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On the trigonometric moment problem in two dimensions

Authors
Journal
Indagationes Mathematicae
0019-3577
Publisher
Elsevier
Volume
23
Issue
4
Identifiers
DOI: 10.1016/j.indag.2012.09.003
Keywords
  • Trigonometric Moment Problem
  • Band Method
  • Gohberg–Semençul
  • Toeplitz
  • Maximal Entropy
  • One Step Extension
Disciplines
  • Mathematics

Abstract

Abstract A finite sequence c0,…,ck of complex numbers is the set of trigonometric moments of a measure dμ=∣Pk(eiθ)∣−2dθ with Pk(z) a polynomial of degree k, zero-free in |z|≤1, providing only that the Hermitian Toeplitz matrix having c0,…,ck as its top row is positive-definite. The analogous representation in two dimensions for a rectangular array of numbers requires additional conditions, discovered by J.S. Geronimo and H.J. Woerdeman. We use orthogonal decomposition to give an alternate proof of their important result.

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