Asymmetric Hölder Spaces of Sign Sensitive Weighted Integrable Functions

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


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.


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

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