Harmonic Analysis on Spaces of Homogeneous Type.

Deng, Donggao; Yongsheng Han

ISBN 10: 354088744X / ISBN 13: 9783540887447

Published by Berlin, Springer., 2009

About this Item

Gr.-8°. XII, 154 p. Softcover. Versand aus Deutschland / We dispatch from Germany via Air Mail. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Stamped/gestempelt. Book may have slight wear to the edges. Lecture Notes in Mathematics, Vol. 1966. Sprache: Englisch.

Seller Inventory # 5089DB

Bibliographic Details

• Title: Harmonic Analysis on Spaces of Homogeneous Type.
• Publisher: Berlin, Springer.
• Publication Date: 2009
• Language: English
• ISBN 10: 354088744X
• ISBN 13: 9783540887447
• Binding: Soft cover
• Seller catalogs:
  • Mathematik

About this title

Synopsis

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.

From the Back Cover

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.