This article is cited in 1 paper

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

V. A. Abilov^a, F. V. Abilova^b, M. K. Kerimov<sup>c

a</sup> Dagestan State University, ul. Gadzhieva 43a, Makhachkala, 367025, Russia
^b Dagestan State Technical University, pr. Kalinina 7a, Makhachkala, 367015, Russia
^c Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

Abstract: 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.

Key words: Fourier transform in $\mathbb{L}_2(\mathbb{R})$, inverse Fourier transform, Steklov function, generalized modulus of continuity, estimates.

UDC: 519.651

Received: 01.04.2013

DOI: 10.7868/S0044466913090020

 English version: Computational Mathematics and Mathematical Physics, 2013, 53:9, 1231–1238

Bibliographic databases:

• 
• 
• 
•