Application of the $A^{\land}$-integration for Fourier transforms

Anter Ali Alsayad

Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 2, pp. 351-357.

Résumé

The following theorem is proved Theorem. Let the function $f(x)$ be a boundary variation on $\mathbb R$ and $f(x)\to0$ ($x\to\pm\infty$). Then its Fourier transform $$ \widehat f(\lambda)=(L^{\land})\int\limits_{-\infty}^{+\infty}f(t)e^{-2\pi i\lambda t}dt $$ exists in case of $\lambda\ne0$ and $f(x)$ recovers by its Fourier transforms by mean of the $A^{\land}$-integral. Further for all $x\in\tilde{A}$, where $f(x)=\dfrac12(f(x+0)+f(x-0))$ (for all $x$, except countable subset) the following holds $$ f(x)=(A^{\land})\int\limits_{-\infty}^{+\infty}\widehat f(\lambda)e^{2\pi i\lambda x}d\lambda. $$