Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts
The paper is dedicated to obtain sharp inequalities of Favard-Akhiezer-Krein type for classes of periodic convolutions with kernels that do not increase oscillation. Let \(W_p^{(r)}\) denote the space of \(2\pi\)-periodic continuous functions \(f\) for which \(f^{(r-1)}\) is absolutely continuous and \(f^{(r)}\in L_p\). Let \({\mathcal T}_{2n-1}\) denote the space of trigonometric polynomials of order at most \(2n-1\). In 1937, Favard and, independently, Akhiezer and Krein constructed a linear approximation method \(X_{n,r}: L_1 \to {\mathcal T}_{2n-1}\) for \(f\in W_p^{(r)}\) (with \(p=\infty\)) such that \[ \|f-X_{n,r}(f)\|_p\leq \frac{{\mathcal K}_r}{n^r}\|f^{(r)}\|_{p}, \qquad p=\infty, \tag{1} \] where \({\mathcal K}_r\) are the in Approximation Theory celebrated Favard constants. This inequality is sharp. S. M. Nikol'skii later extended this result to the case \(p=1\). After that many mathematicians have been working on obtaining sharp results of this type for various classes of functions and various approximating methods. One of the natural ways to generalize this result is to consider classes of convolutions with certain kernels \(G\) (i.e., \(f=\varphi \star G\)) that do not increase oscillation. The author obtains sharp inequalities of the form (1) for approximations of classes of convolutions by subspaces constructed by uniform shifts of a single function. Conditions on the functions generating approximating subspaces are stated in terms of non increasing oscillations (see below for the precise conditions). Denote by \(A_{2n}(J)\) the set of real-valued functions \(G\in L_1\) satisfying the following condition on the number of sign changes \[ \nu(T+G\star \varphi) \leq 2n \] for any functions \(\varphi\) and \(T\) with the following properties: \(T\) is in the linear span of the set \(\{exp(ik(\cdot)): k\in J\}\) (denoted by \({\mathcal T}_J\)), \(\varphi\in L_1\), \(\varphi \perp {\mathcal T}_J\), and the number of essential changes of sign of the function \(\varphi\) is \(\leq 2n\). Further, by \({\mathcal D}_{2n-2}(J)\) denote the set of real-valued functions \(D\in L_1\) satisfying the following condition on the number of sign changes: \[ \nu\Bigg(T+ \sum_{j=0}^{2n-1}\beta_j D \bigg(\cdot -\frac{j \pi}{n}\bigg) \Bigg)\leq 2n-2. \] The main result of this article is formed by the Favard-Akhiezer-Krein type inequalities for classes of periodic convolutions with kernels that do not increase oscillation. The author proved the following fact. Suppose that \(n\in \mathbb N, J\in [1-n,n-1]\cap \mathbb Z, -J=J\), \(G\in {\mathcal A}_{2n} (J)\), \(D\in {\mathcal D}_{2n-2}(J)\), and \(B:=G\star D\in C\) satisfies the condition \[ \sum_{m=0}^{2n-1}B(w+\frac{m\pi}{n})e^{-i\pi l m / n}\neq 0, \quad w\in \mathbb R, \;l\in J_n'. \] Then for the function \[ f(x)=T(x)+G\star \varphi(x) \] one has \[ \|f-X_{n,G,D,J(f)}\|_p\leq {\mathcal K}_{n,G}\|\varphi\|_p, \] where \(X_{n,G,D,J(f)}\) is the approximation constructed in the paper. For \(p=1\) and \(p=\infty\), the constant in the inequality cannot be decreased for \(f=\varphi\star G\) and \(\varphi \perp S_{2n-1, B, J}\). Furthermore, a large class of approximating odd-dimensional subspaces constructed from uniform shifts of one function, which are extremal in the sense of Kolmogorov width and linear width, is specified. As a corollary, sharp Jackson-type inequalities for the second-order modulus of continuity are derived.
scientific article; zbMATH DE number 5628183
scientific article