We consider extremum problems for entire functions of exponential spherical type related to important extremum problems on the optimal point (the Chernykh point) in the sharp jackson inequality in the spaceL2(ℝn) and the connection between codes and designs on the torusTn.

Extremum problems for entire functions of exponential spherical type

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions with geometry and analysis. In particular, we present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, we discuss topological function theory which provides a new insight on oscillation of functions. The material should be accessible to readers with a basic background in algebraic and differential topology.

Topological Persistence in Geometry and Analysis

A?new lower bound is obtained for the cardinality of spherical designs In its dependence on dimension it improves the well-known bound of Delsarte by an exponential factor as the degree of the design tends to infinity

Lower bounds for spherical designs

The value of the sharp constant in the Jackson type inequality in the space (1)is studied for the generalized modulus of continuity The value of the minimum sharp constant in inequality (1) is found.A class of generalized moduli of continuity is introduced which contains the moduli , with even . The relation is proved in this class for all .

On?Jackson's inequality for a?generalized modulus of continuity in?$ L_2$

CONTENTSIntroductionChapter I. Convergence of rectangular partial sums ?1. Results on the almost everywhere convergence of Fourier series of integrable functions ?2. The almost everywhere convergence of Fourier series of functions from ?3. A brief survey of new results on the convergence of rectangular partial sums in various metricsChapter II. New results on rectangular means of multiple Fourier seriesChapter III. Dirichlet kernels ?1. Lebesgue constants and norms of some trigonometric polynomials ?2. Asymptotic behaviour of Dirichlet kernelsChapter IV. Uniqueness problems for multiple trigonometric series (rectangular partial sums)Chapter V. Problems of localization for rectangular partial sumsChapter VI. Special classes of multiple trigonometric series ?1. Series with monotone coefficients ?2. Multiple lacunary series ?3. Fourier series of piecewise monotone functions of several variablesChapter VII. A definition of convergence of multiple seriesChapter VIII. Conjugate series and conjugate functions of several variables ?1. Conjugate series and conjugate functions in , ?2. The conjugation operator in the space ?3. Multidimensional analogues of Plessner's theoremChapter IX. Representation of functions by trigonometric series and correction of functionsChapter X. Fourier coefficients ?1. Cantor-Lebesgue and Denjoy-Luzin theorems ?2. New results on the absolute convergence of Fourier series ?3. Transformations of Fourier coefficients and other resultsReferences

Some problems in the theory of multiple trigonometric series

The multidimensional Jackson theorem

Multidimensional jackson theorem in L2

A lower bound for Lebesgue constants

Spherical sums of Fourier series in Lp

Jackson's theorems in L2

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