Toeplitz Determinants with One Fisher–Hartwig Singularity

✍ Scribed by Torsten Ehrhardt; Bernd Silbermann

• Book ID: 102589431
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 435 KB
• Volume: 148
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3074

✦ Synopsis

Let c be a function defined on the unit circle with Fourier coefficients [c n ] n=& . The Fisher Hartwig conjecture describes the asymptotic behaviour of the determinants of the n_n Toeplitz matrices

for a certain class of functions c. In this paper we prove this conjecture in the case of functions with one singularity. More precisely, we consider functions of the form c(e i% )=b(e i% ) t ; (e i(%&% 1 ) ) u : (e i(%&% 1 ) ).

Here t ; (e i% )=exp(i;(%&?)), 0<%<2?, is a function with a jump discontinuity, u : (e i% )=(2&2 cos %) : is a function which may have a zero, a pole, or a discontinuity of oscillating type, and b is a sufficiently smooth nonvanishing function with winding number equal to zero. The only restriction we impose on the parameters is that 2: is required not to be a negative integer. In the case where Re : &1Â2, i.e., where the corresponding function c is not integrable, we identify c in an appropriate way with a distribution.