Toeplitz Determinants and Szegö's Formula

THEOREM 1. If where un(\jz) and vm(z) are non-zero for \z\ 2: rx and \z\ 5g r2 respectively, and where rx< 1 < r 2 and uo^O^vo, and if Infm>n{z) = for z e Q = {z;r1 < [«| < r. , n) = lk0 (p = 0), and s ^mt then oo (6-0) Ds(fmJ = exp [sko+ lpkm>vkn^vl J>=I

theorem 1. We further note that if, as n -> oo, (1) un I —I -»- u I —I dominatedly for \z\ > rx {rx ^ 1) (6.10) (2) knt_v -> k_p (3) ^p\km>v\ converges for all m, then we may take the limit as n -> oo on both sides of the equation in (6.8) giving * • • © • COROLLARY. The theorem can be extended to the case where un(e~ie) ^ 0 and vm(z) ^ 0 for \z\ ^ 1. PROOF. Since un(e~ie) and vm(eie) are non-zero polynomials which are continuous and bounded and ljvm(z) is analytic for |^| sS 1, it follows that

Theorem 1 can now be used to extend the formula to generating functions which admit a Laurent expansion.

THEOREM 2. / / the function f(z) admits a Laurent expansion ^,%L-aoci>zV which is convergent and non-zero for z eQ = {z; rx < |z| < r2, rx < 1 < r2}, and if ln/(z) = ^ ^ z ' for zeQ and \f(ei0)\ ^ r0 = min (ljrlt r2), then u (I (7.0) Vun.Dm{f)erJat1t* = exp fn->oo PROOF. Since f(z) is analytic and non-zero for z e Q it follows that the principal branch of In f(z) is analytic in the same region and can be expanded in a Laurent series ^?ookJ)z1'. Hence ^L\k_vz-V and 25^i^»*p a r e anatytic functio

theorem 1 can be applied to give for s ^ m (7-5) D.(fmJ = Dm{fmJ = exp [ J ^ m . A,-*] • The condition that f{z) and hi f(z) are analytic inside Q is a very strong assumption which ensures that all the convergence that will be dealt with in this section is uniform and in fact exponentially fast. It should be noted that only the singularities and/or zeros of f(z) on the boundaries of Q do determine the behaviour of Dm(f), in particular if u(l/z) and v(z) have a common zero on Co, then Dm(f) vanis

THEOREM 3. // f{z) e E, then the Toeplitz determinant Dm(f) satisfies Szego's formula (8.1) HmDm(/) • <r-*» = exp f f pkpk_p] . https://doi.org/10.1017/S1446788700005668 Published online by Cambridge University Press

LEMMA 1. // g(z) = ^=oavzp converges and is analytic for \z\ < 1, g{ew) exists and g(reie) ->g(eie) pointwise as r -> 1, and \g(reid)\ ^K(B), for 0 ^ r < 1, where K(6) e L2(0, 2n), then https://doi.org/10.1017/S1446788700005668 Published online by Cambridge University Press

COROLLARY 1. The assumptions on g(z) may be replaced by (i) analytic for \z\ < 1 (ii) continuous for \z\ ^ 1

COROLLARY 2. The results (8.6), (8.7) and (8.8) with r = 1 also hold forg(e-<°). https://doi.org/10.1017/S1446788700005668 Published online by Cambridge University Press

LEMMA 2. / / satisfy the conditions of Lemma 1, 2/sen (8.13) 1 f " ^ M * " " ) * = I «•*-.. PROOF. By Lemma 1 we have that (8.14) g(O ! -0 oo (8.15) h{e~ie) = and that both functions are in L2(0, 2n). By Parseval's theorem the product of the Fourier series may be integrated term by term, giving (8.13).

Lemma 1 and so by Lemma 2 we arrive at (8.25) lim Dm(fm) = exp [ |pk,k_,] < oo. 8 See Ref. 12, p. 390, ex. iv. https://doi.org/10.1017/S1446788700005668 Published online by Cambridge University Press