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Inequivalence of Wavelet Systems in L ( d ) and B V ( d )

Paweł Bechler (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces L ( d ) and B V ( d ) are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in L ( d ) is also shown.

Isomorphisms of AC(σ) spaces

Ian Doust, Michael Leinert (2015)

Studia Mathematica

Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if AC(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.

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