Asymmetric Hölder Spaces of Sign Sensitive Weighted Integrable Functions

Miguel A. Jiménez-Pozo; José M. Hernández-Morales

doi:10.26713/cma.v3i1.143

Authors

Miguel A. Jiménez-Pozo Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Ave. San Claudio y 18 Sur, C. U. de San Manuel, Puebla, Pue.72570
José M. Hernández-Morales Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Ave. San Claudio y 18 Sur, C. U. de San Manuel, Puebla, Pue.72570

DOI:

https://doi.org/10.26713/cma.v3i1.143

Keywords:

Hölder spaces, Lipschitz functions, Sign-sensitive weights, Weighted integrals, 0-equicontinuous set, Equilipschitzian set, Asymmetric norms

Abstract

We consider the space $L( u,v) $ of $2\pi$-periodic real-valued functions which are integrable with respect to a sign sensitive weight ${(u,v)}$. With some necessary hypothesis for this weight, $L( u,v) $ is an asymmetric Banach space. After defining a convenient modulus of smoothness we introduce the corresponding space $\emph{Lip}_{\alpha}(u,v) $ and its subspace $\emph{lip}_{\alpha }( u,v) $ of Hölder (or Lipschitz) functions associated to this modulus. We prove these spaces are asymmetric Banach spaces too and use the result to study approximation problems.

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