On strong summability of polynomial expansions (Q1090881)

A generalized Jacobi weight is any function of the form \(W(x)=\phi)\prod_{1\leq i\leq r}| x-x_ i|^{\alpha_ i},\) \(x\in (-1,1)\), where \(r\geq 2\), \(\alpha_ i>-1\), \(x_ i\in [-1,1]\) and \(\phi\) is a continuous function whose modulus of continuity \(\omega\) satisfies \(\int_{0}\omega (\delta)\delta^{-1}<\infty\). Consider the system of orthonormal polynomials with respect to a generalized Jacobi weight W and denote by \(S_ k(f,\cdot)\) the n-th partial sum of the Fourier series of the function f with respect to this system. Theorem. If \(0<p<\infty\) and \(W^{1/2}(x)(1-x^ 2)^{-1/4}f(x)\) is summable over (-1,1) then \(n^{-1}\sum_{0\leq k\leq n-1}| S_ k(f,x)-f(x)|^ p\to 0\) a.e on (-1,1).