Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials (Q1084283)

The authors present three theorems and one useful lemma. The main result reads as follows: Let \(\sup p(d\alpha)=[-1,1],\alpha '>0\) almost everywhere in [-1,1], and suppose \(0<p\leq \infty\). Put \(v(x)=(\alpha '(x)\sqrt{1-x^ 2})^{1/2}.\) If g is a Lebesgue-measurable function in [-1,1] then \[ (\int^{1}_{-1}| g/v|^ p)^{1/p}\leq \sqrt{\pi}2^{\max \{1/p-1/2,0\}}\liminf_{n\to \infty}(\int^{1}_{- 1}| gp_ n(d\alpha)|^ p)^{1/p}. \] Presumably this inequality will have a significant role in the extensions of Szegö's theory.