Relating coefficients of expansion of a function to its norm (Q739529)

For the expansion of a function by a complete weighted orthonormal system, the paper establishes estimates of the corresponding weighted \(L_p\)-norms of the function in terms of certain series involving the coefficients of the expansion. The proofs are obtained first for two special cases, which are the analogues of the Hausdorff-Young and the Hardy-Littlewood inequalities. Then the general results are proved via Stein's modification of the Riesz-Thorin interpolation theorem. It is shown that the results can be used for a variety of examples of orthogonal expansions.