Extreme Properties of the Rademacher System

The Rademacher series in rearrangement invariant function spaces close to the spaceL∞are considered. In terms of interpolation theory of operators, a correspondence between such spaces and spaces of coefficients generated by them is stated. It is proved that this correspondence is one-to-one. Some examples and applications are presented.

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Applications of Affine Systems of Walsh Type to Generate Smooth Basis

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.12.25 ◽

2021 ◽

pp. 4875-4884

Author(s):

Khaled Hadi ◽

Saad Nagy

Keyword(s):

Generating Function ◽

Riesz Basis ◽

Walsh System ◽

Continuous Periodic Function ◽

Affine Systems ◽

Rademacher System ◽

Almost Everywhere ◽

Affine System ◽

Smooth Basis ◽

Differential Properties

The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.

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The Rademacher System in Function Spaces

10.1007/978-3-030-47890-2 ◽

2020 ◽

Cited By ~ 1

Author(s):

Sergey V. Astashkin

Keyword(s):

Function Spaces ◽

Rademacher System

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SETS OF UNIQUENESS OF SERIES OF STOCHASTICALLY INDEPENDENT FUNCTIONS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500000420 ◽

2002 ◽

Vol 45 (3) ◽

pp. 557-563 ◽

Cited By ~ 1

Author(s):

Francisco J. Freniche ◽

Ricardo Ríos-Collantes-de-Terán

Keyword(s):

Positive Measure ◽

Subject Classification ◽

Secondary 60G50 ◽

Best Constant ◽

Rademacher System ◽

Independent Functions ◽

Set Of Uniqueness ◽

Sets Of Uniqueness ◽

Uniformly Bounded

AbstractIt is shown that, for every sequence $(f_n)$ of stochastically independent functions defined on $[0,1]$—of mean zero and variance one, uniformly bounded by $M$—if the series $\sum_{n=1}^\infty a_nf_n$ converges to some constant on a set of positive measure, then there are only finitely many non-null coefficients $a_n$, extending similar results by Stechkin and Ul’yanov on the Rademacher system. The best constant $C_M$ is computed such that for every such sequence $(f_n)$ any set of measure strictly less than $C_M$ is a set of uniqueness for $(f_n)$.AMS 2000 Mathematics subject classification:Primary 42C25. Secondary 60G50

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