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Inverse asymptotique des matrices de Toeplitz de symbole [formula omitted], [formula omitted], et noyaux intégraux

Authors
Journal
Bulletin des Sciences Mathématiques
0007-4497
Publisher
Elsevier
Publication Date
Volume
134
Issue
2
Identifiers
DOI: 10.1016/j.bulsci.2008.03.001
Keywords
  • Inversion Des Matrices De Toeplitz
  • Symbole Singulier
  • Polynômes Orthogonaux
  • Inversion Of Toeplitz Matrix
  • Singular Symbol
  • Orthogonal Polynomial

Abstract

Abstract This paper provides us two types of results. In a first part we obtain an asymptotic expansion of the terms ( T N ( ( 1 − cos θ ) α f 1 ) ) [ N x ] + 1 , [ N y ] + 1 − 1 for α in ] − 1 2 , 1 2 [ and 0 < x < 1 , 0 < y < 1 , x ≠ y and where f 1 is a sufficiently smooth function. These expressions are given by two different integral kernels H α or G α according to α ∈ ] − 1 2 , 0 [ or α ∈ ] 0 , 1 2 [ . In the other hand we give an asymptotic expansion of the orthogonal polynomials on the unit circle with respect to the weight ( 1 − cos θ ) α f 1 for the same values of α. We denote by Φ N ( e i θ ) = ∑ i = 0 N ω N , u e i θ these polynomials. With this notation our result concern the coefficients ω N , [ N x ] with 0 < x < 1 . We can remark that the expression is the same for all the reals α in ] − 1 2 , 1 2 [ . When α goes to 1 2 we obtain the quantities ( T N ( 1 − cos θ f 1 ) ) [ N x ] + 1 , [ N y ] + 1 − 1 and ( T N ( 1 − cos θ f 1 ) ) [ N x ] + 1 , 1 − 1 that proves a conjecture of Harry Kesten and that allows us to obtain the trace of ( T N 1 − cos θ ) − 1 .

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