Bessel generalized translations and some problems of approximation theory for functions on the half-line

Approximation problems for functions on the half-line $[0,+\infty)$ in a weighted $L_p$-metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of smoothness of arbitrary order which is constructed on the basis of Bessel generalized translation. Equivalence is stated between the modulus of smoothness and the $K$-functional constructed by the Sobolev space corresponding to the Bessel differential operator. A particular class of entire functions of exponential type is used for approximation. The problems under consideration are studied mostly by means of Fourier–Bessel harmonic analysis.