Norm and smoothness of a function related to the coefficients of its expansion (Q2346290)

The goal of the paper is to obtain general inequalities, which contain both Hausdorff-Young type and Hardy-Littlewood type inequalities. As a consequence there are obtained relations between the coefficients of expansion and the smoothness of the functions expanded. Two situations are considered: the expansion with regard to the spherical harmonic polynomials on the unit sphere of a the space \(\mathbb{R}^d\), and the expansion with regard to the Jacobi polynomials on the interval \([-1,1]\). These results are continuations of the author's studies in [J. Approx. Theory; 162, No 5, 980--986 (2010; Zbl 1195.42046)] and [J. Approx. Theory, 164, No. 10, 1369--1389 (2012; Zbl 1257.42015)].