On the Pollard decomposition method applied to some Jacobi-Sobolev expansions (Q2873172)

This contribution deals with the Fourier expansion in terms of the set \(\{q_{n}^{(\alpha,\beta)}\}_{n \geq 0}\) of orthogonal polynomials, called Jacobi-Sobolev polynomials, and associated with the following Sobolev inner product NEWLINE\[NEWLINE\langle f,g\rangle_{S}:=\int_{-1}^{1}f(x)g(x)w^{(\alpha,\beta)}(x)dx+ \int_{-1}^{1}f^\prime(x)g^\prime(x)w^{(\alpha+1,\beta+1)}(x)dx,NEWLINE\]NEWLINE where \(w^{(\alpha,\beta)}(x)=(1-x)^{\alpha}(1+x)^{\beta},\) \(x\in [-1,1]\) and \(\alpha,\beta>-1\). Since the polynomials \(\{q_{n}^{(\alpha,\beta)}\}_{n \geq 0}\) are essentially Jacobi polynomials, the authors use the Pollard decomposition method to study the \(W^{1,p}\left((-1,1), (w^{(\alpha,\beta)},w^{(\alpha+1,\beta+1)})\right)\) norm convergence of the corresponding orthogonal expansion. Also, some numerical examples concerning the comparison between the approximation of functions in the \(L^{2}\) norm and the \(W^{1,2}\) norm are presented.