Asymptotic expanion of Toeplitz symbol matrices (1-cos theta)(alpha)f(1),-1/2 <alpha <= 1/2, and integral nuclei

作者：Rambour Philippe^*; Seghier Abdellatif

来源：Bulletin des Sciences Mathematiques, 2010, 134(2): 155-188.

DOI：10.1016/j.bulsci.2008.03.001

• Inversion of Toeplitz matrix
• Singular symbol
• Orthogonal polynomial

This paper provides us two types of results. In a first part we obtain an asymptotic expansion of the terms (T-N ((1-cos theta)(alpha)f(1)))([Nx]+1.vertical bar Ny vertical bar+1)(-1) for alpha in ]-1/2, 1/2[ and 0 < x < 1,0 < y < 1, x not equal y and where f(1) is a sufficiently smooth function. These expressions are given by two different integral kernels H-alpha or G(alpha) according to alpha is an element of ]-1/2 , 0[ or alpha is an element of ]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 theta)(alpha) f(1) for the same values of a. We denote by Phi(N)(e(i theta)) = Sigma(N)(i=0) omega N,u(ei theta) these polynomials. With this notation our result concern the coefficients omega(N,[Nx]) with 0 < x < 1. We can remark that the expression is the same for all the reals alpha in ]-1/2, 1/2[. When a goes to we obtain the quantities (T-N(root 1-cos theta f(1)))([Nx]+1.[ny]+1)(-1) and (T-N(root 1-cos theta f(1)))([Nx]+1,1)(-1) that proves a conjecture of Harry Kesten and that allows us to obtain the trace of (T-N root 1 - cos theta)(-1).

• 出版日期2010-3