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

Anter Ali Alsayad

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 307-312

Résumé

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 $$ (A)\!\int\limits_{0}^{2\pi}f(x)g(x)\,dx =\sum_{m=-\infty}^{+\infty}\alpha_m\beta_{-m}. $$