On The Convergence of Series in Orthogonal Polynomials

Abstract

In his recent paper [1] LP. Natanson has proved that for any summable positive weight function p(x) the series in orthogonal polynomials converges almost everywhere to the function f (x), where f admits such a series expansion, provided that the latter function satisfies a Lipschitz condition with exponent a greater than ½. We are going to show that this result remains valid for every function satisfying a Lipschitz condition with arbitrarily small positive exponent a. We establish in fact a somewhat stronger result; namely,the above-mentioned series converges almost everywhere if there exist constantsCand e >0 such that $$\left| {f\left( {x''} \right)} \right| - f\left( {x'} \right) \leqslant \frac{C}{{\left| {\log } \right|{{{\left. {x'' - x'} \right\|}}^{{1 + \in }}}}}$$ (1) for any x” ≠ x’.