FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2002, VOLUME 8, NUMBER 1, PAGES 307-312
Anter Ali Alsayad
Abstract
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The following theorems are proved.
\textbf{Theorem 1.}
Let $f$
be a function of bounded variation
on $\mathbb R$ ,
$f(x) \to 0$
($x \to \pm\infty$ ), and
$\varphi \in L(\mathbb R)$
be a bounded function. Then
$$
(A^{\land})\int\limits_{\mathbb R}
\hat f(x) \Bar{\Hat\varphi}(x)\,dx =
(L)\int\limits_{\mathbb R} f(x) \bar\varphi(x)\,dx.
$$
\textbf{Theorem 2.} Let
$f(x) = \sum\limits_{n=-\infty}^{+\infty}
\alpha_k e^{ikx}$ , where
$\alpha_k \in \mathbb C$ ,
$\{ \alpha_k \}$ is a sequence with
bounded variation,
$\alpha_k \to 0$
($k \to \pm\infty$ ), and let
$g(x) = \sum\limits_{j=-\infty}^{+\infty}
\beta_j e^{ijx}$ , where
$\sum\limits_{j=-\infty}^{+\infty}
|\beta_j| < \infty$ .
Then
$$
(A)\int\limits_{0}^{2\pi} f(x) \bar g(x)\,dx =
\sum_{m=-\infty}^{+\infty} \alpha_m \bar\beta_m
$$
$$
(A)\int\limits_{0}^{2\pi} f(x) g(x)\,dx =
\sum_{m=-\infty}^{+\infty} \alpha_m \beta_{-m}.
$$
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Last modified: July 5, 2002