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Fundamentalnaya i Prikladnaya Matematika, 1997, Volume 3, Issue 2, Pages 351–357
(Mi fpm219)
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This article is cited in 1 scientific paper (total in 1 paper)
Application of the $A^{\land}$-integration for Fourier transforms
Anter Ali Alsayad M. V. Lomonosov Moscow State University
Abstract:
The following theorem is proved
Theorem.
Let the function $f(x)$ be a boundary variation on $\mathbb R$ and $f(x)\to0$ ($x\to\pm\infty$). Then its Fourier transform
$$
\widehat f(\lambda)=(L^{\land})\int\limits_{-\infty}^{+\infty}f(t)e^{-2\pi i\lambda t}dt
$$
exists in case of $\lambda\ne0$ and $f(x)$ recovers by its Fourier transforms by mean of the $A^{\land}$-integral. Further for all $x\in\tilde{A}$, where $f(x)=\dfrac12(f(x+0)+f(x-0))$ (for all $x$, except countable subset) the following holds
$$
f(x)=(A^{\land})\int\limits_{-\infty}^{+\infty}\widehat f(\lambda)e^{2\pi i\lambda x}d\lambda.
$$
Received: 01.01.1996
Citation:
Anter Ali Alsayad, “Application of the $A^{\land}$-integration for Fourier transforms”, Fundam. Prikl. Mat., 3:2 (1997), 351–357
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https://www.mathnet.ru/eng/fpm219 https://www.mathnet.ru/eng/fpm/v3/i2/p351
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