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

Authors: FRANCISCO MARCELLÁN, YAMILET QUINTANA, ALEJANDRO URIELES

Abstract: Let {q_n^{(\alpha,\beta)}}_{n \geq 0} be the sequence of polynomials orthonormal with respect to the Sobolev inner product \langle f,g\rangle_S:=\int_{-1}^1f(x)g(x)w^{(\alpha,\beta)}(x)dx+\int_{-1}^1f'(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}\left((-1,1), (w^{(\alpha,\beta)},w^{(\alpha+1,\beta+1)})\right) norm of the Fourier expansion in terms of {q_n^{(\alpha,\beta)}}_{n\geq 0} with 1< p<\infty, 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.

Keywords: Sobolev orthogonal polynomials, weighted Sobolev spaces, Fourier expansions, Sobolev--Fourier expansions

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