Eigenvalues, Embeddings and Generalised Trigonometric Functions

1. Front Matter

   Pages i-xi

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2. Basic Material

   • Jan Lang, David Edmunds

   Pages 1-31

3. Trigonometric Generalisations

   • Jan Lang, David Edmunds

   Pages 33-48

4. The Laplacian and Some Natural Variants

   • Jan Lang, David Edmunds

   Pages 49-63

5. Hardy Operators

   • Jan Lang, David Edmunds

   Pages 65-71

6. s-Numbers and Generalised Trigonometric Functions

   • Jan Lang, David Edmunds

   Pages 73-104

7. Estimates of s-Numbers of Weighted Hardy Operators

   • Jan Lang, David Edmunds

   Pages 105-128

8. More Refined Estimates

   • Jan Lang, David Edmunds

   Pages 129-151

9. A Non-Linear Integral System

   • Jan Lang, David Edmunds

   Pages 153-182

10. Hardy Operators on Variable Exponent Spaces

    • Jan Lang, David Edmunds

    Pages 183-209

11. Back Matter

    Pages 211-220

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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.

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

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)

• Department of Mathematics, Ohio State University, Columbus, USA

  Jan Lang

• Department of Mathematics, University of Sussex, Brighton, United Kingdom

  David Edmunds

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