A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Non-discrete Jacobi-sobolev Inner Product

Let {Q n (x)}n≥0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product ⟨f, g⟩ = ∫ 1 −1 f(x)g(x)dμα,β(x) + λ ∫ 1 −1 f (x)g(x)dμα+1,β(x) where λ > 0 and dμα,β(x) = (1− x)α(1 + x)βdx with α > −1, β > −1. In this paper we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials {Q n (x)}n. Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally, the failure of a.e. convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved.