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
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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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DOI: https://doi.org/10.1007/978-3-642-18429-1_5
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18267-9
Online ISBN: 978-3-642-18429-1
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