Fourier series of Jacobi-Sobolev polynomial

TL;DR

This paper studies the Fourier series expansion of Jacobi-Sobolev orthogonal polynomials, establishing conditions for the boundedness and convergence of partial sums in Sobolev spaces.

Contribution

It provides necessary and sufficient conditions for the uniform boundedness and convergence of Fourier partial sums of Jacobi-Sobolev polynomials in Sobolev spaces.

Findings

Conditions for boundedness of partial sum operators are established.

Convergence of Fourier series in Sobolev space norms is proved.

Results extend classical Fourier analysis to Sobolev orthogonal polynomial expansions.

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 $W_{\alpha,\beta}^{p,m}$.