Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii-Besov spaces.

*(English. Ukrainian original)*Zbl 07308113
J. Math. Sci., New York 252, No. 4, 508-525 (2021); translation from Ukr. Mat. Visn. 17, No. 3, 372-395 (2020).

Nine new theorems are proved, and the statement of six known further ones are written in this eighteen pages paper with thirty one references, whose study requires several pages with notations, definitions and auxiliary assertions.

Here it is a resume of the paper written from its introduction:

The authors continue the study (see [A. S. Romanyuk and V. S. Romanyuk, “Estimates of some approximative characteristics of classes of periodic functions of one variable and many ones”, Ukr. Mat. Zh., 71, No. 8, 1102–1115 (2019)]) on the approximation of the Nikol’skii-Besov \(B_{p,\theta}^r\) and Sobolev \(\mathbb{W}_{p,\alpha}^r\) classes of periodic functions of one and many variables in the space \(B_{\infty ,1}\).

A specific feature of the space \(B_{\infty ,1}\), as a linear subspace in \(L_{\infty }\), is that its normalization is carried out on the basis of the expansion of functions from \(L_{\infty }\) in Fourier series in the trigonometric system \(\left\{e^{i(k,x)}\right\}_{k\in \mathbb{Z}^d}\), and the corresponding norm in this space is stronger, than the \(L_{\infty}\)-norm. As was noted in the aforementioned reference, the study of definite approximative characteristics just in the space \(B_{\infty ,1}\) is favored by the following circumstance. The solution of problems of determination of their ordinal values in the space \(L_{\infty }\), in particular, for \(d\geq 3\), with the help of the available methods meets a number of obstacles that cannot be yet removed. At the same time, the authors note that some approximative characteristics of certain functional classes in the space \(B_{\infty ,1}\) were studied in the first four references of the paper, these including the aforesaid one.

The results of the present work are presented, in fact, in two items. In the first one, the authors consider the orthowidths of the classes \(B_{p,\theta}^r\) and \(\mathbb{W}_{p,\alpha}^r\) (and the quantities similar by definition) from the viewpoint of the determination of the ordinal values of those approximative characteristics. In the second item, the objects of studies are linear operators that are optimum in the problem of exact-by-order values of the best approximations of the classes \(B_{p,\theta}^r\) in the space \(B_{\infty ,1}\) with the help of the trigonometric polynomials generated by the system \(\left\{e^{i(k,x)}\right\}_{k\in \mathbb{Q}_n}\). Here, the sets \(\mathbb{Q}_n\), \(n\in \mathbb{N}\), are the so-called step hyperbolic crosses in \(\mathbb{Z}^d\). One of the key questions concerns the sequence of the norms of such operators (the operators that act from \(L_{\infty}\) into \(L_{\infty}\)), namely, the question of the boundedness or unboundedness of this sequence.

Here it is a resume of the paper written from its introduction:

The authors continue the study (see [A. S. Romanyuk and V. S. Romanyuk, “Estimates of some approximative characteristics of classes of periodic functions of one variable and many ones”, Ukr. Mat. Zh., 71, No. 8, 1102–1115 (2019)]) on the approximation of the Nikol’skii-Besov \(B_{p,\theta}^r\) and Sobolev \(\mathbb{W}_{p,\alpha}^r\) classes of periodic functions of one and many variables in the space \(B_{\infty ,1}\).

A specific feature of the space \(B_{\infty ,1}\), as a linear subspace in \(L_{\infty }\), is that its normalization is carried out on the basis of the expansion of functions from \(L_{\infty }\) in Fourier series in the trigonometric system \(\left\{e^{i(k,x)}\right\}_{k\in \mathbb{Z}^d}\), and the corresponding norm in this space is stronger, than the \(L_{\infty}\)-norm. As was noted in the aforementioned reference, the study of definite approximative characteristics just in the space \(B_{\infty ,1}\) is favored by the following circumstance. The solution of problems of determination of their ordinal values in the space \(L_{\infty }\), in particular, for \(d\geq 3\), with the help of the available methods meets a number of obstacles that cannot be yet removed. At the same time, the authors note that some approximative characteristics of certain functional classes in the space \(B_{\infty ,1}\) were studied in the first four references of the paper, these including the aforesaid one.

The results of the present work are presented, in fact, in two items. In the first one, the authors consider the orthowidths of the classes \(B_{p,\theta}^r\) and \(\mathbb{W}_{p,\alpha}^r\) (and the quantities similar by definition) from the viewpoint of the determination of the ordinal values of those approximative characteristics. In the second item, the objects of studies are linear operators that are optimum in the problem of exact-by-order values of the best approximations of the classes \(B_{p,\theta}^r\) in the space \(B_{\infty ,1}\) with the help of the trigonometric polynomials generated by the system \(\left\{e^{i(k,x)}\right\}_{k\in \mathbb{Q}_n}\). Here, the sets \(\mathbb{Q}_n\), \(n\in \mathbb{N}\), are the so-called step hyperbolic crosses in \(\mathbb{Z}^d\). One of the key questions concerns the sequence of the norms of such operators (the operators that act from \(L_{\infty}\) into \(L_{\infty}\)), namely, the question of the boundedness or unboundedness of this sequence.

Reviewer: Daniel Cárdenas Morales (Jaén)

##### MSC:

42A10 | Trigonometric approximation |

41A46 | Approximation by arbitrary nonlinear expressions; widths and entropy |

42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

##### Keywords:

Nikol’skii-Besov classes; Sobolev classes; orthowidth; linear bounded operator; step hyperbolic cross
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\textit{A. S. Romanyuk} and \textit{V. S. Romanyuk}, J. Math. Sci., New York 252, No. 4, 508--525 (2021; Zbl 07308113); translation from Ukr. Mat. Visn. 17, No. 3, 372--395 (2020)

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##### References:

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