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 = +¥ ak eikx, where ak Î C, {ak} is a sequence with bounded variation, ak → 0 (k → ± ¥), and let g(x)= å j = +¥ bj eijx , where å j = +¥ |bj| < ¥. Then

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

and

(A)ò02p f(x) g(x) dx = åm = +¥ am b-m.

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Last modified: July 8, 2002