FPM 2002, vol. 8, no. 1, pp. 307-312

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 1, PAGES 307-312

$$ A^{\land} $$ -integration of Fourier transformations

Anter Ali Alsayad

Abstract

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The following theorems are proved.

Theorem 1. Let $f$ be a function of bounded variation on R, $f(x) → 0$ ( $x →$ ±¥), and f Î L(R) be a bounded function. Then

$$ (A^{\land })\int_{\mathbb R}
\hat f(x) \Bar{\Hat \varphi}(x) dx = (L)\int_{\mathbb R} f(x)
\bar\varphi(x) dx.
$$

Theorem 2. Let $f(x)=$ å _{n = -¥}^+¥ a_k e^ikx, where a_k Î C, ${$ a_k} is a sequence with bounded variation, a_k → 0 ( $k →$ ± ¥), and let $g(x)=$ å _{j = -¥}^+¥ b_j e^ijx , where å _{j = -¥}^+¥ |b_j| < ¥. Then

$$ (A)\int_{0}^{2\pi}
f(x) \bar g(x) dx = \sum_{m=-\infty}^{+\infty} \alpha_m \bar\beta _m
$$

and

(A)

ò₀^2p f(x) g(x) dx = å_{m = -¥}^+¥ a_m b_-m.

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$ A^{\land} $-integration of Fourier transformations

$$ A^{\land} $$ -integration of Fourier transformations