Discrete-Continuous Jacobi-Sobolev Spaces and Fourier Series

2019/11/28 by Abel Díaz-González, Díaz-González, Abel, Francisco Marcellán-Español +5
Mathematics · #33C45 #42C05 #42C10 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #math.CA #math.CV #msc:33C45 #msc:42C05 #msc:42C10

paper · pdf · doi:10.48550/arxiv.1911.12746

arxiv created 2020/02/07 · arxiv updated 2020/02/11

Abstract

Let p≥ 1, ℓ∈ \NN, α,β>-1 and \varpi=(ω₀,ω₁, …, ω_ℓ-1)∈ \RR^ℓ. Given a suitable function f, we define the discrete-continuous Jacobi-Sobolev norm of f as: \normSpf:= (∑_k=0^ℓ-1 |f^(k)(ω_k)|^p + ∫_-1¹ |f^(ℓ)(x)|^p d\Jm(x))^(1)/(p), where d\Jm(x)=(1-x)^α (1+x)^βdx. Obviously, \normSp[2]⋅= √\IpS⋅⋅, where \IpS⋅⋅ is the inner product. \IpSfg:= ∑_k=0^ℓ-1 f^(k)(ω_k) g^(k)(ω_k) + ∫_-1¹ f^(ℓ)(x) g^(ℓ)(x) d\Jm(x). In this paper, we summarize the main advances on the convergence of the Fourier-Sobolev series, in norms of type L^p, cases continuous and discrete. We study the completeness of the Sobolev space of functions associated with the norm \normSp⋅ and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in \normSp⋅ norm of the partial sum of the Fourier-Sobolev series of orthogonal polynomials with respect to \IpS⋅⋅ .

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