Geodesic
Some new estimates of the Fourier–Bessel transform in the space $\mathbb{L}_2(\mathbb{R}_+)$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 10, pp. 1622-1628

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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. 
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     author = {V. A. Abilov and F. V. Abilova and M. K. Kerimov},
     title = {Some new estimates of the {Fourier{\textendash}Bessel} transform in the space $\mathbb{L}_2(\mathbb{R}_+)$},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     volume = {53},
     number = {10},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_10_a2/}
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V. A. Abilov; F. V. Abilova; M. K. Kerimov. Some new estimates of the Fourier–Bessel transform in the space $\mathbb{L}_2(\mathbb{R}_+)$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 10, pp. 1622-1628. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_10_a2/