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

Let {qn((alpha,beta))}(n >= 0) be the sequence of polynomials orthonormal with respect to the Sobolev inner product < f,g > s := integral(1)(-1) f(x)g(x)w((alpha,beta)) (x)dx + integral(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 is an element of [-1,1] and alpha,beta > -1. This paper explores the convergence in the W-1,W-p ((-1,1),(w((alpha,beta)), w((alpha+1,beta+1)))) norm of the Fourier expansion in terms of {q(n)((alpha,beta))}(n >= 0) with 1 <p < infinity, using the Pollard decomposition method. Numerical examples concerning the comparison between the approximation of functions in L-2 norm and W-1,W-2 norm are also presented.