Some new estimates of the Fourier-Bessel transform in the space mathbb{L}_2 left( {mathbb{R}_ + } right)

The Fourier-Bessel integral transform $$gleft( x right) = Fleft[ f right]left( x right) = frac{1} {{2^p Gamma left( {p + 1} right)}}intlimits_0^{ + infty } {t^{2p + 1} fleft( x right)j_p left( {xt} right)dt}$$ is considered in the space $mathbb{L}_2 left( {mathbb{R}_ + } right)$ . Here, j _p (u) = ((2 ^p Γ(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 $$delta _N^2 left( f right) = intlimits_N^{ + infty } {x^{2p + 1} g^2 left( x right)dx, N > 0,}$$ in $mathbb{L}_2 left( {mathbb{R}_ + } right)$ for some classes of functions characterized by a generalized modulus of continuity.