Box-splines orthogonal projections

TL;DR

This paper generalizes a known result about orthogonal projections of polynomial functions to the broader context of box-splines, enabling new ways to define Sobolev space seminorms via projection error asymptotics.

Contribution

It extends Sweldens and Piessens's result from polynomial splines to box-splines, linking projection errors to Sobolev space seminorms.

Findings

Generalization of Bernoulli polynomial relation to box-splines

New characterization of Sobolev seminorms via projection error asymptotics

Potential applications in approximation theory and numerical analysis

Abstract

Let $P$ be orthogonal projection on B-splines of degree $r-1$ with equally spaced knots. Sweldens and Piessens proved that $P(x^r)-x^r$ is Bernoulli polynomial. We generalize Sweldens ans Piessens's result for box-splines. It gives the opportunity to define the seminorm of Sobolev space in terms of the asymptotic formula for the error in orthogonal projection.

Equations

$$\Vert f{\Vert }_{k,p}=\sum _{|\beta |\le k}\Vert D^{\beta }f{\Vert }_p,$$

$$D^{\beta }f=\frac{{\partial }^{|\beta |}f}{\partial x_1^{{\beta }_1}\cdots \partial x_d^{{\beta }_d}},\beta =({\beta }_1,\cdots ,{\beta }_d),$$

$$\beta !={\beta }_1!\cdot ...\cdot {\beta }_d!|\beta |={\beta }_1+\cdots +{\beta }_d.$$

$$\Vert f{\Vert }_p={\left({\int }_{{\mathbb{R}}^d}|f{|}^p\right)}^{1/p}.$$

$$\mathrm{span}\{V\}={\mathbb{R}}^d.$$

$${\int }_{{\mathbb{R}}^d}f(x)B_V(x)dx={\int }_{[0,1{]}^n}f(Vu)du$$

$$Vu=u_1{v}_1+\cdots u_n{v}_n.$$

$$\hat{f}(\xi )={\int }_{{\mathbb{R}}^d}f(t)e^{-2\pi i\xi \cdot t}dt.$$

$${\hat{B}}_V(x)=\prod _{v\in V}g(x\cdot v),$$

$$g(t)=\frac{1-e^{-2\pi it}}{2\pi it}.$$

$${\varrho }_V=\max \{r:\mathrm{span}\{V\setminus W\}={\mathbb{R}}^{d}for\; allW\subset V,\#W=r,\}.$$

$$B_V(\cdot )\in C^{{\varrho }_V-1}({\mathbb{R}}^d)\setminus C^{{\varrho }_V}({\mathbb{R}}^d).$$

$$S_{L^2}(hV)=\overline{span}\{B_V(\cdot /h-\alpha ):\alpha \in {\mathbb{Z}}^d\},$$

$$P_h={\sigma }_h\circ P\circ {\sigma }_{1/h},$$

$${\sigma }_{h}f(x)=f(x/h).$$

$$[{]}^{\beta }(x)=x^{\beta }$$

$$\gamma \le \beta \mathrm{iff}{\gamma }_j\le {\beta }_j,j=1,\dots ,d.$$

$$L_{\beta }(x)=P([{]}^{\beta })(x)-x^{\beta }$$

$$L_{\beta }(x)={\left(\frac{1}{2\pi i}\right)}^{|\beta |}\sum _{\alpha \in {\mathbb{Z}}^d,\alpha \ne 0}{D}^{\beta }\widehat{B_V}(\alpha )e^{2\pi i\alpha \cdot x}.$$

$$\Lambda =\{U\subset V:\#U={\varrho }_V+1,\mathrm{span}\{V\setminus U\}\ne {\mathbb{R}}^d\}.$$

$$v\cdot \alpha =0$$

$$v\cdot \alpha \ne 0\mathrm{for}\mathrm{all}v\in U.$$

$$B(V,U)(x)=\sum _{\alpha \ne 0,\alpha \perp (V\setminus U)}\prod _{v\in U}\frac{1}{2\pi i\alpha \cdot v}{e}^{2\pi i\alpha \cdot x},$$

$$U_{\alpha }=\{v\in V:\alpha \cdot v\ne 0\}$$

$$D^{\beta }\widehat{B_V}(x)=\sum _{\gamma \le \beta }\left(\genfrac{}{}{0ex}{}{\beta }{\gamma }\right)D^{\gamma }\prod _{v\in U_{\alpha }}g(x\cdot v)D^{\beta -\gamma }\prod _{v\in V\setminus U_{\alpha }}g(x\cdot v)$$

$$D^{\gamma}\prod_{v\in U_{\alpha}}g(x\cdot v)\Big{|}_{x=\alpha}=0.$$

$$D^{\beta}\widehat{B_{V}}(\alpha)=D^{\beta}\prod_{v\in U_{\alpha}}g(x\cdot v)\Big{|}_{x=\alpha}\neq 0,$$

$$L_{\beta }(x)=P([{]}^{\beta })(x)-x^{\beta }=\sum _{U\in \Lambda }C(\beta ,U)B(V,U)(x)$$

$$C(\beta ,U)=D^{\beta }(\prod _{v\in U}(x\cdot v))$$

$$L_{\beta }(x)={\left(\frac{1}{2\pi i}\right)}^{{\varrho }_V+1}\sum _{U\in \Lambda }\sum _{\alpha \ne 0,\alpha \perp (V\setminus U)}{D}^{\beta }\widehat{B_V}(\alpha )e^{2\pi i\alpha \cdot x}.$$

Full text

Box-splines orthogonal

projections

M. Beśka, K. Dziedziul

Abstract. Let $P$ be orthogonal projection on B-splines of degree $r-1$ with equally spaced knots. Sweldens and Piessens proved that $P(x^{r})-x^{r}$ is Bernoulli polynomial. We generalize Sweldens ans Piessens’s result for box-splines. It gives the opportunity to define the seminorm of Sobolev space in terms of the asymptotic formula for the error in orthogonal projection. 41A15, 41A35, 41A60. Keywords: box spline, Bernoulli spline, asymptotic formula, orthogonal projection, function of polynomials.

1. Introduction

In this year (2018) a beautiful paper [10] on box-spline was published. So I would like to present some of the results from a paper published in East Journal of Approximation EJA, vol 10 (4) 2004, since EJA is rather difficult to reach. I hope that our results will be interesting.

The second part of the paper published in EJA is devoted to sets with finite perimeter through perspective of Botschkarev’s theorem and this research was never entirely finished.

We start by recalling some basic facts and terminology.

Let $W^{k}_{p}(\mathbb{R}^{d})$ denote the Sobolev spaces, $1\leq p<\infty$ with the norm

[TABLE]

where

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[TABLE]

and

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Let $V=\{v_{1},v_{2},\cdots,v_{n}\}$ denote a set of not necessarily distinct, non zero vectors in $\mathbb{Z}^{d}\setminus\{0\}$, such that

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We call such set admissible. The box spline denoted by $B_{V}(\cdot)$ corresponding to $V$ is defined by requiring that

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holds for any continuous function $f$ on $\mathbb{R}^{d}$, see reference [5]. As usual

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The Fourier transform is given by

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Here and subsequently ”$\cdot$” denotes the scalar product in $\mathbb{R}^{d}$. From (1) by simple calculation we get that

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where

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We denote by $\#V$ the cardinality of the set $V$. For an admissible set $V$ let

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This parameter determines the smoothness of a box splines

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Let us define

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where $h>0$ and the closure is taken in $L^{2}(\mathbb{R}^{d})$. The orthogonal projection from $L^{2}(\mathbb{R}^{d})$ onto $S_{L^{2}}(hV)$ is denoted by $P_{h}$. Denoting by $(\cdot,\cdot)$ the inner product in $L^{2}(\mathbb{R}^{d})$, the orthogonal projection onto $S_{L^{2}}(hV)$ can be written by $(P=P_{1})$:

[TABLE]

where

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A family $V\subset\mathbb{Z}^{d}$ is unimodular if for all $W\subset V$ with $\#W=d$ we have $|\det W|\leq 1$. Set

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and

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2. Box-spline orthogonal projections

Let us define

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The above definition is a little misleading. We simply can also enlarge a definition of $P$ for a class of functions with polynomial growth. Note that in the univariate case $L_{\beta}$ is a Bernoulli spline for $|\beta|=\varrho_{V}+1$, see [14]. In [8] it was proved in a particular case that $L_{\beta}$ is linear combination of Bernoulli splines. In this section we generalize this results, see Theorem 2.4 below. Applying this result we simplify the asymptotic formula for orthogonal projection calculated in [4]. Our method in the case of $L^{2}$ implies Theorem 2.2 of [2].

We know that $L_{\beta}$ is a periodic piecewise polynomial and from Lemma 3.4 in [8] we have:

Lemma 2.1**.**

Let $|\beta|\leq\varrho_{V}+1$. Then

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The series converges in every point of continuity of $L_{\beta}$.

In fact the problem of the convergence appears only for box splines with $\varrho_{V}=0$ and on the boundary of the support of that box-splines. By Theorem 2.6 and Remark 2.9 we write $L_{\beta}$ for $|\beta|=1$ as a linear combination of a Bernoulli spline $B^{1}=x-1/2$, where the Fourier series of $B^{1}$ converges also in the point of discontinuity of $B^{1}$ (i.e $x=0$) to zero.

Define a set $\Lambda$,

[TABLE]

Let $U\in\Lambda$. If for all $v\in V\setminus U$

[TABLE]

we will denote that $\alpha\bot(V\setminus U)$. Note that the vectors from $V\setminus U$ span a hyperplane in $\mathbb{R}^{d}$ i.e. $d-1$-dimensional subspace. From definition of the set $\Lambda$ we get that for all $\alpha\neq 0$ such that $\alpha\bot(V\setminus U)$

[TABLE]

Definition 2.1**.**

Let us define Bernoulli splines [11] for $U\in\Lambda$ by

[TABLE]

$\alpha\in\mathbb{Z}^{d}$ and $x\in\mathbb{R}^{d}$.

Lemma 2.2**.**

Let $V$ be unimodular and let $\alpha\in\mathbb{Z}^{d}\setminus\{0\}$. Let $D^{\beta}\widehat{B_{V}}(\alpha)\neq 0$ for given $|\beta|=\varrho_{V}+1$. Then if $\#U_{\alpha}\geq\varrho_{V}+1$ where

[TABLE]

then $\#U_{\alpha}=\varrho_{V}+1$.

Proof.

Note that by (2)

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Since $\#U_{\alpha}\geq|\beta|=\varrho_{V}+1$ then for $\gamma<\beta$

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since $v\cdot\alpha\neq 0$ for $v\in U_{\alpha}$ and $g(v\cdot\alpha)=0$. Since $g(0)=1$, (9) shows that

[TABLE]

which implies theorem. ∎

Lemma 2.3**.**

Let $V$ be unimodular and let $\alpha\in\mathbb{Z}^{d}\setminus\{0\}$. Let $D^{\beta}\widehat{B_{V}}(\alpha)\neq 0$ for given $|\beta|=\varrho_{V}+1$. Then $U_{\alpha}\in\Lambda$.

Proof.

Note that $U_{\alpha}\neq\emptyset$ since $V$ spans $\mathbb{R}^{d}$. If $\#U_{\alpha}\geq\varrho_{V}+1$ then from Lemma 1.4 we get that $\#U_{\alpha}=\varrho_{V}+1$. Moreover $\alpha\bot(V\setminus U_{\alpha})$, hence $span\{V\setminus U_{\alpha}\}\neq\mathbb{R}^{d}$, it follows that $U_{\alpha}\in\Lambda$.

Let us assume that $\#U_{\alpha}<\varrho_{V}+1$. But $\alpha\bot(V\setminus U_{\alpha})$, hence $span\{V\setminus U_{\alpha}\}\neq\mathbb{R}^{d}$. This is a contradiction with definition of $\varrho_{V}$ see (3). ∎

Theorem 2.4**.**

Let $V$ be unimodular. Then $L_{\beta}$ is a linear combination of Bernoulli splines

[TABLE]

where

[TABLE]

Proof.

By Lemma 2.1 and Lemma 2.5 we get

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By (10) we get that

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Note that

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Consequently

[TABLE]

∎

Let us recall the results form [4] and [8].

Theorem 2.5**.**

Let $1\leq p<\infty$. Let $V$ be unimodular. Let $f\in W_{p}^{\varrho_{V}+1}(\mathbb{R}^{d})$. Then

[TABLE]

[TABLE]

Now we want to examine the right part of (13).

Theorem 2.6**.**

Let $V$ be unimodular then

[TABLE]

where

[TABLE]

and $D_{v}$ is the directional derivative.

Proof.

From Lemma 2.1 and 2.5 we get

[TABLE]

[TABLE]

[TABLE]

Note that the sets $V\setminus U$, where $U\in\Lambda$ are disjoint. Tedious calculation shows that

[TABLE]

Consequently using (12), (11) and (14) we get the theorem. ∎

Remark 2.1*.*

Let $V$ be unimodular. Then

1. the functions $B(V,U)$, $U\in\Lambda$ are orthogonal.

2. Since all norms in a finite dimension space are equivalent $"\asymp"$ we get

[TABLE]

[TABLE]

[TABLE]

For $p=2$ we have equality and we obtain the Theorem 2.2 [2], com. [7].

Remark 2.2*.*

Note also that for all $U\in\Lambda$ there is a vector $\alpha_{U}\in\mathbb{Z}^{d}\setminus\{0\}$ such that

[TABLE]

Thus

[TABLE]

where $B^{k}$ is Bernoulli polynomial

[TABLE]

Consequently changing the variable we get

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[TABLE]