Harmonic Analysis on Spaces of Homogeneous Type

Donggao Deng

ISBN 10: 354088744X / ISBN 13: 9783540887447

Editore: Springer Berlin Heidelberg, 2008

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Druck auf Anfrage Neuware - Printed after ordering - This book could have been entitled 'Analysis and Geometry.' The authors are addressing the following issue: Is it possible to perform some harmonic analysis on a set Harmonic analysis on groups has a long tradition. Here we are given a metric set X with a (positive) Borel measure and we would like to construct some algorithms which in the classical setting rely on the Fourier transformation. Needless to say, the Fourier transformation does not exist on an arbitrary metric set. This endeavor is not a revolution. It is a continuation of a line of research whichwasinitiated,acenturyago,withtwofundamentalpapersthatIwould like to discuss brie y. The rst paper is the doctoral dissertation of Alfred Haar, which was submitted at to University of Gottingen in July 1907. At that time it was known that the Fourier series expansion of a continuous function may diverge at a given point. Haar wanted to know if this phenomenon happens for every 2 orthonormal basis of L [0,1]. He answered this question by constructing an orthonormal basis (today known as the Haar basis) with the property that the expansion (in this basis) of any continuous function uniformly converges to that function.

Codice articolo 9783540887447

Dati bibliografici

• Titolo: Harmonic Analysis on Spaces of Homogeneous Type
• Editore: Springer Berlin Heidelberg
• Data di pubblicazione: 2008
• Lingua: Inglese
• ISBN 10: 354088744X
• ISBN 13: 9783540887447
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 10 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 10 millimetri di profondità
• Peso: 277 grammi

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Riassunto

The dramatic changes that came about in analysis during the twentieth century are truly amazing. In the thirties, complex methods and Fourier series played a seminal role. After many improvements, mostly achieved by the Calderón-Zygmund school, the action today is taking place in spaces of homogeneous type. No group structure is available and the Fourier transform is missing, but a version of harmonic analysis is still available. Indeed the geometry is conducting the analysis. The authors succeed in generalizing the construction of wavelet bases to spaces of homogeneous type. However wavelet bases are replaced by frames, which in many applications serve the same purpose.