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On Fourier Series on the Torus and Fourier Transforms
R. M. Trigub Donetsk National University
Abstract:
The question of the representability of a continuous function on $\mathbb R^d$ in the form of the Fourier integral of a finite Borel complex-valued measure on $\mathbb R^d$ is reduced in this article to the same question for a simple function. This simple function is determined by the values of the given function on the integer lattice $\mathbb R^d$. For $d=1$, this result is already known: it is an inscribed polygonal line. The article also describes applications of the obtained theorems to multiple trigonometric Fourier series.
Keywords:
Fourier series of a measure on the torus $\mathbb T^d$ and functions from $L_1(\mathbb T^d)$, variation of a measure, Wiener Banach algebras, positive definite functions, exponential entire functions, $(C,1)$-means of Fourier series, Vitali variation, Banach–Alaoglu theorem.
Received: 02.06.2021 Revised: 30.06.2021
Citation:
R. M. Trigub, “On Fourier Series on the Torus and Fourier Transforms”, Mat. Zametki, 110:5 (2021), 766–772; Math. Notes, 110:5 (2021), 767–772
Linking options:
https://www.mathnet.ru/eng/mzm13178https://doi.org/10.4213/mzm13178 https://www.mathnet.ru/eng/mzm/v110/i5/p766
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Abstract page: | 412 | Full-text PDF : | 79 | References: | 53 | First page: | 19 |
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