On the Pollard decomposition method applied to some Jacobi–Sobolev expansions

Abstract

Let ${q_n^{(alpha,beta)}}_{ngeq0}$ be the sequence of polynomials orthonormal with respect to the Sobolev inner product$langle f,g rangle s := int_{-1}^1 f(x)g(x)w^{(alpha,beta)}(x)dx + int_{-1}^1 f'(x)g'(x)w^{(alpha+1,beta+1)}(x)dx $,where $w^{(alpha,beta)} (x) = (1-x)^{alpha} (1+x)^{beta}, x in [-1,1]$ and $alpha, beta > −1$ . This paper explores the convergence in the $W^{1-p} biggl((-1,1),(w^{(alpha,beta)}, w^{(alpha +1,beta +1)})biggr)$ norm of the Fourier expansion in terms of ${q_n^{(alpha,beta)}}_{ngeq0}$with 1 < p < ∞ , using the Pollard decomposition method. Numerical examples concerning the comparison between the approximation of functions in $L^2$ norm and $W^{1,2}$ norm are also presented.