FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1997, VOLUME 3, NUMBER 2, PAGES 351-357

Application of the AÙ-integration for Fourier transforms

A. Anter

Abstract

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The following theorem is proved
\begin{theorem}
Let the function $f(x)$ be a boundary variation on $\mathbb R$ and $f(x)\to 0$ ($x\to \pm\infty$). 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 $f(x)=\dfrac12(f(x+0)+f(x-0))$ \textup{(}for all $x$, except countable subset\textup{)} the following holds
$$
f(x)=(A^{\land})
\int\limits_{-\infty}^{+\infty}\widehat f(\lambda) e^{2\pi i\lambda x}d\lambda.
$$
\end{theorem}

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