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Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

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Abstract

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain

    Eugenio Hernández & Daniel Vera

Authors
  1. Eugenio Hernández
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  2. Daniel Vera
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Corresponding author

Correspondence to Eugenio Hernández.

Additional information

Communicated by K. Gröchenig.

This research was supported by GrantS MTM2007-60952 and MTM2010-16518 of Spain.

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Hernández, E., Vera, D. Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation. Monatsh Math 169, 187–217 (2013). https://doi.org/10.1007/s00605-012-0425-6

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  • DOI: https://doi.org/10.1007/s00605-012-0425-6

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