Optimal subspaces for mean square approximation of classes of differentiable functions with boundary conditions

TL;DR

This paper identifies optimal subspaces for $L_2$ approximation of certain Sobolev space functions with boundary conditions, using equidistant shifts of a single function, including spline spaces of various degrees.

Contribution

It introduces a method to determine optimal subspaces generated by shifts of a single function for approximating Sobolev space functions with boundary conditions.

Findings

Optimal subspaces are generated by equidistant shifts of a single function.

Spline spaces of all degrees $d \,\geqslant\, r-1$ are optimal.

The results apply to functions with boundary conditions in Sobolev spaces.

Abstract

In this paper, we specify a set of optimal subspaces for $L_2$ approximation of three classes of functions in the Sobolev space $W^{(r)}_2$, defined on a segment and subject to certain boundary conditions. All of these subspaces are generated by equidistant shifts of a single function. In particular, we indicate optimal spline spaces of all degrees $d\geqslant r-1$ with uniform knots.

Equations

$$\Vert f{\Vert }_p={\left({\int }_{-\pi }^{\pi }|f{|}^p\right)}^{1/p},\Vert f{\Vert }_{L_p[a,b]}={\left({\int }_a^b|f{|}^p\right)}^{1/p},$$

$$E(f,\mathfrak{N}{)}_p=\underset{T\in \mathfrak{N}}{\inf }\Vert f-T{\Vert }_p$$

$$c_k(f)=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}f(t)e^{-ikt}dt,\widehat{{\beta }_l}=\sum _{k=0}^{2n-1}{\beta }_k{e}^{-\frac{ilk\pi }{n}}.$$

$$f^e=\frac{f+f(-\cdot )}{2},f^o=\frac{f-f(-\cdot )}{2}.$$

$$d_n(A;X)=\underset{X_n}{\inf }\underset{x\in A}{\sup }\underset{y\in X_n}{\inf }\Vert x-y{\Vert }_X,$$

$$H_0^r$$

$$H_1^r$$

$$H_2^r$$

$$A_i^r$$

$$A_2^r$$

$$d_n(A_0^r)=\frac{1}{(n+1{)}^r},d_n(A_1^r)=\frac{1}{n^r},d_n(A_2^r)=\frac{1}{(2n+1{)}^r},$$

$$\mathrm{span}\{x\mapsto \sin kx{\}}_{k=1}^n,\mathrm{span}\{x\mapsto \cos kx{\}}_{k=0}^{n-1},\mathrm{span}\{x\mapsto \sin (2k-1)x{\}}_{k=1}^n$$

$$S_{d,0}$$

$$S_{d,1}$$

$$S_{d,2}$$

$${\tau }_0$$

$${\tau }_1$$

$${\tau }_2$$

$$s(x)=\sum _{j=0}^{2n-1}{\beta }_{j}B\left(x-\frac{j\pi }{n}\right),$$

$$\sum _{j=0}^{2n-1}(-1{)}^j{\beta }_j=0.$$

$$s(x)\sim \sum _{j=0}^{2n-1}{\beta }_j\sum _{l\in \mathbb{Z}}{c}_l(B)e^{il\left(x-\frac{j\pi }{n}\right)}=\sum _{l\in \mathbb{Z}}{c}_l(B){\hat{\beta }}_l{e}^{ilx}\sim \sum _{l=0}^{2n-1}{\hat{\beta }}_l{\Phi }_{B,l}(x),$$

$${\Phi }_{B,l}(x)={\Phi }_{B,n,l}(x)=\frac{1}{2n}\sum _{j=0}^{2n-1}{e}^{\frac{ilj\pi }{n}}B\left(x-\frac{j\pi }{n}\right)\sim \sum _{\nu \in \mathbb{Z}}{c}_{l+2n\nu }(B)e^{i(l+2n\nu )x}.$$

$$\frac{1}{2\pi }\Vert {\Phi }_{B,l}{\Vert }_2^2=D_{B,l}=D_{B,n,l}=\sum _{\nu \in \mathbb{Z}}|c_{l+2n\nu }(B){|}^2.$$

$$D_{n-1}(t)=\sum _{k=1-n}^{n-1}{e}^{ikt},$$

$$B_{n,\mu }(t)=\sum _{k\in \mathbb{Z}}{\left(\frac{e^{i\frac{\pi }{n}k}-1}{i\frac{\pi }{n}k}\right)}^{\mu +1}{e}^{ikt},\mu \in {\mathbb{Z}}_{+}$$

$${\Phi }_{B_{n,\mu },l}(x)={\left(\frac{e^{i\frac{\pi }{n}l}-1}{i\frac{\pi }{n}}\right)}^{\mu +1}\sum _{\nu \in \mathbb{Z}}\frac{e^{i(l+2n\nu )x}}{(l+2n\nu {)}^{\mu +1}},$$

$$c_{-l-2nk}(B)={\gamma }_l{c}_{l+2nk}(B)\text{for all k∈Z.}$$

$${\Phi }_{B,-l}(-x)=\sum _{j=1-m}^{m-1}{\gamma }_j{\Phi }_{B,j}(x)$$

$${\Phi }_{B,-l}(-x)={\gamma }_l{\Phi }_{B,l}(x).$$

$${\mathbb{S}}_{B,n,m}^{\times }=\mathrm{span}\{{\Phi }_{B,0}\}\oplus \mathrm{span}\{{\Phi }_{B,l}^e{\}}_{l=1}^{m-1}\oplus \mathrm{span}\{{\Phi }_{B,l}^o{\}}_{l=1}^{m-1}.$$

$${\Phi }_{B,l}^e(\pi -x)=(-1{)}^l{\Phi }_{B,l}^e(x),{\Phi }_{B,l}^o(\pi -x)=(-1{)}^{l+1}{\Phi }_{B,l}^o(x).$$

$$E\bigl{(}f,\mathbb{S}^{\times}_{B,n,m}\bigr{)}_{2}\leqslant\frac{1}{m^{r}}\|f^{(r)}\|_{2}.$$

$$c_l(B)\ne 0$$

$$c_{2n\nu }(B)=0$$

Full text

This work is supported by the Russian Science Foundation under grant No. 18-11-00055.

1. Introduction

1.1. Notation

In what follows, $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Z}_{+}$, $\mathbb{N}$ are the sets of complex, real, integer, nonnegative integer, and natural numbers, respectively; $[a:b]=[a,b]\cap\mathbb{Z}$. Unless otherwise follows from the context, all the functional spaces under consideration can be real or complex. If $p\in[1,+\infty)$, $L_{p}$ is the space of $2\pi$-periodic, measurable, and $p$-integrable functions; $L_{p}[a,b]$ is the space of measurable on $[a,b]$, $p$-integrable functions. The norms in these spaces are defined by

[TABLE]

respectively. Furthermore, $W^{(r)}_{2}[a,b]$ is the space of functions $f$ in $L_{2}[a,b]$ such that $f^{(r-1)}$ is absolutely continuous and $f^{(r)}\in L_{2}[a,b]$; the class $W^{(r)}_{2}$ of periodic functions is defined similarly.

The symbol $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ denotes the inner product in the Hilbert space $\mathcal{H}$;

[TABLE]

is best approximation of $f$ in $L_{p}$ by the set $\mathfrak{N}\subset L_{p}$.

The Fourier coefficients of the function $f$ and the discrete Fourier transform of the finite sequence $\{\beta_{k}\}_{k=0}^{2n-1}$ are defined by the equalities

[TABLE]

The notation $f(x)\sim\sum\limits_{k\in\mathbb{Z}}c_{k}e^{ikx}$ means that the series on the right-hand side is the Fourier series of $f$.

For $n\in\mathbb{N}$ and $\mu\in\mathbb{Z}_{+}$, let $\mathbf{S}_{n,\mu}$ denote the $2n$-dimensional space of $2\pi$-periodic splines of degree $\mu$ and defect $1$ with knots at the points $\dfrac{k\pi}{n}$, $k\in\mathbb{Z}$; $\mathcal{T}_{2n-1}$ is the $(2n-1)$-dimensional space of trigonometric polynomials of degree at most $n-1$.

The symbols $f^{e}$ and $f^{o}$ denote the even and the odd parts of $f$, respectively, i.e.

[TABLE]

Let $X$ be a normed linear space and $A$ a subset of $X$. The Kolmogorov $n$-width of $A$ in $X$ is given by

[TABLE]

where the external lower bound is taken over all subspaces $X_{n}$ of the space $X$, whose dimension does not exceed $n$.

1.2. An overview of the results

In [1], Floater and Sande studied the $L_{2}$ approximation of three classes of functions in $W^{(r)}_{2}[0,1]$, defined by certain boundary conditions. With slightly different scaling (which we hereafter adhere to) these classes are given by

[TABLE]

The authors considered the function classes

[TABLE]

and described the $n$-widths and certain optimal subspaces for $A^{r}_{i}$. Specifically, they showed that

[TABLE]

and the spaces

[TABLE]

are optimal $n$-dimensional spaces for, respectively, $A^{r}_{0}$, $A^{r}_{1}$, and $A^{r}_{2}$. The result for $A^{1}_{1}$ was proved by Kolmogorov [4]. In addition, the authors proved that the spaces $A^{r}_{i}$ admit optimal spline subspaces, which are defined as follows.

Let $\tau=(\tau_{1},\ldots,\tau_{m})$ be a knot vector such that $0<\tau_{1}<\ldots<\tau_{m}<P_{i}$, where $P_{0}=P_{1}=\pi$ and $P_{2}=\pi/2$. Denote by $S_{d,\tau,i}$ the space of splines of degree $d$ and defect $1$ on $[0,P_{i}]$ and consider its $n$-dimensional subspaces defined by

[TABLE]

where the knot vectors $\tau_{i}$ for $i=0,1,2$ are given as

[TABLE]

It was proved in [1] that for any $d\geqslant r-1$ the spline spaces $S_{d,i}$ are optimal $n$-dimensional spaces for the set $A^{r}_{i}$, $i=0,1,2$.

As we can see, all the spaces $S_{d,i}$ have equidistant knots, but the form of the knot vector is determined by the degree $d$. In this paper, we show that the classes $A^{r}_{i}$ admit optimal spline subspaces with both types of knots indicated in the definition of $\tau_{i}$ independently of the degree. Of course, boundary conditions should be relaxed or some extra conditions should be added in the remaining cases to ensure the dimension to equal $n$.

Our technique consists in the following. In [8], we studied the periodic case and found a wide set of optimal subspaces generated by shifts of a single function, including spline spaces. The conditions of optimality in the periodic case were formulated in terms of Fourier coefficients of a function mentioned. Now we reduce the problems for functions defined on a segment to similar problems for periodic functions and, using the results of [8], specify a set of optimal subspaces in the nonperiodic situation.

2. Spaces of shifts

For $n\in\mathbb{N}$ and $B\in L_{1}$, let $\mathbb{S}_{B,n}$ be the space of functions $s$ defined on $\mathbb{R}$ and representable in the form

[TABLE]

and $\mathbb{S}^{\times}_{B,n}$ be the space of functions in $\mathbb{S}_{B,n}$ that can be represented as (2) with the additional condition

[TABLE]

Substituting the Fourier series expansion of $B$ into (2), we obtain

[TABLE]

where

[TABLE]

Clearly, $\Phi_{B,l}=\Phi_{B,l+2n}$ and condition (3) is equivalent to $\widehat{\beta}_{n}=0$. Thus, the spaces $\mathbb{S}_{B,n}$ and $\mathbb{S}^{\times}_{B,n}$ coincide with linear spans of the sets $\{\Phi_{B,l}\}_{l=0}^{2n-1}$ (or, equivalently, $\{\Phi_{B,l}\}_{l=1-n}^{n}$) and $\{\Phi_{B,l}\}_{l=1-n}^{n-1}$. For $m\in[1:n]$, denote by $\mathbb{S}^{\times}_{B,n,m}$ the linear span of the set $\{\Phi_{B,l}\}_{l=1-m}^{m-1}$.

Note that the functions $\Phi_{B,l}$ are orthogonal: $\langle\Phi_{B,l},\Phi_{B,j}\rangle_{L_{2}}=0$ for $l\neq j$ and

[TABLE]

The linear independence of the sets $\{B\left(\cdot-\frac{j\pi}{n}\right)\}_{j=0}^{2n-1}$ and $\{B\left(\cdot-\frac{j\pi}{n}\right)\}_{j=1-n}^{n-1}$ is equivalent to the fact that the functions $\Phi_{B,l}$ are nonzero for $l\in[1-n:n]$ and $l\in[1-n:n-1]$, respectively. In this case, the systems $\{\Phi_{B,l}\}_{l=1-n}^{n}$ and $\{\Phi_{B,l}\}_{l=1-n}^{n-1}$ form orthogonal bases in the spaces $\mathbb{S}_{B,n}$ and $\mathbb{S}^{\times}_{B,n}$. Orthonormal bases are constituted by the functions $\frac{1}{\sqrt{2\pi D_{B,l}}}\Phi_{B,l}$.

When $B$ is the Dirichlet kernel

[TABLE]

we have $\mathbb{S}_{B,n}=\mathbb{S}^{\times}_{B,n}=\mathcal{T}_{2n-1}$, $\mathbb{S}^{\times}_{B,n,m}=\mathcal{T}_{2m-1}$, $\Phi_{B,n}=0$, and for $|l|<n$, the functions $\Phi_{B,l}$ are ordinary exponents.

If $B$ is the $B$-spline

[TABLE]

(henceforth, the fraction is assumed to equal $1$ whenever $k=0$), we find that $\mathbb{S}_{B,n}$ is the space of splines $\mathbf{S}_{n,\mu}$. The functions

[TABLE]

which form an orthogonal basis in this space, are called exponential splines (by convention, we have $\Phi_{B_{n,\mu},0}(x)=1$). The linear span of the system $\{\Phi_{B_{n,\mu},l}\}_{l=1-n}^{n-1}$ is denoted by $\mathbf{S}^{\times}_{n,\mu}$.

In more general nonperiodic situation exponential splines were introduced by Schoenberg; the basics of the theory and historical remarks can be found in [5]. The orthogonality of periodic exponential splines was noted by many authors; apparently, the earliest studies on this topic were [2, 3]. The spaces $\mathbf{S}^{\times}_{n,\mu}$ and $\mathbb{S}^{\times}_{B,n}$ were considered by Vinogradov [6, 7].

The following lemma describes symmetry properties of spaces of shifts in terms of Fourier coefficients.

Lemma 1**.**

Let $n,m\in\mathbb{N}$, $m\leqslant n$, and $B\in L_{1}$. Then the following statements are equivalent.

1. (1)

The inclusion $s\in\mathbb{S}^{\times}_{B,n,m}$ implies $s(-\cdot)\in\mathbb{S}^{\times}_{B,n,m}$. 2. (2)

For any $l\in[0:m-1]$ there exists $\gamma_{l}\in\mathbb{C}\setminus\{0\}$ such that $\gamma_{0}\in\{-1,1\}$ and

[TABLE]

Moreover, if $\gamma_{0}=-1$, then $c_{0}(B)=0$.

Proof.

The first statement means that $\Phi_{B,l}(-\cdot)\in\mathbb{S}^{\times}_{B,n,m}$ for all $l\in[1-m:m-1]$. Replacing $l$ with $-l$ for convenience, rewrite the inclusion as

[TABLE]

for some $\gamma_{j}$. Since $\Phi_{B,j}$ is orthogonal to $\Phi_{B,-l}(-\cdot)$ for $j\neq l$, we have

[TABLE]

Equating the Fourier coefficients, we get (4). Replacing $l$ with $-l$ and $k$ with $-k$, we also have $c_{l+2nk}(B)=\gamma_{-l}c_{-l-2nk}(B)$. If $\gamma_{l}=0$ for some $l$, then $c_{l+2nk}(B)=c_{-l-2nk}(B)=0$ and the equalities are trivially satisfied with an arbitrary $\gamma_{l}$. So, we can take $\gamma_{l}\neq 0$.

On the other hand, if (4) is valid for a number $l$ with $\gamma_{l}\neq 0$, it is also valid for a number $-l$ with $\frac{1}{\gamma_{l}}$. So, it is sufficient to consider $l\in[\,0:m-1]$.

Putting $l=0$, we conclude that $c_{-2nk}(B)=\gamma_{0}c_{2nk}(B)$ for all $k$. Replacing $k$ by $-k$, we also have $c_{2nk}(B)=\gamma_{0}c_{-2nk}(B)$. If $c_{2nk}(B)=0$ for all $k$, we can take $\gamma_{0}=1$. If $c_{2nk}(B)\neq 0$ for some $k$, then $c_{-2nk}(B)\neq 0$ for the same $k$, and so $\gamma_{0}=\pm 1$. Obviously, $\gamma_{0}=-1$ implies $c_{0}(B)=0$. ∎

Note that all even functions (and, in particular, the Dirichlet kernel) satisfy the second condition of Lemma 1 with $\gamma_{l}=1$ for all $l$.

For the $B$-spline, we have $\gamma_{l}=e^{-i\frac{\pi}{n}l(\mu+1)}$, and for the shifted $B$-spline \widetilde{B}_{n,\mu}={B}_{n,\mu}\bigl{(}\cdot-\frac{\pi}{2n}\bigr{)}, the identity $c_{k}(\widetilde{B}_{n,\mu})=e^{-\frac{ik\pi}{2n}}c_{k}(B_{n,\mu})$ yields $\gamma_{l}=e^{-\frac{il\pi\mu}{n}}$.

Remark 1**.**

For $l\in[1:m-1]$, we have $\Phi^{e}_{B,-l}=\gamma_{l}\Phi^{e}_{B,l}$ and $\Phi^{o}_{B,-l}=-\gamma_{l}\Phi^{o}_{B,l}$. Therefore, the space $\mathbb{S}^{\times}_{B,n,m}$ can be represented as

[TABLE]

Remark 2**.**

If $s\in\mathbb{S}^{\times}_{B,n,m}$, then $s(\cdot+\pi)\in\mathbb{S}^{\times}_{B,n,m}$ and, since $\Phi_{B,l}(x+\pi)=(-1)^{l}\Phi_{B,l}(x)$, we have

[TABLE]

3. Main results

The following theorem was established in [8, Theorem 1].

Theorem 1**.**

Let $r,n,m\in\mathbb{N}$, $m\leqslant n$, and $B\in L_{2}$. Then the following statements are equivalent.

1. (1)

For any function $f\in W^{(r)}_{2}$, the following inequality holds:

[TABLE] 2. (2)

The Fourier coefficients of $B$ satisfy the conditions

[TABLE]

We need the following simple corollary of Theorem 1.

Corollary 1**.**

Let $r,n,m\in\mathbb{N}$, $m\leqslant n$, and $B\in L_{2}$. Then the following statements are equivalent.

1. (1)

For any function $f\in W^{(r)}_{2}$ such that $c_{0}(f)=0$, the following inequality holds:

[TABLE] 2. (2)

For all $|l|\in[1:m-1]$, we have $c_{l}(B)\neq 0$ and

[TABLE]

Proof.

For $g\in L_{2}$, put $g_{0}(x)=\sum\limits_{\nu\in\mathbb{Z}}c_{2n\nu}(g)e^{i2n\nu x}$. Obviously, the inequality (6) on the whole class $W^{(r)}_{2}$ is equivalent to the system

[TABLE]

[TABLE]

$(1)\implies(2)$. Let $\widetilde{B}\in L_{2}$ be such a function that $c_{0}(\widetilde{B})=1$, $c_{2n\nu}(\widetilde{B})=0$ for all $\nu\in\mathbb{Z}\setminus\{0\}$, and $c_{k}(\widetilde{B})=c_{k}(B)$ for $k\neq 2n\nu$. For any $f\in W^{(r)}_{2}$ we have

[TABLE]

Here (10) is trivial because $\mathbb{S}^{\times}_{\widetilde{B}_{0},n,m}$ contains constants, while (11) holds due to assumption. So, (6) is valid. By Theorem 1, the Fourier coefficients of $\widetilde{B}$ satisfy its second assertion, so the same holds for $c_{k}(B)$ when $k\neq 2n\nu$.

$(2)\implies(1)$. Let $f\in W^{(r)}_{2}$, $c_{0}(f)=0$. Define $\widetilde{B}$ as above. Then, by Theorem 1, (6) holds for $\widetilde{B}$. By (10) and (11), it also holds for $B$. ∎

Consider the following functional classes:

[TABLE]

Evidently, every function $u\in\widetilde{H}^{r}_{i}$ belongs to $H^{r}_{i}$. Conversely, according to the boundary conditions in the definition of the classes $H_{i}^{r}$, the $2\pi$-periodization of the odd extension of $u\in H_{0}^{r}$ to the interval $[-\pi,0]$ belongs to $\widetilde{H}_{0}^{r}$. Similarly, the $2\pi$-periodization of the even extension of $u\in H_{1}^{r}$ to the interval $[-\pi,0]$ is in $\widetilde{H}_{1}^{r}$. Consecutively extending $u\in H_{2}^{r}$ to an even (with respect to $\pi/2$) function on $[0,\pi]$ and to an odd function on $[-\pi,\pi]$, after $2\pi$-periodization we get a function belonging to $\widetilde{H}_{2}^{r}$.

Therefore, putting

[TABLE]

we derive

[TABLE]

Thus, we can reduce problems for nonperiodic classes to those for periodic classes and apply Theorem 1. We will formulate our results for periodic classes (denoted with tildes).

Remark 3**.**

Let $\mathsf{S}$ be a closed subspace of $L_{2}$ such that the condition $s\in\mathsf{S}$ implies $s(-\cdot)\in\mathsf{S}$. Then an element of best approximation of any function $u\in\widetilde{H}^{r}_{0}$ in $L_{2}$ by the space $\mathsf{S}$ is odd. Indeed, if $\|u-s\|_{2}=\inf\limits_{T\in\mathsf{S}}\|f-T\|_{2}$, we can write

[TABLE]

This means that all the inequalities in this chain turn into equalities. In particular, we have $\|u-s\|_{2}=\|u-s^{o}\|_{2}$. By the uniqueness of an element of best approximation in $L_{2}$, we conclude that $s$ is odd.

For the same reason, an element of best approximation of any function $u\in\widetilde{H}^{r}_{1}$ by the space $\mathsf{S}$ is even. If, in addition, the space $\mathsf{S}$ is invariant under the shift by $\pi$, an element of best approximation of $u\in\widetilde{H}^{r}_{2}$ by the space $\mathsf{S}$ possesses the same symmetry properties as the function $u$ itself.

Consider the following $m$-dimensional spaces:

[TABLE]

In the following three theorems we give sufficient conditions of extremality of these spaces.

Theorem 2**.**

Let $r,n,m\in\mathbb{N}$, $m+1\leqslant n$, and suppose that the Fourier coefficients of a function $B\in L_{2}$ satisfy the following conditions.

1. (1)

For any $l\in[1:m]$ there exists $\gamma_{l}\in\mathbb{C}\setminus\{0\}$ such that for all $k\in\mathbb{Z}$ $c_{-l-2nk}(B)=\gamma_{l}c_{l+2nk}(B)$. 2. (2)

For all $\nu\in\mathbb{N}$, $c_{2n\nu}(B)=c_{-2n\nu}(B)$. 3. (3)

For all $l\in[1:m]$, we have $c_{l}(B)\neq 0$ and

[TABLE]

Then for any $u\in\widetilde{H}_{0}^{r}$,

[TABLE]

Proof.

Since every $u\in\widetilde{H}_{0}^{r}$ has zero mean and $B$ satisfies the conditions of the second proposition of Corollary 1, we can write

[TABLE]

By Lemma 1, the space $\mathbb{S}^{\times}_{B,n,m+1}$ contains each function $s$ with its odd and even parts. Thus, an element of best approximation of $u\in\widetilde{H}_{0}^{r}$ by the space $\mathbb{S}^{\times}_{B,n,m+1}$ is odd. This impies that the space $\mathbb{S}^{\times}_{B,n,m+1}$ on the left-hand side of (13) can be substituted for its subspace consisting of odd functions. Since $\Phi_{B,0}$ is even (by condition 2), it follows from decomposition (5) that the desired approximating subspace coincides with $\widetilde{\mathcal{S}}^{0}_{B,n,m}$. ∎

Theorem 3**.**

Let $r,n,m\in\mathbb{N}$, $m\leqslant n$, and suppose that the Fourier coefficients of a function $B\in L_{2}$ satisfy the following conditions.

1. (1)

For any $l\in[1:m-1]$ there exists $\gamma_{l}\in\mathbb{C}\setminus\{0\}$ such that for all $k\in\mathbb{Z}$ $c_{-l-2nk}(B)=\gamma_{l}c_{l+2nk}(B)$. 2. (2)

For all $l\in[0:m-1]$, $c_{l}(B)\neq 0$. 3. (3)

For all $\nu\in\mathbb{Z}\setminus\{0\}$, $c_{2n\nu}(B)=0$. 4. (4)

For all $l\in[1:m-1]$, we have

[TABLE]

Then for any $u\in\widetilde{H}_{1}^{r}$,

[TABLE]

Proof.

Applying Theorem 1 to $u\in\widetilde{H}_{1}^{r}$, we can write

[TABLE]

Using the same argument as in the proof of Theorem 2, we conclude that the space $\mathbb{S}^{\times}_{B,n,m}$ on the left-hand side of the last inequality can be substituted for the subspace of even functions, i.e. for $\widetilde{\mathcal{S}}^{1}_{B,n,m}$. ∎

Theorem 4**.**

Let $r,n,m\in\mathbb{N}$, $2m+1\leqslant n$, and suppose that the Fourier coefficients of a function $B\in L_{2}$ satisfy the following conditions.

1. (1)

For any $l\in[1:2m]$ there exists $\gamma_{l}\in\mathbb{C}\setminus\{0\}$ such that for all $k\in\mathbb{Z}$ $c_{-l-2nk}(B)=\gamma_{l}c_{l+2nk}(B)$. 2. (2)

For all $\nu\in\mathbb{N}$, $c_{2n\nu}(B)=c_{-2n\nu}(B)$. 3. (3)

For all $l\in[1:2m]$, we have $c_{l}(B)\neq 0$ and

[TABLE]

Then for any $u\in\widetilde{H}_{2}^{r}$,

[TABLE]

Proof.

Because $\widetilde{H}_{2}^{r}$ is a subspace of $\widetilde{H}_{0}^{r}$, by Theorem 2, for every $u\in\widetilde{H}_{2}^{r}$, we have

[TABLE]

Since $u$ satisfies the equality $u=u(\pi-\cdot)$, we can restrict the space $\widetilde{\mathcal{S}}^{0}_{B,n,2m}$ to a subspace of functions possessing this property. By Remark 2, we obtain that the desired subspace coincides with $\widetilde{\mathcal{S}}^{2}_{B,n,m}$. ∎

Remark 4**.**

It follows easily that under the assumptions of Theorem 4 the space $\mathrm{span}\,\{\Phi^{e}_{B,2l-1}\}_{l=1}^{m}$ is extremal for the set defined by interchanging the roles of $k$ and $l$ in $H^{r}_{2}$ (or, equivalently, the symmetry conditions in $\widetilde{H}_{2}^{r}$).

Remark 5**.**

Note that the conditions of Theorems 2–4 are invariant under the shift of $B$ by $\frac{\pi}{2n}$.

Remark 6**.**

Inequalities (12), (14) and (15) turn into equalities for the functions $x\mapsto\sin(m+1)x$, $x\mapsto\cos mx$ and $x\mapsto\sin(2m+1)x$, respectively.

The estimates from Theorems 2–4 can be strengthened in a standard way by replacing their right-hand sides with best approximations.

Corollary 2**.**

Let $B\in W^{(r)}_{2}$ under the assumptions of Theorem 2.

1. (1)

If $r$ is even, then for any $u\in\widetilde{H}_{0}^{r}$,

[TABLE] 2. (2)

If $r$ is odd, then for any $u\in\widetilde{H}_{0}^{r}$,

[TABLE]

Proof.

If $r$ is even, denote by $s$ an element of best approximation of the function $u^{(r)}$ by the space $\widetilde{\mathcal{S}}^{0}_{B^{(r)},n,m}$. For odd $r$, let $s$ be an element of best approximation of $u^{(r)}$ by the space $\mathrm{span}\,\{\Phi^{e}_{B^{(r)},l}\}_{l=1}^{m}$. Since $c_{0}(s)=0$, the function $s$ has $2\pi$-periodic $r$th primitive, which we denote by $s_{r}$. For any $l\in[1:m]$,

[TABLE]

and hence $s_{r}\in\widetilde{\mathcal{S}}^{0}_{B,n,m}$. Applying Theorem 2 to the function $u-s_{r}$, we obtain

[TABLE]

∎

The proof of the two following statements goes exactly the same way as in Corollary 2 and therefore is omitted.

Corollary 3**.**

Let $B\in W^{(r)}_{2}$ under the assumptions of Theorem 3.

1. (1)

If $r$ is even, then for any $u\in\widetilde{H}_{1}^{r}$,

[TABLE] 2. (2)

If $r$ is odd, then for any $u\in\widetilde{H}_{1}^{r}$,

[TABLE]

Corollary 4**.**

Let $B\in W^{(r)}_{2}$ under the assumptions of Theorem 4.

1. (1)

If $r$ is even, then for any $u\in\widetilde{H}_{2}^{r}$,

[TABLE] 2. (2)

If $r$ is odd, then for any $u\in\widetilde{H}_{2}^{r}$,

[TABLE]

4. Examples

In [8, Theorem 2], we specified an easily verifiable condition that is sufficient for the fulfilment of inequality (9) (and, therefore, of the corresponding conditions of Theorems 2–4). Namely, inequality (9) holds for all functions $B$ possessing the property

[TABLE]

Among examples of functions $B$ satisfying this condition for all $m\leqslant n$ are the functions with coefficients of the form

[TABLE]

where $|\eta_{l+2nk}|\leqslant|\eta_{l}|$ and $\eta_{l}\neq 0$ for $|l|<n$. For $\eta_{k}=1$ we get the $B$-spline. If $\eta_{k}$ are the Fourier coefficients of the function $K\in L_{1}$, the function $B$ is the Steklov average of order $\mu+1$ of $K$. For example, $K$ can be the Poisson kernel ($\eta_{k}=e^{-\alpha|k|}$, $\alpha>0$), the heat kernel ($\eta_{k}=e^{-\alpha k^{2}}$, $\alpha>0$), the kernels of some differential operators ($\eta_{k}=\frac{1}{P(ik)}$, where $P$ is a polynomial with only real roots), and the generalized Bernoulli kernel ($\eta_{k}=|k|^{-s}e^{-i\beta\mathop{\mathrm{sign}}{k}}$, $s>0$, $\beta\in\mathbb{R}$); in the latter two examples, $\eta_{0}$ is assumed to equal $1$.

Taking the Dirichlet kernels of appropriate order as a function $B$ in Theorems 2–4, we get the optimal subspaces of trigonometric polynomials (1).

Now we describe spline spaces arising from Theorems 2–4 and show that the results of [1] follow from these theorems.

Recall that for a given space of periodic functions with appropriate symmetry conditions, we denote the space of their restrictions to $[\,0,\pi]$ or to $[\,0,\pi/2]$ by the same letter but without tilde.

1. Replace $n$ with $n+1$ in Theorem 2 and take $m=n$, $B=B_{n+1,d}$. Then our space $\widetilde{\mathcal{S}}^{0}_{B,n+1,n}$ is the $n$-dimensional space of odd splines from $\mathbb{S}^{\times}_{B,n+1}$. Consider the space $Q_{d,1}$ of splines $s$ which have knots $\left\{\frac{k\pi}{n+1}\right\}_{k=1}^{n}$ and satisfy the boundary conditions

[TABLE]

Its dimension equals $n$ for $d$ odd and equals $n+1$ for $d$ even. So, for $d$ odd we have $\mathcal{S}^{0}_{B,n+1,n}=Q_{d,1}=S_{d,0}$. For $d$ even, $\mathcal{S}^{0}_{B,n+1,n}$ is an $n$-dimensional subspace of $Q_{d,1}$.

2. Replace $n$ with $n+1$ in Theorem 2 and take $m=n$, B=B_{n+1,d}\bigl{(}\cdot-\frac{\pi}{2(n+1)}\bigr{)}. Then our space $\widetilde{\mathcal{S}}^{0}_{B,n+1,n}$ is the $n$-dimensional space of odd splines from $\mathbb{S}^{\times}_{B,n+1}$. Consider the space $Q_{d,2}$ of splines $s$ which have knots $\left\{\frac{k\pi}{n+1}+\frac{\pi}{2(n+1)}\right\}_{k=0}^{n}$ and satisfy the boundary conditions

[TABLE]

Its dimension equals $n$ for $d$ even and equals $n+1$ for $d$ odd. Note that [math] and $\pi$ are not the knots. So, for $d$ even we have $\mathcal{S}^{0}_{B,n+1,n}=Q_{d,2}=S_{d,0}$. For $d$ odd, $\mathcal{S}^{0}_{B,n+1,n}$ is an $n$-dimensional subspace of $Q_{d,2}$.

3. Take $m=n$, $B=B_{n,d}$ in Theorem 3. Then our space $\widetilde{\mathcal{S}}^{1}_{B,n,n}$ is the $n$-dimensional space of even splines from $\mathbb{S}^{\times}_{B,n}$. Consider the space $Q_{d,3}$ of splines $s$ which have knots $\left\{\frac{k\pi}{n}\right\}_{k=1}^{n-1}$ and satisfy the boundary conditions

[TABLE]

Its dimension equals $n$ for $d$ even and equals $n+1$ for $d$ odd. So, for $d$ even we have $\mathcal{S}^{1}_{B,n,n}=Q_{d,3}=S_{d,1}$. For $d$ odd, $\mathcal{S}^{1}_{B,n,n}$ is an $n$-dimensional subspace of $Q_{d,3}$.

4. Take $m=n$, B=B_{n,d}\bigl{(}\cdot-\frac{\pi}{2n}\bigr{)} in Theorem 3. Then our space $\widetilde{\mathcal{S}}^{1}_{B,n,n}$ is the $n$-dimensional space of even splines from $\mathbb{S}^{\times}_{B,n}$. Consider the space $Q_{d,4}$ of splines $s$ which have knots $\left\{\frac{k\pi}{n}+\frac{\pi}{2n}\right\}_{k=0}^{n-1}$ and satisfy the boundary conditions

[TABLE]

Its dimension equals $n$ for $d$ odd and equals $n+1$ for $d$ even. Note that [math] and $\pi$ are not the knots. So, for $d$ odd we have $\mathcal{S}^{1}_{B,n,n}=Q_{d,4}=S_{d,1}$. For $d$ even, $\mathcal{S}^{1}_{B,n,n}$ is an $n$-dimensional subspace of $Q_{d,4}$.

5. Replace $n$ with $2n+1$ in Theorem 4 and take $m=n$, $B=B_{2n+1,d}$. Consider the space $Q_{d,5}$ of splines $s$ which have knots $\left\{\frac{k\pi}{2n+1}\right\}_{k=1}^{n}$ and satisfy the boundary conditions

[TABLE]

Its dimension equals $n$ for $d$ odd and equals $n+1$ for $d$ even. So, for $d$ odd we have $\mathcal{S}^{2}_{B,2n+1,n}=Q_{d,5}=S_{d,2}$. For $d$ even, $\mathcal{S}^{2}_{B,2n+1,n}$ is an $n$-dimensional subspace of $Q_{d,5}$.

6. Replace $n$ with $2n+1$ in Theorem 4 and take $m=n$, B=B_{2n+1,d}\bigl{(}\cdot-\frac{\pi}{2(2n+1)}\bigr{)}. Consider the space $Q_{d,6}$ of splines $s$ which have knots $\left\{\frac{k\pi}{2n+1}+\frac{\pi}{2(2n+1)}\right\}_{k=0}^{n-1}$ and satisfy the boundary conditions

[TABLE]

Its dimension equals $n$ for $d$ even and equals $n+1$ for $d$ odd. So, for $d$ even we have $\mathcal{S}^{2}_{B,2n+1,n}=Q_{d,6}=S_{d,2}$. For $d$ odd, $\mathcal{S}^{2}_{B,2n+1,n}$ is an $n$-dimensional subspace of $Q_{d,6}$.

Taking other values of $m$ and $n$ and taking the $B$-spline (shifted or not) as a function $B$ in Theorems 2–4 we get new families of optimal spline subspaces with equidistant knots.