TOEPLITZ DETERMINANTS WITH ONE FISHER-HARTWIG SINGULARITY

摘要

Let c be a function defined on the unit circle with Fourier coefficients {c(n)}(n=-infinity)(infinity). The Fisher-Hartwig conjecture describes the asymptotic behaviour of the determinants of the n x n Toeplitz matrices D-n(c) = det[c(i-j)](i,j=0)(n-1) 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(i0)) = b(e(i0)) t(beta)(e(i(0-0i))) u(alpha)(e(i(0-0i))). Here t(beta)(e(i0)) = exp(i beta(theta - pi)), 0 < theta < 2 pi, is a function with a jump discontinuity, u(alpha)(e(i0)) = (2 - 2 cos theta)(alpha) 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 alpha is required not to be a negative integer. In the case where Re alpha less than or equal to - 1/2, i.e., where the corresponding function c is not integrable, we identify c in an appropriate way with a distribution. (C) 1997 Academic Press. [References: 18]