This article is cited in 1 paper

Some new estimates of the Fourier–Bessel transform in the space $\mathbb{L}_2(\mathbb{R}_+)$

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: The Fourier–Bessel integral transform
$$ g(x)=F[f](x)=\frac1{2^p\Gamma(p+1)}\int_0^{+\infty}t^{2p+1}f(x)j_p(xt)dt $$
is considered in the space $\mathbb{L}_2(\mathbb{R}_+)$. Here, $j_p(u)=((2^p\Gamma(p+1))/(u^p))J_p(u)$ and $J_p(u)$ is a Bessel function of the first kind. New estimates are proved for the integral
$$ \delta^2_N(f)=\int_N^{+\infty}x^{2p+1}g^2(x)dx,\quad N>0, $$
in $\mathbb{L}_2(\mathbb{R}_+)$ for some classes of functions characterized by a generalized modulus of continuity.

Key words: Fourier–Bessel integral transform, Bessel operator, shift operator, generalized modulus of continuity, Fourier-Bessel transform estimates.

UDC: 519.651

Received: 11.05.2013

DOI: 10.7868/S0044466913100025

 English version: Computational Mathematics and Mathematical Physics, 2013, 53:10, 1440–1446

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