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Trace theorems of Szegő type in the singular case. (Théorémes de trace de type Szegő dans le cas singulier.) (French) Zbl 1089.47028

The authors consider \(N\)-order Toeplitz matrices \(T_{N,f}\) generated by functions \(f\) defined on the unit circle of the form \(f(e^{i\theta})=| 1-e^{i\theta}| ^{2\alpha} f_1(e^{i\theta})\), where \(\theta\in (-1/2,1/2)\) and \(f_1\) is a smooth and strictly positive function on the unit circle. The main purpose is to derive asymptotic formulas for the diagonal entries and for the trace of the inverse \(T_{N,f}^{-1}\) as \(N\rightarrow \infty\). The obtained result distinguishes between the cases \(\alpha\in (-1/2,0)\) and \(\alpha\in (0,1/2)\). This paper can be seen as the continuation of the papers by the same authors [Integral Equations Oper. Theory 50, No. 1, 83–114 (2004; Zbl 1069.47027)] and [C. R., Math., Acad. Sci. Paris 335, No. 8, 705–710 (2002; Zbl 1012.65025)].

MSC:

47B39 Linear difference operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
65F05 Direct numerical methods for linear systems and matrix inversion
15A15 Determinants, permanents, traces, other special matrix functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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