappa }_n}{|\det (\mathbf{A})|}\right).$$

$$|\det (\mathbf{B})|=|\det (\mathbf{A})|=n!\mathrm{vol}(S)\le n!{\sigma }_n.$$

$$\frac{n!{\sigma }_n}{|\det (\mathbf{B})|}\ge 1.$$

$$\|P\|_{B_{n}}=\max_{x\in B_{n}}\sum_{j=1}^{n+1}|\lambda_{j}(x)|=\max\Bigl{\{}\sum_{j=1}^{n+1}|\beta_{j}|:\,\sum_{j=1}^{n+1}\beta_{j}=1,\,\sum_{j=1}^{n+1}\beta_{j}x^{(j)}\in B_{n}\Bigr{\}}.$$

$$\|P\|_{B_{n}}=\max\Bigl{\{}\sum_{j=1}^{n}\Bigl{|}\beta_{j}\Bigr{|}+\Bigl{|}1-\sum_{j=1}^{n}\beta_{j}\Bigr{|}\Bigr{\}}.$$

$$x=F(\beta ):=\sum _{j=1}^n{\beta }_j\left(x^{(j)}-x^{(n+1)}\right).$$

$${\gamma }^{\ast }:={\chi }_n^{-1}\left(\frac{n!{\varkappa }_n}{|\det (\mathbf{B})|}\right).$$

$${\chi }_n({\gamma }^{\ast })=\frac{n!{\varkappa }_n}{|\det (\mathbf{B})|}.$$

$$E_{n,\gamma}=\Bigl{\{}\beta=(\beta_{1},\ldots,\beta_{n})\in{\mathbb{R}}^{n}:\sum_{j=1}^{n}\left|\beta_{j}\right|+\Bigl{|}1-\sum_{j=1}^{n}\beta_{j}\Bigr{|}\leq\gamma\Bigr{\}}.$$

$$\mathrm{me}{\mathrm{s}}_{\mathrm{n}}(F(E_{n,\gamma }))<\mathrm{me}{\mathrm{s}}_{\mathrm{n}}(F(E_{n,{\gamma }^{\ast }}))=|\det \mathbf{B}|\cdot \mathrm{me}{\mathrm{s}}_{\mathrm{n}}(E_{n,{\gamma }^{\ast }})=$$

$$=|\det \mathbf{B}|\cdot \frac{{\chi }_n({\gamma }^{\ast })}{n!}={\varkappa }_n.$$

$$x^{(\varepsilon)}=\sum_{j=1}^{n}\beta_{j}^{(\varepsilon)}\bigl{(}x^{(j)}-x^{(n+1)}\bigr{)}\in B^{\prime},\quad\sum\limits_{j=1}^{n}\Bigl{|}\beta_{j}^{(\varepsilon)}\Bigr{|}+\Bigl{|}1-\sum\limits_{j=1}^{n}\beta_{j}^{(\varepsilon)}\Bigr{|}\geq\gamma^{*}-\varepsilon.$$

$$\Vert P\Vert \ge {\gamma }^{\ast }={\chi }_n^{-1}\left(\frac{n!{\varkappa }_n}{|\det (\mathbf{B})|}\right)={\chi }_n^{-1}\left(\frac{n!{\varkappa }_n}{|\det (\mathbf{A})|}\right)={\chi }_n^{-1}\left(\frac{{\varkappa }_n}{\mathrm{vol}(S)}\right).$$

$${\theta }_n(B_n)\ge {\chi }_n^{-1}\left(\frac{{\varkappa }_n}{{\sigma }_n}\right).$$

$$\Vert P{\Vert }_{B_n}\ge {\chi }_n^{-1}\left(\frac{{\varkappa }_n}{\mathrm{vol}(S)}\right)\ge {\chi }_n^{-1}\left(\frac{{\varkappa }_n}{{\sigma }_n}\right).$$

$${\varkappa }_n=\frac{{\pi }^{\frac{n}{2}}}{\Gamma \left(\frac{n}{2}+1\right)},{\sigma }_n=\frac{1}{n!}\sqrt{n+1}{\left(\frac{n+1}{n}\right)}^{\frac{n}{2}},$$

$${\varkappa }_{2k}=\frac{{\pi }^k}{k!},{\varkappa }_{2k+1}=\frac{2^{k+1}{\pi }^k}{(2k+1)!!}=\frac{2(k!)(4\pi {)}^k}{(2k+1)!}.$$

$${\theta }_n(B_n)\ge {\chi }_n^{-1}\left(\frac{{\pi }^{\frac{n}{2}}n!}{\Gamma \left(\frac{n}{2}+1\right)\sqrt{n+1}{\left(\frac{n+1}{n}\right)}^{\frac{n}{2}}}\right).$$

$${\theta }_{2k}(B_{2k})\ge {\chi }_{2k}^{-1}\left(\frac{{\pi }^k(2k)!}{k!\sqrt{2k+1}{\left(\frac{2k+1}{2k}\right)}^k}\right).$$

$${\theta }_{2k+1}(B_{2k+1})\ge {\chi }_{2k+1}^{-1}\left(\frac{2(k!)(4\pi {)}^k}{\sqrt{2k+2}{\left(\frac{2k+2}{2k+1}\right)}^{\frac{2k+1}{2}}}\right).$$

Full text

Keywords: simplex, ball, linear interpolation, projector, norm, estimate

1 Main Definitions

We always suppose $n\in{\mathbb{N}}$. An element $x\in{\mathbb{R}}^{n}$ will be written in the form $x=(x_{1},\ldots,x_{n}).$ By definition,

[TABLE]

The notation $L(n)\asymp M(n)$ means that there exist $c_{1},c_{2}>0$ not depending on $n$ such that $c_{1}M(n)\leq L(n)\leq c_{2}M(n)$. By $\Pi_{1}\left({\mathbb{R}}^{n}\right)$ we mean the set of polynomials in $n$ variables $\leq 1$, i. e., the set of linear functions upon ${\mathbb{R}}^{n}$.

Let $S$ be a nondegenerate simplex in ${\mathbb{R}}^{n}$ with vertices x^{(j)}=\bigl{(}x_{1}^{(j)},\ldots,x_{n}^{(j)}\bigr{)}, $1\leq j\leq n+1$. Consider the vertex matrix (or the node matrix)

[TABLE]

We have the equality ${\rm vol}(S)=\frac{|\det({\bf A}|}{n!}$. Put ${\bf A}^{-1}$ $=(l_{ij})$. Let us define $\lambda_{j}$ as polynomials from $\Pi_{1}\left({\mathbb{R}}^{n}\right)$ whose coefficients form the columns of ${\bf A}^{-1}$, i. e.,  $\lambda_{j}(x)$ $:=$ $l_{1j}x_{1}+\ldots+l_{nj}x_{n}+l_{n+1,j}.$ We call $\lambda_{j}$ the basic Lagrange polynomials corresponding to this simplex. The numbers $\lambda_{j}(x)$ are the barycentric coordinates of a point $x\in{\mathbb{R}}^{n}$ with respect to $S$.

For a convex body $\Omega\subset{\mathbb{R}}^{n}$, denote by $C(\Omega)$ the space of continuous functions $f:\Omega\to{\mathbb{R}}$ with the uniform norm

[TABLE]

We say that an interpolation projector $P:C(\Omega)\to\Pi_{1}({\mathbb{R}}^{n})$ corresponds to a simplex $S\subset\Omega$ if interpolation nodes of $P$ coincide with vertices $x^{(j)}$ of this simplex. Projector $P$ is given by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ We have the following analogue of the Lagrange interpolation formula:

[TABLE]

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

[TABLE]

If $\Omega$ is a convex polytope in ${\mathbb{R}}^{n}$ (e. g., $\Omega=Q_{n})$, this equality is equivalent to the formula

[TABLE]

where ${\rm ver}(\Omega)$ is the set of vertices of $\Omega$.

Let us define $\theta_{n}(\Omega)$ as the minimal value of $\|P\|_{\Omega}$ under the conditions $x^{(j)}\in\Omega$. In the case $\Omega=Q_{n}$ various relations for the numbers $\theta_{n}(\Omega)$, including the equivalence $\theta_{n}(\Omega)\asymp\sqrt{n}$, were obtained by the author earlier. This results are systematized in [1]. Further on some estimates were improved (see [2], [3], [4] and references in these papers).

This paper deals with the case $\Omega=B_{n}$. We describe the approach when the norm of an interpolation projector $P:C(B_{n})\to\Pi_{1}({\mathbb{R}}^{n})$ can be estimated from below through the volume of the corresponding simplex. The essential feature of this approach is the application of the classical Legendre polynomials. We prove that $\theta_{n}(B_{n})\asymp\sqrt{n}$. In other words, the interpolation projector corresponding to a regular simplex inscribed into the boundary sphere has the norm equivalent to the minimal possible.

2 Estimation of $\|P\|_{B_{n}}$ via the Volume of $S$

The standardized Legendre polynomial of degree $n$ is the function

[TABLE]

(the Rodrigues formula). For properties of $\chi_{n}$, see [6], [7]. The Legendre polynomials are orthogonal on $[-1,1]$ with respect to the weight $w(t)=1.$ It is known that $\chi_{n}(1)=1$; if $n\geq 1$, then $\chi_{n}(t)$ strictly increases for $t\geq 1$. Denote by $\chi_{n}^{-1}$ the function inverse to $\chi_{n}$ on the halfline $[1,+\infty)$.

The appearance of Legendre polynomials in our questions is connected with such their property. For $\gamma\geq 1$, consider the set

[TABLE]

In 2003 the author established that

[TABLE]

(the proof is also given in [1]). Utilizing this equality he managed to obtain the lower estimates for proector’s norms related to linear interpolation on the unit cube $Q_{n}$. The following relations take place:

[TABLE]

Here $\nu_{n}$ is the maximum volume of a simplex contained in $Q_{n}$ and $h_{n}$ is the maximum value of $0/1$-determinant of order $n$. Applying the properties of $\chi_{n}$ the author got from (3) some more visible inequalities, e. g., $\theta_{n}(Q_{n})>\frac{1}{e}\sqrt{n-1}.$ This estimate occured to be sharp concerning to $n$ which led to the equivalence $\theta_{n}(Q_{n})\asymp\sqrt{n}.$

Further on we will extend this approach to linear interpolation of functions given on the unit ball $B_{n}$. Denote $\varkappa_{n}:={\rm vol}(B_{n})$. By $\sigma_{n}$ we mean the volume of a regular simplex inscribed into $B_{n}$.

Theorem 1. Assume $P:C(B_{n})\to\Pi_{1}({\mathbb{R}}^{n})$ is an arbitrary interpolation projector. Then for the corresponding simplex $S\subset B_{n}$ and the node matrix ${\bf A}$ we have

[TABLE]

Proof. As known, a regular simplex which is inscribed into a ball has the maximum possible volume among all simplices being contained in this ball. Therefore, ${\rm vol}(S)=\frac{|\det({\bf A})}{n!}\leq\sigma_{n}$. For each $i=1,$ $\ldots,$ $n$, let us subtract from the $i$th row of ${\bf A}$ its $(n+1)$th row. Denote by ${\bf B}$ the submatrix of order $n$ which stands in the first $n$ rows and columns of the result matrix. Then

[TABLE]

In other words,

[TABLE]

Let $x^{(j)}$ be the vertices and $\lambda_{j}$ be the basic Lagrange polynomials of the simplex $S$. Since $\lambda_{1}(x),$ $\ldots$, $\lambda_{n+1}(x)$ are the barycentric coordinates of a point $x$, we have

[TABLE]

Let us replace $\beta_{n+1}$ with the equal value $1-\sum\limits_{j=1}^{n}\beta_{j}.$ The condition $\sum\limits_{j=1}^{n+1}\beta_{j}x^{(j)}$ $\in B_{n}$ is equivalent to $\sum\limits_{j=1}^{n}\beta_{j}(x^{(j)}-x^{(n+1)})\in B^{\prime}:=B_{n}-x^{(n+1)}.$ This means

[TABLE]

Maximum in (6) is taken upon $\beta_{j}$ such that $\sum\limits_{j=1}^{n}\beta_{j}(x^{(j)}-x^{(n+1)})\in B^{\prime}.$

Consider the nondegenerate linear operator $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ which maps a point $\beta=(\beta_{1},\ldots,\beta_{n})$ into a point $x=(x_{1},\ldots,x_{n})$ according to the rule

[TABLE]

We have the matrix equality $F(\beta)=(\beta_{1},\ldots,\beta_{n}){\bf B},$ where $\bf B$ is the above introduced $(n\times n)$-matrix with the elements $b_{ij}=x_{j}^{(i)}-x_{j}^{(n+1)}.$ Put

[TABLE]

Since $\varkappa_{n}\geq\sigma_{n}$, it follows from (5) that $\gamma^{*}$ is defined correctly. Note the equality

[TABLE]

Now suppose $1\leq\gamma<\gamma^{*}$,

[TABLE]

Let us show that $B^{\prime}\not\subset F(E_{n,\gamma}).$ It is sufficient to get the inequality ${\rm mes_{n}}(F(E_{n,\gamma}))<{\rm mes_{n}}(B^{\prime})=\varkappa_{n}.$ This is really so:

[TABLE]

[TABLE]

We have applied (2) and (7). Thus, for every $\varepsilon>0$, there exists a point $x^{(\varepsilon)}$ with the properties

[TABLE]

In view of (6) this gives $\|P\|_{B_{n}}\geq\gamma^{*}-\varepsilon.$ Since $\varepsilon>0$ is an arbitrary, we obtain

[TABLE]

The theorem is proved. $\Box$

Corollary 1. For each $n$,

[TABLE]

Proof. Let $P:C(B_{n})\to\Pi_{1}({\mathbb{R}}^{n})$ be an arbitrary interpolation projector. Since the volume of the corresponding simplex is not greater than $\sigma_{n}$, inequality (4) yields

[TABLE]

This gives (8). $\Box$

It is known that

[TABLE]

[TABLE]

Therefore, the estimate (8) can be made more concrete.

Corollary 2. For every $n$,

[TABLE]

If $n=2k$, then (11) is equivalent to the inequality

[TABLE]

For $n=2k+1$ we have

[TABLE]

Proof. It is sufficient to apply (8), (9), and (10). $\Box$

Corollary 3. Suppose $P$ is a minimal interpolation projector, i. e., $\|P\|_{B_{n}}=\theta_{n}(B_{n})$. Then

[TABLE]

Proof. It was proved in [5] that $\theta_{n}(B_{n})\leq\sqrt{n+1}$. Hence, if $P$ is a minimal projector, then

[TABLE]

We have made use of (4). It remains to compare the boundary values and take into account (9). $\Box$

Since $|\det({\bf A})|=n!{\rm vol}(S)$, relation (14) also implies an estimate for the determinant of the node matrix corresponding to a minimal projector. Namely, $|\det({\bf A})|\geq\varepsilon_{n},$ where $\varepsilon_{n}$ is $n!$ times more than the right-hand part of (14). Without giving the value of $\varepsilon_{n}$ this restriction was used in [5].

3 The Relation $\theta_{n}(B_{n})\asymp\sqrt{n}$

The Stirling formula $n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}e^{\frac{\zeta_{n}}{12n}}$, $0<\zeta_{n}<1,$ yields

[TABLE]

Also we will need the following estimates which were proved in [1, Section 3.4.2]:

[TABLE]

Theorem 2. There exist a constant $c>0$ not depending on $n$ such that

[TABLE]

The inequality (17) takes place, e. g., with the constant

[TABLE]

Proof. First let $n=2k$ be even. Making use of (12) and (16), we get

[TABLE]

[TABLE]

Let us estimate $k!$ from below applying (15):

[TABLE]

Since $n=2k,$ for each even $n$,

[TABLE]

We have utilized the inequalities $\frac{\pi n}{n+1}>1$, $\frac{n}{n+1}>\frac{2}{3}$. Remark that $\sqrt{\frac{\pi}{3e}}=0.6206...$

Now suppose $n$ is odd, i. e., $n=2k+1$. In this case (13) and (16) give

[TABLE]

[TABLE]

[TABLE]

From (15), it follows that

[TABLE]

[TABLE]

[TABLE]

Also

[TABLE]

[TABLE]

Consequently,

[TABLE]

Now let us estimate $B^{\frac{1}{2k+1}}$:

[TABLE]

[TABLE]

[TABLE]

Therefore,

[TABLE]

Since $n=2k+1$, we have $\sqrt{k+1}=\frac{1}{\sqrt{2}}\sqrt{2k+2}>\frac{1}{\sqrt{2}}\sqrt{n}.$ For every odd $n$,

[TABLE]

The constant $0.2135...$ from the right-hand part of (19) is less than the constant from the above inequality (18) for even $n$. Hence, (19) is true for all $n\in{\mathbb{N}}$. This completes the proof. $\Box$

Corollary 4. $\theta_{n}(B_{n})\asymp\sqrt{n}$*. *

Proof. In [5] it was proved that $\theta_{n}(B_{n})\leq\sqrt{n+1}.$ Consequently, the lower estimate $\theta_{n}(B_{n})>c\sqrt{n}$ is precise with respect to dimension $n$. $\Box$

Corollary 5. *Assume $P:C(B_{n})\to\Pi_{1}({\mathbb{R}}^{n})$ is the interpolation projector whose nodes coincide with vertices of a regular simplex being inscribed into the boundary sphere $\|x\|=1$. When $\|P\|_{B_{n}}\asymp\theta_{n}(B_{n})$. *

Proof. As it was shown in [5], $\sqrt{n}\leq\|P\|_{B_{n}}\ \leq\sqrt{n+1}$. It remains to utilize the previous corollary. $\Box$

Our results mean that any interpolation projector $P$ corresponding to an inscrided regular simplex has the norm equivalent to the minimal possible. The equality $\|P\|_{B_{n}}=\theta_{n}(B_{n})$ remains proved only for $1\leq n\leq 4$.

The author is grateful to A. Yu. Ukhalov for some useful computer calculations.

References

Nevskii, M. V., Geometricheskie ocenki v polinomialnoi interpolyacii (Geometric Estimates in Polynomial Interpolation), Yaroslavl’: Yarosl. Gos. Univ., 2012 (in Russian).

• 2.

Nevskii, M. V., and Ukhalov, A. Yu., On numerical charasteristics of a simplex and their estimates, Model. Anal. Inform. Sist., 2016, vol. 23, no. 5, pp. 603–619 (in Russian). English transl.: Aut. Control Comp. Sci., 2017, vol. 51, no. 7, pp. 757–769.

• 3.

Nevskii, M. V., and Ukhalov, A. Yu., New estimates of numerical values related to a simplex, Model. Anal. Inform. Sist., 2017, vol. 24, no. 1, pp. 94–110 (in Russian). English transl.: Aut. Control Comp. Sci., 2017, vol. 51, no. 7, pp. 770–782.

• 4.

Nevskii, M. V., and Ukhalov, A. Yu., On optimal interpolation by linear functions on an $n$-dimensional cube, Model. Anal. Inform. Sist., 2018, vol. 25, no. 3, pp. 291–311 (in Russian). English transl.: Aut. Control Comp. Sci., 2018, vol. 52, no. 7, pp. 828–842.

• 5.

Nevskii, M. V., and Ukhalov, A. Yu., Linear interpolation on a Euclidean ball in ${\mathbb{R}}^{n}$ (to appear).

• 6.

Szego, G., Orthogonal Polynomials, New York: American Mathematical Society, 1959.

• 7.

Suetin, P. K. Klassicheskie ortogonal’nye mnogochleny, Moscow: Nauka, 1979 (in Russian).