(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

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

Abstract

View as HTML     View as gif image    View as LaTeX source

The following theorems are proved.

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

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

Theorem 2. Let $f\left(x\right)=$å n = +¥ ak eikx, where ak Î C, $\left\{$ak} is a sequence with bounded variation, ak → 0 ($k \to$± ¥), and let $g\left(x\right)=$å j = +¥ bj eijx , where å j = +¥ |bj| < ¥. Then

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

and

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

All articles are published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k02/k021/k02124h.htm