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Über dieses Buch

Classical harmonic analysis is an important part of modern physics and mathematics, comparable in its significance with calculus. Created in the 18th and 19th centuries as a distinct mathematical discipline it continued to develop, conquering new unexpected areas and producing impressive applications to a multitude of problems. It is widely understood that the explanation of this miraculous power stems from group theoretic ideas underlying practically everything in harmonic analysis. This book is an unusual combination of the general and abstract group theoretic approach with a wealth of very concrete topics attractive to everybody interested in mathematics. Mathematical literature on harmonic analysis abounds in books of more or less abstract or concrete kind, but the lucky combination as in this volume can hardly be found.



Chapter 1. Convolution and Translation in Classical Analysis

If the question arises about the most important object of commutative harmonic analysis, priority would be given to the convolution.
V. P. Havin, N. K. Nikolski

Chapter 2. Invariant Integration and Harmonic Analysis on Locally Compact Abelian Groups

Two important achievements of eighteenth-century mathematics — one connected to the appearance of harmonic analysis as a theory of trigonometric series and integrals, the other connected to the discovery of the role of multiplicative characters in number theory — turned out to be special cases of harmonic analysis on LCA groups created in the thirties of this century. The development of this theory was determined by two discoveries made during this period: the existence and uniqueness of invariant measures on arbitrary locally compact groups and the duality theory for locally compact abelian groups. Both discoveries had their roots in the mathematics of the end of the nineteenth century, above all in the work of Poincaré.
V. P. Havin, N. K. Nikolski


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