TOEPLITZ DETERMINANTS WITH ONE FISHER-HARTWIG SINGULARITY

Let c be a function defined on the unit circle with Fourier coefficien ts {c(n)}(n=-infinity)(infinity). The Fisher-Hartwig conjecture descri bes 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 func tions c. In this paper we prove this conjecture in the case of functio ns 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))). Her e t(beta)(e(i0)) = exp(i beta(theta - pi)), 0 < theta < 2 pi, is a fun ction with a jump discontinuity, u(alpha)(e(i0)) = (2 - 2 cos theta)(a lpha) is a function which may have a zero, a pole, or a discontinuity of oscillating type, and b is a sufficiently smooth nonvanishing funct ion 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 int eger. In the case where Re alpha less than or equal to - 1/2, i.e., wh ere the corresponding function c is not integrable, we identify c in a n appropriate way with a distribution. (C) 1997 Academic Press.