Fourier series of Jacobi-Sobolev polynomial

Abstract

Let $\{q_n^{(\alpha,\beta,m)}(x)\}_{n\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \begin{equation*} \langle f,g\rangle_{\alpha,\beta,m}=\sum_{k=0}^m \int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\, dw_{\alpha+k,\beta+k}(x), \quad \alpha,\beta>-1, \quad m\ge 1, \end{equation*} where $dw_{a,b}(x)=(1-x)^{a}(1+x)^b\, dx$. We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space $W_{\alpha,\beta}^{p,m}$. As a consequence we deduce the convergence of such partial sums in the norm of $W_{\alpha,\beta}^{p,m}$.