Fourier series of Jacobi-Sobolev polynomial

2018/06/21 by Óscar Ciaurri, Ciaurri, Óscar, Judit Mínguez +1
Mathematics · #33C47 #42A20 #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics #math.FA #msc:33C47 #msc:42A20

paper · pdf · doi:10.48550/arxiv.1806.08105

openalex publication_date 2018/06/21 · arxiv created 2018/06/22 · arxiv updated 2018/06/25 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let \q_n^(α,β,m)(x)\_{n≥ 0} be the orthonormal polynomials respect to the Sobolev-type inner product ⟨ f,g⟩_α,β,m=∑_k=0^m ∫_-1¹f^(k)(x)g^(k)(x) dw_α+k,β+k(x), α,β>-1, m≥ 1, 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_α,β^p,m. As a consequence we deduce the convergence of such partial sums in the norm of W_α,β^p,m.

Citations

• Mean convergence of Jacobi series

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