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An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation

Lars Inge Hedberg Linköping University, Linköping, Sweden
Yuri Netrusov University of Bristol, Bristol, UK
Available Formats:
Electronic ISBN: 978-1-4704-0486-4
Product Code: MEMO/188/882.E
List Price: $66.00 MAA Member Price:$59.40
AMS Member Price: $39.60 Click above image for expanded view An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation Lars Inge Hedberg Linköping University, Linköping, Sweden Yuri Netrusov University of Bristol, Bristol, UK Available Formats:  Electronic ISBN: 978-1-4704-0486-4 Product Code: MEMO/188/882.E  List Price:$66.00 MAA Member Price: $59.40 AMS Member Price:$39.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1882007; 97 pp
MSC: Primary 46; Secondary 26; 31; 41;

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.

• Chapters
• Introduction. Notation
• 1. A class of function spaces
• 2. Differentiability and spectral synthesis
• 3. Luzin type theorems
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Volume: 1882007; 97 pp
MSC: Primary 46; Secondary 26; 31; 41;

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.

• Chapters
• Introduction. Notation
• 1. A class of function spaces
• 2. Differentiability and spectral synthesis
• 3. Luzin type theorems
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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