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© 2011

Eigenvalues, Embeddings and Generalised Trigonometric Functions

  • Review of recent developments in approximation theory for Hardy-type operators and Sobolev embeddings (description of the exact values of s-numbers and widths)

  • A special chapter devoted to the theory of generalized trigonometric functions (presented for the first time in a book)

  • Description of connections between optimal approximations, eigenvalues for the p-Laplacian and generalized trigonometric functions

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2016)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Jan Lang, David Edmunds
    Pages 1-31
  3. Jan Lang, David Edmunds
    Pages 33-48
  4. Jan Lang, David Edmunds
    Pages 49-63
  5. Jan Lang, David Edmunds
    Pages 65-71
  6. Jan Lang, David Edmunds
    Pages 73-104
  7. Jan Lang, David Edmunds
    Pages 105-128
  8. Jan Lang, David Edmunds
    Pages 129-151
  9. Jan Lang, David Edmunds
    Pages 153-182
  10. Jan Lang, David Edmunds
    Pages 183-209
  11. Back Matter
    Pages 211-220

About this book

Introduction

The main theme of the book is the study, from the standpoint of s-numbers, of integral operators of Hardy type and related Sobolev embeddings. In the theory of s-numbers the idea is to attach to every bounded linear map between Banach spaces a monotone decreasing sequence of non-negative numbers with a view to the classification of operators according to the way in which these numbers approach a limit: approximation numbers provide an especially important example of such numbers. The asymptotic behavior of the s-numbers of Hardy operators acting between Lebesgue spaces is determined here in a wide variety of cases. The proof methods involve the geometry of Banach spaces and generalized trigonometric functions; there are connections with the theory of the p-Laplacian.

Keywords

41A35, 41A46, 47B06, 33E30, 47G10, 35P05 eigenvalues and eigenfunctions generalized trigonometric functions p-Laplacian s-numbers spectral theory on Banach spaces

Authors and affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA
  2. 2.Department of MathematicsUniversity of SussexBrightonUnited Kingdom

Bibliographic information

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Reviews

From the reviews:

“This well-written book deals with asymptotic behavior of the s-numbers of Hardy operators on Lebesgue spaces via methods of geometry of Banach spaces and generalized trigonometric functions. … This book contains many interesting results that are proved in detail and are usually preceded by technical lemmas. The list of references is very rich and up to date. Many open problems are pointed out. We warmly recommend it.” (Sorina Barza, Mathematical Reviews, Issue 2012 e)