RT Journal Article
T1 Discrete-continuous Jacobi-Sobolev spaces and Fourier series
A1 Díaz González, Abel
A1 Marcellán Español, Francisco José
A1 Pijeira Cabrera, Héctor Esteban
A1 Urbina, Wilfredo
AB 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.
PB Springer Nature
SN 0126-6705
YR 2021
FD 2021-03
LK https://hdl.handle.net/10016/33956
UL https://hdl.handle.net/10016/33956
LA eng
NO Authors thank the valuable comments by the referees. Their suggestions have contributed to improve the presentation of this manuscript. The research of F. Marcellán and H. Pijeira-Cabrera was partially supported by Spanish State Research Agency, under Grant PGC2018-096504-B-C33. The research of A. Díaz-González was supported by the Research Fellowship Program, Ministry of Economy and Competitiveness of Spain, under grant BES-2016-076613.
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RD 27 jul. 2024