Haar wavelet characterization of dyadic Lipschitz regularity

• Hugo Aimar ^[1] ; Carlos Exequiel Arias ^[1] ; Ivana Gómez ^[1]
  1. [1] Instituto de Matem´atica Aplicada del Litoral “Dra. Eleonor Harboure”, Santa Fe, Argentina
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 68, Nº. 1, 2025, págs. 49-54
• Idioma: inglés
• DOI: 10.33044/revuma.3574
• Enlaces
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• Resumen

  • We obtain a necessary and sufficient condition on the Haar coefficients of a real function ff defined on R+R+ for the Lipschitz αα regularity of ff with respect to the ultrametric δ(x,y)=inf{|I|:x,y∈I;I∈D}δ(x,y)=inf{|I|:x,y∈I;I∈D}, where DD is the family of all dyadic intervals in R+R+ and αα is positive. Precisely, f∈Lipδ(α)f∈Lipδ(α) if and only if |⟨fhjk⟩|≤C2−(α+1/2)j|⟨fhkj⟩|≤C2−(α+1/2)j for some constant CC, every j∈Zj∈Z and every k=0,1,2,…k=0,1,2,… Here, as usual, hjk(x)=2j/2h(2jx−k)hkj(x)=2j/2h(2jx−k) and h(x)=X[0,1/2)(x)−X[1/2,1)(x)h(x)=X[0,1/2)(x)−X[1/2,1)(x).

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