Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 307-312
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Anter Ali Alsayad. $A^{\land}$-integration of Fourier transformations. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 307-312. http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a23/
@article{FPM_2002_8_1_a23,
author = {Anter Ali Alsayad},
title = {$A^{\land}$-integration of {Fourier} transformations},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {307--312},
year = {2002},
volume = {8},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a23/}
}
TY - JOUR
AU - Anter Ali Alsayad
TI - $A^{\land}$-integration of Fourier transformations
JO - Fundamentalʹnaâ i prikladnaâ matematika
PY - 2002
SP - 307
EP - 312
VL - 8
IS - 1
UR - http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a23/
LA - ru
ID - FPM_2002_8_1_a23
ER -
%0 Journal Article
%A Anter Ali Alsayad
%T $A^{\land}$-integration of Fourier transformations
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2002
%P 307-312
%V 8
%N 1
%U http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a23/
%G ru
%F FPM_2002_8_1_a23
The following theorems are proved. Theorem 1. Let $f$ be a function of bounded variation on $\mathbb R$, $f(x)\to0$ ($x\to\pm\infty$), and $\varphi\in L(\mathbb R)$ be a bounded function. Then $$ (A^{\land})\!\int\limits_{\mathbb R}\hat f(x)\bar{\hat\varphi}(x)\,dx =(L)\!\int\limits_{\mathbb R}f(x)\bar\varphi(x)\,dx. $$ Theorem 2. Let $f(x)=\sum\limits_{n=-\infty}^{+\infty}\alpha_ke^{ikx}$, where $\alpha_k\in\mathbb C$, $\{\alpha_k\}$ is a sequence with bounded variation, $\alpha_k\to0$ ($k\to\pm\infty$), and let $g(x)=\sum\limits_{j=-\infty}^{+\infty} \beta_j e^{ijx}$, where $\sum\limits_{j=-\infty}^{+\infty}|\beta_j|\infty$. Then $$ (A)\!\int\limits_{0}^{2\pi}f(x)\bar g(x)\,dx =\sum_{m=-\infty}^{+\infty}\alpha_m\bar\beta_m $$ and $$ (A)\!\int\limits_{0}^{2\pi}f(x)g(x)\,dx =\sum_{m=-\infty}^{+\infty}\alpha_m\beta_{-m}. $$
