Some new estimates of the Fourier transform in $\mathbb{L}_2(\mathbb{R})$

V. A. Abilov ; F. V. Abilova ; M. K. Kerimov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 9, pp. 1419-1426

Résumé

Given a function $\mathbb{L}_2(\mathbb{R})$, its Fourier transform $$ g(x)=\hat{f}(x)=F[f](x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}f(x)e^{-ixt}dt,\quad f(t)=F^{-1}[g](t)=\frac1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}g(x)e^{ixt}dx $$ and the inverse Fourier transform are considered in the space $f\in\mathbb{L}_2(\mathbb{R})$. New estimates are presented for the integral $ \int_{|t|\geqslant N}|g(t)|^2dt=\int_{|t|\geqslant N}|\hat{f}(t)|^2dt, \quad N\geqslant1 $, in the vase of $f\in\mathbb{L}_2(\mathbb{R})$ characterized by the generalized modulus of continuity of the $k$th order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.