Volume 74, Nº 4 (2019)

Ano: 2019
Artigos: 10
URL: https://journal-vniispk.ru/0042-1316/issue/view/7511

Integral norm discretization and related problems

Dai F., Prymak A., Temlyakov V., Tikhonov S.

Resumo

The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated for elements of finite-dimensional spaces. Also, discretization of the uniform norm of functions in a given finite-dimensional subspace of continuous functions is studied. Special attention is given to the case of multivariate trigonometric polynomials with frequencies (harmonics) in a finite set with fixed cardinality. Both new results and a survey of known results are presented.Bibliography: 47 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(4):3-58

3-58

(Rus)

(JATS XML)

Critical configurations of solid bodies and the Morse theory of MIN functions

Ogievetskii O., Shlosman S.

Resumo

This paper studies the manifold of clusters of non-intersecting congruent solid bodies, all touching the central ball $B\subset\mathbb{R}^{3}$ of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. The notion of critical cluster is introduced, and several critical clusters of balls and of cylinders are studied. In the case of cylinders, some of the critical clusters here are new. The paper also establishes criticality properties of clusters introduced earlier by Kuperberg [7].

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Uspekhi Matematicheskikh Nauk. 2019;74(4):59-86

59-86

(Rus)

(JATS XML)

Sobolev-orthogonal systems of functions and some of their applications

Sharapudinov I.

Resumo

Systems of functions are considered which are associated with a given orthogonal system and are orthogonal with respect to an inner product of Sobolev type involving terms with masses concentrated at a point. Special attention is paid to such systems generated by classical orthogonal systems such as the cosine system, the Haar system, and the systems of Legendre, Jacobi, and Laguerre polynomials. The approximation properties of Fourier series in Sobolev-orthogonal systems are investigated in several cases. For (generally speaking, non-linear) systems of differential equations deep connections between Sobolev-orthogonal systems and the Cauchy problem are considered.Bibliography: 54 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(4):87-164

87-164

(Rus)

(JATS XML)

Vladimir Andreevich Uspensky (27/11/1930–27/6/2018)

Adian S., Andreev N., Beklemishev L., Goncharov S., Ershov Y., Matiyasevich Y., Osipov Y., Pentus M., Plungyan V., Rahilina E., Sadovnichii V., Semenov A., Tatevosov S., Tikhomirov V., Shen' A.

Uspekhi Matematicheskikh Nauk. 2019;74(4):165-180

165-180