Some characterizations of Hausdorff matrices and their application to Fourier approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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The Problem of ‘Total Translativity’ for Hausdorff Matrices

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-20.3.525 ◽

1970 ◽

Vol s3-20 (3) ◽

pp. 525-536 ◽

Cited By ~ 1

Author(s):

B. E. Rhoades

Keyword(s):

Hausdorff Matrices

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Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

Open Mathematics ◽

10.1515/math-2021-0089 ◽

2021 ◽

Vol 19 (1) ◽

pp. 1047-1055

Author(s):

Zhihua Zhang

Keyword(s):

Fourier Series ◽

Trigonometric Polynomial ◽

Algebraic Polynomial ◽

Fourier Coefficients ◽

Random Processes ◽

Qualitative Theory ◽

Trigonometric Polynomials ◽

Algebraic Polynomials ◽

Fourier Approximation ◽

Random Differential Equations

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

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