FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1997, VOLUME 3, NUMBER 2, PAGES 351-357
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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