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s-Numbers and Generalised Trigonometric Functions

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

Abstract

This chapter gives a review of the various s-numbers and n-widths, their properties and relations; an important subclass of s-numbers, the strict s-numbers, is identified. Then we focus on the Hardy operator (with functions u and v both identically equal to 1) and certain first-order Sobolev embeddings, and show that the generalised trigonometric functions play an essential rôle in the derivation of estimates of s-numbers of these maps.In particular, for the Hardy operator \(T\,:\,L_p(I) \longrightarrow L_p(I)\) where \(1\,< \,P\,<\,\infty,\) I is a bounded interval in \(\mathbb{R}\) and \(T \,f(x)\,=\,\int\limits_a^x f(t)dt,\) it is shown that all the strict s-numbers of T coincide and are given by an explicit formula.

Keywords

  • Banach Space
  • Trigonometric Function
  • Extension Property
  • Approximation Number
  • Sobolev Embedding

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Jan Lang .

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© 2011 Springer-Verlag Berlin Heidelberg

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Lang, J., Edmunds, D. (2011). s-Numbers and Generalised Trigonometric Functions. In: Eigenvalues, Embeddings and Generalised Trigonometric Functions. Lecture Notes in Mathematics(), vol 2016. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18429-1_5

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