Discrete-continuous Jacobi-Sobolev spaces and Fourier series

Abstract

Let p≥1,ℓ∈N,α,β>−1 and ϖ=(ω0,ω1,…,ωℓ−1)∈Rℓ. Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as:

∥f∥s,p:=(∑k=0ℓ−1∣∣f(k)(ωk)∣∣p+∫1−1∣∣f(ℓ)(x)∣∣pdμα,β(x))1p, where dμα,β(x)=(1−x)α(1+x)βdx. Obviously, ∥⋅∥s,2=⟨⋅,⋅⟩s−−−−√, where ⟨⋅,⋅⟩s is the inner product

⟨f,g⟩s:=∑k=0ℓ−1f(k)(ωk)g(k)(ωk)+∫1−1f(ℓ)(x)g(ℓ)(x)dμα,β(x). In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type Lp, in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm ∥⋅∥s,p and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in ∥⋅∥s,p norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to ⟨⋅,⋅⟩s.