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Extreme and exposed points of shift-invariant spaces generated by Gaussian kernel and hyperbolic secant

Markus Valås Hagen, Alexander Ulanovskii, Denis Zelent, Ilya Zlotnikov

Abstract

We characterize the extreme and exposed points of the unit ball (with respect to the $L^1$-norm) in the shift-invariant space generated by the Gaussian function, as well as in the quasi shift-invariant space generated by the hyperbolic secant.

Extreme and exposed points of shift-invariant spaces generated by Gaussian kernel and hyperbolic secant

Abstract

We characterize the extreme and exposed points of the unit ball (with respect to the -norm) in the shift-invariant space generated by the Gaussian function, as well as in the quasi shift-invariant space generated by the hyperbolic secant.
Paper Structure (16 sections, 12 theorems, 89 equations)

This paper contains 16 sections, 12 theorems, 89 equations.

Table of Contents

  1. Introduction and Main Results
  2. Extreme and exposed points
  3. Shift-invariant spaces and quasi shift-invariant spaces
  4. Main results
  5. Preliminaries
  6. Extreme and exposed points criteria
  7. A corollary of Hayman's theorem
  8. Gaussian kernel: Proof of Theorem \ref{['th:gauss_ext']}
  9. Auxiliary lemmas
  10. Proof of Theorem \ref{['th:gauss_ext']} (i)
  11. Proof of Theorem \ref{['th:gauss_ext']} (ii)
  12. Hyperbolic secant: Proof of Theorem \ref{['th:hyp_ext']}
  13. Auxiliary results
  14. Proof of Theorem \ref{['th:hyp_ext']} (i)
  15. Proof of Theorem \ref{['th:hyp_ext']} (ii)
  16. ...and 1 more sections

Key Result

Theorem 1.2

(Gaussian kernel) (i) The set ${\rm Ext}(V^1(G_a))$ consists of those functions $f\in V^1(G_a)$ satisfying $\|f\|_{1} = 1$ and such that (ii) The set $\,{\rm Exp}(V^1(G_a))$ consists of those functions $f\in {\rm Ext}(V^1(G_a))$ for which the following two conditions hold:

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Example 1.7
  • Lemma 2.1: Extreme point criterion
  • Lemma 2.2: Exposed point criterion
  • Lemma 2.3
  • ...and 14 more