离散-连续Jacobi-Sobolev空间与傅里叶级数
经典分析与常微分方程
2020-02-11 v2 复变函数
摘要
设p ≥ 1 p\geq 1 p ≥ 1 ℓ ∈ \NN \ell\in \NN ℓ ∈ \NN α , β > − 1 \alpha,\beta>-1 α , β > − 1 ϖ = ( ω 0 , ω 1 , … , ω ℓ − 1 ) ∈ \RR ℓ \varpi=(\omega_0,\omega_1, \dots, \omega_{\ell-1})\in \RR^{\ell} ϖ = ( ω 0 , ω 1 , … , ω ℓ − 1 ) ∈ \RR ℓ f f f f f f \normSp f : = ( ∑ k = 0 ℓ − 1 ∣ f ( k ) ( ω k ) ∣ p + ∫ − 1 1 ∣ f ( ℓ ) ( x ) ∣ p d \Jm ( x ) ) 1 p , \normSp{f}:= \left(\sum_{k=0}^{\ell-1} \left|f^{(k)}(\omega_{k})\right|^{p} + \int_{-1}^{1} \left|f^{(\ell)}(x)\right|^{p} d\Jm(x)\right)^{\frac{1}{p}}, \normSp f := ( k = 0 ∑ ℓ − 1 f ( k ) ( ω k ) p + ∫ − 1 1 f ( ℓ ) ( x ) p d \Jm ( x ) ) p 1 , d \Jm ( x ) = ( 1 − x ) α ( 1 + x ) β d x d\Jm(x)=(1-x)^{\alpha} (1+x)^{\beta}dx d \Jm ( x ) = ( 1 − x ) α ( 1 + x ) β d x \normSp [ 2 ] ⋅ = \IpS ⋅ ⋅ \normSp[2]{\cdot}= \sqrt{\IpS{\cdot}{\cdot}} \normSp [ 2 ] ⋅ = \IpS ⋅ ⋅ \IpS ⋅ ⋅ \IpS{\cdot}{\cdot} \IpS ⋅ ⋅ \IpS f g : = ∑ k = 0 ℓ − 1 f ( k ) ( ω k ) g ( k ) ( ω k ) + ∫ − 1 1 f ( ℓ ) ( x ) g ( ℓ ) ( x ) d \Jm ( x ) . \IpS{f}{g}:= \sum_{k=0}^{\ell-1} f^{(k)}(\omega_{k}) \, g^{(k)}(\omega_{k}) + \int_{-1}^{1} f^{(\ell)}(x) \,g^{(\ell)}(x) d\Jm(x). \IpS f g := k = 0 ∑ ℓ − 1 f ( k ) ( ω k ) g ( k ) ( ω k ) + ∫ − 1 1 f ( ℓ ) ( x ) g ( ℓ ) ( x ) d \Jm ( x ) . L p L^p L p \normSp ⋅ \normSp{\cdot} \normSp ⋅ \IpS ⋅ ⋅ \IpS{\cdot}{\cdot} \IpS ⋅ ⋅ \normSp ⋅ \normSp{\cdot} \normSp ⋅
引用
@article{arxiv.1911.12746,
title = {Discrete-Continuous Jacobi-Sobolev Spaces and Fourier Series},
author = {Abel Díaz-González and Francisco Marcellán-Español and Héctor Pijeira-Cabrera and Wilfredo Urbina-Romero},
journal= {arXiv preprint arXiv:1911.12746},
year = {2020}
}