# Function Spaces and Wavelets on Domains

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*Hans Triebel*

A publication of the European Mathematical Society

Wavelets have emerged as an important tool in analyzing functions
containing discontinuities and sharp spikes. They were developed independently
in the fields of mathematics, quantum physics, electrical
engineering, and seismic geology. Interchanges between these fields
during the last ten years have led to many new wavelet applications such
as image compression, turbulence, human vision, radar, earthquake
prediction, and pure mathematics applications such as solving partial
differential equations.

This book develops a theory of wavelet bases and wavelet frames for
function spaces on various types of domains. Starting with the usual spaces on
Euclidean spaces and their periodic counterparts, the exposition moves on to
so-called thick domains (including Lipschitz domains and snowflake domains).
Specifically, wavelet expansions and extensions to corresponding spaces on
Euclidean \(n\)-spaces are developed. Finally, spaces on smooth and
cellular domains and related manifolds are treated.

Although the presentation relies on the recent theory of function spaces,
basic notation and classical results are repeated in order to make the text
self-contained.

This book is addressed to two types of readers: researchers in the theory of
function spaces who are interested in wavelets as new effective building blocks
for functions and scientists who wish to use wavelet bases in classical
function spaces for various applications. Adapted to the second type of
reader, the preface contains a guide on where to find basic definitions and
key assertions.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in analysis.