An Axiomatic Approach To Function Spaces Spectral Synthesis And Luzin Approximation
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An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation
Author: Lars Inge Hedberg
language: en
Publisher: American Mathematical Soc.
Release Date: 2007
The authors define axiomatically a large class of function (or distribution) spaces on $N$-dimensional Euclidean space. The crucial property postulated is the validity of a vector-valued maximal inequality of Fefferman-Stein type. The scales of Besov spaces ($B$-spaces) and Lizorkin-Triebel spaces ($F$-spaces), and as a consequence also Sobolev spaces, and Bessel potential spaces, are included as special cases. The main results of Chapter 1 characterize our spaces by means of local approximations, higher differences, and atomic representations. In Chapters 2 and 3 these results are applied to prove pointwise differentiability outside exceptional sets of zero capacity, an approximation property known as spectral synthesis, a generalization of Whitney's ideal theorem, and approximation theorems of Luzin (Lusin) type.
An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation
Author: Lars Inge Hedberg
language: en
Publisher: American Mathematical Soc.
Release Date: 2007
The authors define axiomatically a large class of function (or distribution) spaces on $N$-dimensional Euclidean space. The crucial property postulated is the validity of a vector-valued maximal inequality of Fefferman-Stein type.
Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration
Author: Hans Triebel
language: en
Publisher: European Mathematical Society
Release Date: 2010
The first chapters of this book deal with Haar bases, Faber bases and some spline bases for function spaces in Euclidean $n$-space and $n$-cubes. These are used in the subsequent chapters to study sampling and numerical integration preferably in spaces with dominating mixed smoothness. The subject of the last chapter is the symbiotic relationship between numerical integration and discrepancy, measuring the deviation of sets of points from uniformity. This book is addressed to graduate students and mathematicians who have a working knowledge of basic elements of function spaces and approximation theory and who are interested in the subtle interplay between function spaces, complexity theory and number theory (discrepancy).