Toeplitz Determinants and Szegö's Formula

Theorem 2. If the function f(z) admits a Laurent expansion $\sum_{p=-\infty}^{\infty} c_p z^p$ which is convergent and non-zero for $z \in \Omega = \{z; r_1 < |z| < r_2, r_1 < 1 < r_2\}$ , and if $\ln f(z) = \sum_{-\infty}^{\infty} k_p z^p$ for $z \in \Omega$ and $|f(e^{i\theta})| \le r_0 = \min(1/r_1, r_2)$ , then (7.0) $$\lim_{m\to\infty} D_m(f)e^{-mk_0} = \exp\left[\sum_{p=1}^{\infty} pk_p k_{-p}\right].$$ PROOF. Since f(z) is analytic and non-zero for $z \in \Omega$ it follows that the principal branch of $\ln f(z)$ is analytic in the same region and can be expanded in a Laurent series $\sum_{-\infty}^{\infty} k_p z^p$ . Hence $\sum_{p=1}^{\infty} k_{-p} z^{-p}$ and $\sum_{p=1}^{\infty} k_p z^p$ are analytic functions for $|z| > r_1$ and $|z| < r_2$ respectively. The function f(z) can now be written in the form (7.1) $$f(z) = \frac{u\left(\frac{1}{z}\right)}{v(z)} = \frac{u_0}{v_0} \cdot \left[\frac{u\left(\frac{1}{z}\right)/u_0}{v(z)/v_0}\right]$$ by defining (7.2) and $$u\left(\frac{1}{z}\right) / u_0 = \exp\left[\sum_{p=1}^{\infty} k_{-p} z^{-p}\right]$$ $$v(z)/v_0 = \exp\left[-\sum_{p=1}^{\infty} k_p z^p\right]$$ together with (7.3) $$\frac{u_0}{v_0} = e^{k_0} \qquad (u_0 \neq 0 \neq v_0),$$ from which it is clear that u(1/z) and v(z) are analytic for $|z| > r_1$ and $|z| < r_2$ respectively. It is sometimes more convenient to take the form of f(z) in (7.1), rather than the existence of the Laurent expansion of $\ln f(z)$ , as the starting point of the investigation. Expanding the series in their respective regions of convergence gives (7.4) $$u\left(\frac{1}{z}\right) = u_0 + \frac{u_1}{z} + \dots + \frac{u_n}{z^n} + \dots \qquad (|z| > r_1)$$ $$v(z) = v_0 + v_1 z + \dots + v_m z^m + \dots \qquad (|z| < r_2).$$ From (7.1) and (1.4) it is clear that without loss of generality we may assume that $u_0 = v_0 = 1$ . For sufficiently large m and n, i.e. for all m, $n > N_0$ , the partial sums $u_n(1/z)$ and $v_m(z)$ will, by Hurwitz's theorem [12], be non-zero for $|z| > r_1$ and $|z| < r_2$ respectively, and hence on forming the rational approximant $$f_{m,n}(z) = \frac{u_n\left(\frac{1}{z}\right)}{v_m(z)},$$ theorem 1 can be applied to give for $s \ge m$ (7.5) $$D_s(f_{m,n}) = D_m(f_{m,n}) = \exp\left[\sum_{p=1}^{\infty} p k_{m,p} k_{n,-p}\right].$$ The condition that f(z) and $\ln f(z)$ are analytic inside $\Omega$ 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 $\Omega$ do determine the behaviour of $D_m(f)$ , in particular if u(1/z) and v(z) have a common zero on $C_0$ , then $D_m(f)$ vanishes as $m \to \infty$ . The conditions of (6.10) are clearly satisfied since u(1/z) is analytic for $|z| > r_1$ and the coefficients $k_{m,p}$ and $k_p$ are the Laurent coefficients of functions analytic inside $\Omega$ with $r_1 < 1 < r_2$ and $r_0 > 1$ . We have in fact that (7.6) (i) $$|k_p| = O\left(\frac{1}{r_0^p}\right), \quad |k_{m,p}| = O\left(\frac{1}{r_0^p}\right),$$ $$(7.7) (ii) k_{m,p} = k_p (p \le m),$$ (7.8) $$|k_{m,p} - k_p| = O\left(\frac{1}{r_0^p}\right)$$ $(p > m).$ The second relation is a direct consequence of the definition (7.9) $$\ln v_m(z) = \sum_{q=1}^{\infty} \frac{(-1)^q}{q} (v_1 z + \dots + v_m z^m)^q = -\sum_{p=1}^{\infty} k_{m,p} z^p,$$ and the third relation follows from the fact that (7.10) $$k_{m,p} - k_p = \frac{1}{2\pi i} \int_C \sum_{q=1}^{\infty} \frac{1}{q} \left[ \frac{Q_{m+1}(z)}{v(z)} \right]^q \frac{dz}{z^{p+1}},$$ where $Q_{m+1}(z) = v(z) - v_m(z)$ and C is in $\Omega$ . After interchanging the summation and integration it is seen that $k_{m,p} - k_p$ equals, to the first order, the coefficient of $z_p$ in $$\frac{Q_{m+1}(z)}{v(z)},$$ i.e. (7.11) $$|k_{m,p}-k_p|=O\left(|v_p|\right)=O\left(\frac{1}{r_p^p}\right).$$ Because rx < 1 it is easily seen that (7.6), (7.7) and (7.8) also hold for K,-, and £\_". Whence by (6.11) we arrive at (7.12) $$D_{m}(f_{m}) = \exp \left[ \sum_{n=1}^{\infty} p k_{m, p} k_{-p} \right] = e^{A_{m}}$$ which clearly converges by (7.6) and where $$f_m(z) = \frac{u\left(\frac{1}{z}\right)}{v_m(z)}$$ Taking the limit as tn -> oo gives with aid of (7.8) the required result (7.13) $$\lim_{m\to\infty} D_m(f_m) = \exp\left[\sum_{p=1}^{\infty} p k_p k_{-p}\right] = e^{A}$$ which also converges by (7.6). Alternatively we can employ the integral in (6.6) and show, using the uniformity of the convergence, that the double limit lim^.^ lim,,^.,,,, may be taken under the integral sign, and that the series expansion of the integrand may be integrated term by term to give (7.13). There remains to be shown that [Dm{fJ—Dm(J)] -+0 as m -+ oo, and for this use will be made of the Landsberg-P61ya representation which enables a Toeplitz determinant of order n to be written as an »-fold multiple integral. The Toeplitz determinant Dn(j) generated by the function F(6) = f(eie), which has a Fourier series, may be written [13] as $$(7.14) \quad D_n(f) = \frac{2^{n(n-1)}}{n!} \cdot \frac{1}{(2\pi)^n} \int_0^{2\pi} \cdots \int_0^{2\pi} \prod_{\mu < \nu}^{1 \cdots n} \sin^2 \left[ \frac{\theta_{\mu} - \theta_{\nu}}{2} \right] \prod_{r=1}^n F(\theta_r) d\theta_r$$ On putting F(0) = 1, i.e. c0 = 1 and cr = c\_r = 0, this gives that $$(7.15) \quad 1 = \frac{2^{n(n-1)}}{n!} \cdot \frac{1}{(2\pi)^n} \int_0^{2\pi} \cdots \int_0^{2\pi} \prod' \sin^2 \left[ \frac{\theta_{\mu} - \theta_{\nu}}{2} \right] d\theta_1 \cdots d\theta_n,$$ where JJ' stands for the product ranging over (i, v = 1, • • •, n with fi < v. Hence it is clear that $$(7.16) |D_n(f)| \le D_n(|f|) \le [\max_{\theta \in I} |F(\theta)|]^n,$$ where Ix is the closed interval [0, 2n]. Since $$\frac{2^{n(n-1)}}{n!} \sim \left(\frac{2^n}{n}\right)^n \to \infty$$ this rules out any inequality using $$\frac{1}{2\pi}\int_0^{2\pi}|F(\theta)|d\theta,$$ mainly because no suitable upperbound, apart from unity, can be found for the product of sine terms. It is also obvious that if $|F(\theta)| < 1$ then $D_n(f)$ decays exponentially to zero. The consequences of these results are that all the functions which will be dealt with in the remaining part of this paper will be assumed to satisfy either: (i) $|F(\theta)| \le r_0$ or (ii) $F(\theta)$ is real and positive. Application of (7.14) to the function $$F(\theta) = f(e^{i\theta}) = \frac{u(e^{-i\theta})}{v(e^{i\theta})}$$ and its rational approximant $$F_m(\theta) = \frac{u(e^{-i\theta})}{v_m(e^{i\theta})},$$ both with $u_0 = v_0 = 1$ , and putting (7.17) $$column{2} c c c c c c c c c c c c c c c c c c c$$ gives that $$|\zeta_{m}| = |k'_{m} \int_{0}^{2\pi} \cdots \int_{0}^{2\pi} \left[ \prod_{r=1}^{m} F_{m}(\theta_{r}) - \prod_{r=1}^{m} F(\theta_{r}) \right]$$ $$\times \prod' \sin^{2} \left[ \frac{\theta_{\mu} - \theta_{\nu}}{2} \right] d\theta_{1} \cdots d\theta_{m}$$ $$\leq \max_{\theta_{r} \in I_{1}} \left| \prod_{r=1}^{m} F_{m}(\theta_{r}) - \prod_{r=1}^{m} F(\theta_{r}) \right|$$ $$\leq M^{m-1} \max_{\theta_{r} \in I_{1}} \sum_{r=1}^{m} |F_{m}(\theta_{r}) - F(\theta_{r})|$$ $$\leq m \cdot M^{m-1} \max_{\theta \in I_{1}} |F_{m}(\theta) - F(\theta)|,$$ $$(7.19) \qquad \leq m \cdot M^{m-1} \max_{\theta \in I_{1}} |F_{m}(\theta) - F(\theta)|,$$ where $$k'_{m} = \frac{2^{m(m-1)}}{(2\pi)^{m} m!} \quad \text{and} \quad M = \max_{\theta \in I_{1}} \{|F_{m}(\theta)|, |F(\theta)|\}.$$ To obtain an estimate of (7.19) the following equality (omitting the argument z) will be used: ^&lt;sup>7</sup> An alternative method is to use the fact that $c_p(m) = c_p + O(r_0^{-m})$ , and that the entries in $C^{-1}$ are bounded. This allows one to omit the last condition of theorem 2. $$(7.20) |f_m-f|=\left|\frac{u}{v_m}-\frac{u}{v}\right|=|f|\left|\frac{v-v_m}{v_m}\right|=|f|\left|\frac{Q_{m+1}}{v_m}\right|.$$ Now since $$|Q_{m+1}(z)| = O\left(\frac{1}{r_0^{m+1}}\right) \text{ for } z \in \Omega$$ and $F(\theta)$ is bounded, say $|F(\theta)| \leq a$ , then it is clear from (7.20) that $$(7.21) |F_m(\theta) - F(\theta)| \leq O\left(\frac{1}{r_0^{m+1}}\right) \text{and} M \leq a \left[1 + O\left(\frac{1}{r_0^{m+1}}\right)\right].$$ Substituting this in (7.19) gives that (7.22) $$|\zeta_m| \leq m \cdot a^{m-1} \left[ 1 + O\left(\frac{1}{r_0^{m+1}}\right) \right]^{m-1} O\left(\frac{1}{r_0^{m+1}}\right),$$ which tends to zero provided that $|F(\theta)| \le a < r_0 \ (r_0 > 1)$ . Alternatively it may be written as (7.23) $$\begin{aligned} |\zeta_{m}| & \leq \max_{\theta \in I_{1}} \left| \prod_{r=1}^{m} \frac{F_{m}(\theta_{r})}{F(\theta_{r})} - 1 \right| \cdot D_{m}(|f|) \\ & \leq \left[ \max_{\theta \in I_{1}} \left\{ \left| \frac{F_{m}(\theta)}{F(\theta)} \right|, 1 \right\} \right]^{m-1} \cdot m \cdot \max_{\theta \in I_{1}} \left| \frac{F_{m}(\theta)}{F(\theta)} - 1 \right| \cdot D_{m}(|f|). \end{aligned}$$ Since $$\left|\frac{F_m(\theta)}{F(\theta)}\right| \leq 1 + O\left(\frac{1}{r_0^{m+1}}\right),$$ it follows that (7.24) $$|\zeta_m| \leq O\left(\frac{m}{r_0^{m+1}}\right) \left[1 + O\left(\frac{1}{r_0^{m+1}}\right)\right]^{m-1} \cdot D_m(|f|),$$ which tends to zero provided that $D_m(|f|)$ remains bounded or increases slower than $r_0^{m+1}/m$ . When the determinants are real it can be deduced that (7.25) $$D_m(f) = D_m(f_m) \left[ 1 \pm O\left(\frac{ma^{m-1}}{r_0^{m+1}}\right) \right].$$ From (7.12), (7.13) and (7.8) we see that $$(7.26) |e^{\Lambda_m} - e^{\Lambda}| \leq O(|\Lambda_m - \Lambda|) \leq O\left(\frac{1}{r_0^{2m+2}}\right).$$ This combined with (7.22) shows that $D_m(f)$ tends to $e^A$ at a far slower rate than $D_m(f_m)$ does, and hence brings out the rather crude estimate of (7.22). In general when uQ\vQ ^ 1 it follows directly that (7.27) $$\lim_{m\to\infty} D_m(f) \cdot e^{-mk_0} = e^A.$$

Theorem 3. / / f{z) e E, then the Toeplitz determinant Dm(f) satisfies Szego's formula (8.1) $$\lim_{m\to\infty} D_m(f) \cdot e^{-mk_0} = \exp\left[\sum_{p=1}^{\infty} p k_p k_{-p}\right].$$ PROOF. Without loss of generality we can assume that $u_0 = v_0 = 1$ . If this is not the case we simply have to replace $D_m(f)$ by $D_m(f) \cdot e^{-mk_0}$ in the final result. Conditions (b) and (c) imply that $\ln u(z)$ and $\ln v(z)$ are analytic for |z| < 1 and hence can be expanded in power series as: (8.2) $$\ln u(z) = \sum_{p=1}^{\infty} k_{-p} z^p$$ and $\ln v(z) = -\sum_{p=1}^{\infty} k_p z^p$ , where the principal branch of the logarithm is taken to ensure single valuedness. By assumption (f) the partial sums $v_m(z)$ converge uniformly for $|z| \leq 1$ to v(z) and hence (8.3) v(z) is uniformly continuous, bounded and non-zero for $|z| \le 1$ . In particular, there exists $\delta > 0$ , such that $|v(e^{i\theta})| \ge \delta > 0$ for all $\theta$ . Furthermore we have that $\ln v(z)$ is also uniformly continuous and bounded for $|z| \leq 1$ . This follows from the inequality $$|\ln z_1 - \ln z_2| \le \left(\varepsilon + \frac{1}{\mu}\right) |z_1 - z_2|,$$ where $z_1, z_2 \neq 0$ , $|z_1-z_2| < \delta_1$ and $\mu = \min\{|z_1|, |z_2|\}$ . For sufficiently large m the partial sums $v_m(z)$ are non-zero for $|z| \leq 1$ . For |z| < 1 this follows immediately from Hurwitz's theorem [12], whereas for |z| = 1 it can easily be deduced from (c) and (8.3). Thus, for all $m > N_1$ $$(8.5) v_m(e^{i\theta}) \neq 0.$$ It will now be shown that the above three results also hold for u(z) and its partial sums $u_n(z)$ . From the existence of $u'(e^{i\theta})$ and the assumption (b) it follows that u(z) and $\ln u(z)$ are uniformly continuous and hence bounded for $|z| \leq 1$ . Since both $u(e^{-i\theta})$ and $1/v(e^{i\theta})$ are bounded it follows that the Fourier coefficients $c_p$ of $$F(\theta) = \frac{u(e^{-i\theta})}{v(e^{i\theta})}$$ exist and are finite. In order to show that the partial sums $u_n(e^{i\theta})$ are non-zero for sufficiently large n, we first have to prove the uniformity of their convergence. To do this we will make use of the following Lemmas.

Lemma 1. If $g(z) = \sum_{p=0}^{\infty} a_p z^p$ converges and is analytic for |z| < 1, $g(e^{i\theta})$ exists and $g(re^{i\theta}) \to g(e^{i\theta})$ pointwise as $r \to 1$ , and $|g(re^{i\theta})| \le K(\theta)$ , for $0 \le r < 1$ , where $K(\theta) \in L^2(0, 2\pi)$ , then (8.6) (i) $$\sum_{p=0}^{\infty} a_p e^{ip\theta} \text{ is the Fourier series of } g(e^{i\theta}),$$ (8.7) (ii) $$g(e^{i\theta}) \in L^2(0, 2\pi),$$ (8.8) (iii) $$\frac{1}{2\pi} \int_0^{2\pi} |g(re^{i\theta})|^2 d\theta = \sum_{r=0}^{\infty} |a_r|^2 r^{2r}$$ for $0 \le r \le 1$ . PROOF. (i) From the power series expansion it is clear that $$(8.9) a_p r^p = \frac{1}{2\pi} \int_0^{2\pi} g(re^{i\theta}) e^{-ip\theta} d\theta \text{ for } 0 \leq r < 1.$$ Now $g(re^{i\theta})e^{-ip\theta}$ is dominated by $K(\theta) \in L^2(0, 2\pi)$ , which by Schwarz's inequality is also in $L^1(0, 2\pi)$ . Lebesgue's theorem on dominated convergence shows that it is permitted to take the limit as $r \to 1$ inside the integral, which indicates that $a_p$ is the Fourier coefficient of $g(e^{i\theta})$ , since $g(re^{i\theta}) \to g(e^{i\theta})$ pointwise as $r \to 1$ . (ii) For |z| < 1 it follows directly on multiplication of g(z) by its complex conjugate, and by the uniformity of the convergence, that (8.8) holds for r < 1. Again applying the theorem on dominated convergence, but this time to $|g(re^{i\theta})|^2$ , which is dominated by $K^2(\theta)$ , gives that $$(8.10) \quad \lim_{r\to 1} \frac{1}{2\pi} \int_0^{2\pi} |g(re^{i\theta})|^2 d\theta = \lim_{r\to 1} \sum_{p=0}^{\infty} |a_p|^2 r^{2p} = \frac{1}{2\pi} \int_0^{2\pi} |g(e^{i\theta})|^2 d\theta < \infty,$$ which shows that $g(e^{i\theta})$ is square integrable. (iii) Using Fatou's Lemma it can be seen that (8.11) $$\sum_{p=0}^{\infty} |a_p|^2 \leq \lim_{r \to 1} \sum_{p=0}^{\infty} |a_p|^2 r^{2p} = \frac{1}{2\pi} \int_0^{2\pi} |g(e^{i\theta})|^2 d\theta < \infty,$$ and hence by Abel's theorem it is clear that (8.12) $$\lim_{r\to 1} \sum_{p=0}^{\infty} |a_p|^2 r^{2p} = \sum_{p=0}^{\infty} |a_p|^2,$$ and so (8.8) also holds for r = 1.

Corollary 1. The assumptions on g(z) may be replaced by - (i) analytic for |z| < 1 - (ii) continuous for $|z| \leq 1$

Corollary 2. The results (8.6), (8.7) and (8.8) with r=1 also hold for $g(e^{-i\theta})$ .

Lemma 2. / / $$g(z) = \sum_{n=0}^{\infty} a_n z^n$$ and $h(z) = \sum_{n=0}^{\infty} b_{-n} z^n$ satisfy the conditions of Lemma 1, 2/sen (8.13) $$\frac{1}{2\pi} \int_{0}^{2\pi} g(e^{i\theta}) h(e^{-i\theta}) d\theta = \sum_{p=0}^{\infty} a_{p} b_{-p}.$$ PROOF. By Lemma 1 we have that $$g(e^{i\theta}) = \sum_{p=0}^{\infty} a_p e^{ip\theta}$$ (8.15) $$h(e^{-i\theta}) = \sum_{p=0}^{\infty} b_{-p} e^{-ip\theta}$$ 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). Application of Lemma 1 to the function u'(z) shows that u'(eie) is square integrable and has Fourier series (8.16) $$\sum_{n=1}^{\infty} p u_n e^{i(p-1)\ell}$$ with partial sums $$u'_n(e^{i\theta}) = \sum_{p=1}^n p u_p e^{i(p-1)\theta},$$ and which may be integrated to give (8.17) $$u(e^{i\theta}) = \sum_{p=0}^{\infty} u_p e^{ip\theta}.$$ Moreover 2a^o Pa \ u p\a converges and hence by Schwarz's inequality 2^.o \^u v\ ^so converges, implying that un{z) converges uniformly to u(z) for all \z\ ^ 1. Hence we deduce, as in (8.5), that 3NZ, such that ujz) ^ 0 for all n> N2 and \z\ ^ 1. These results are obviously also true for «" (e~ie). Let No = max {N-^, N2), then for all m, n > No, we can form the rational approximant un(llz)/vm(z), and use theorem 1, which gives that $$(8.18) D_m(f_{m,n}) = \exp\left[\frac{1}{2\pi}\int_0^{2\pi} \ln v_m(e^{i\theta}) \frac{u_n'(e^{-i\theta})}{u_n(e^{-i\theta})} (-e^{-i\theta}) d\theta\right].$$ Under the assumptions made and with aid of the Lemmas we can now show that the double limit limn^^ lim,^.^, may be taken under the integral sign in (8.18), and that it gives the required result. By the completeness theorem of Fourier series it follows that the partial sums in (8.16) converge in mean square to $u'(e^{i\theta})$ . The same trivially holds for $u'_n(e^{-i\theta})$ and so (8.19) $$\frac{1}{2\pi} \int_0^{2\pi} |u'_n(e^{-i\theta}) - u'(e^{-i\theta})|^2 d\theta \to 0 \text{ as } n \to \infty.$$ Similarly we have for $m > N_0$ that $$(8.20) \quad \frac{1}{2\pi} \int_0^{2\pi} \left| \ln v_m(e^{i\theta}) \left[ \frac{1}{u_n(e^{-i\theta})} - \frac{1}{u(e^{-i\theta})} \right] \right|^2 d\theta \to 0 \text{ as } n \to \infty.$$ Using the continuity of the inner product in the $L^2$ space⁸, and the fact that $u'(e^{-i\theta})$ and $\ln v_m(e^{i\theta})/u(e^{-i\theta})$ are in $L^2$ , we obtain that as $n \to \infty$ $$(8.21) \quad \frac{1}{2\pi} \int_0^{2\pi} \ln v_m(e^{i\theta}) \, \frac{d}{d\theta} \ln u_n(e^{-i\theta}) \, d\theta \to \frac{1}{2\pi} \int_0^{2\pi} \ln v_m(e^{i\theta}) \, \frac{d}{d\theta} \ln u(e^{-i\theta}) \, d\theta.$$ It is a simple matter to verify from (8.3) and the assumptions (d) and (e), that for |z| < 1 and $m > N_0$ , the functions (8.22) $$\ln v_m(z) = -\sum_{p=1}^{\infty} k_{m,p} z^p \text{ and } \frac{u'(z)}{u(z)} = \sum_{p=1}^{\infty} p k_{-p} z^p$$ satisfy the conditions of Lemma 1. Whence by Lemma 2 it immediately follows from (8.18) and (8.21) that (8.23) $$\lim_{n\to\infty} D_m(f_{m,n}) = D_m(f_m) = \exp\left[\sum_{p=1}^{\infty} p k_{m,p} k_{-p}\right] < \infty.$$ Since $u'(e^{-i\theta})/u(e^{-i\theta})$ is in $L^2$ and $\ln v_m(e^{i\theta})$ $(m>N_0)$ converges boundedly (and in fact uniformly) and hence in mean square to $\ln v(e^{i\theta})$ , we can again apply the continuity of the inner product, (or the theorem on dominated convergence), which shows that the limit as $m\to\infty$ may be taken inside the integral (8.21) giving: (8.24) $$\lim_{m\to\infty} D_m(f_m) = \lim_{m\to\infty} \frac{1}{2\pi_i} \int_0^{2\pi} \ln v_m(e^{i\theta}) \frac{d}{d\theta} \ln u(e^{-i\theta}) d\theta$$ $$= \frac{1}{2\pi_i} \int_0^{2\pi} \ln v(e^{i\theta}) \frac{d}{d\theta} \ln u(e^{-i\theta}) d\theta.$$ As in (8.22) we have that $\ln v(z)$ and u'(z)/u(z) satisfy the conditions of Lemma 1 and so by Lemma 2 we arrive at (8.25) $$\lim_{m\to\infty} D_m(f_m) = \exp\left[\sum_{p=1}^{\infty} p k_p k_{-p}\right] < \infty.$$ ^&lt;sup>8</sup> See Ref. 12, p. 390, ex. iv. It is noteworthy that Lemma 1 not only gives that $k_{-p}$ is the Fourier coefficient of $\ln u(e^{i\theta})$ and that $$|k_{-p}| = o\left(\frac{1}{p}\right),$$ but also that the series for $\ln u(z)$ in (8.2) converges for all $|z| \leq 1$ . We note that this cannot be deduced for the series for $\ln v(z)$ , and hence the Fourier series $\sum_{-\infty}^{\infty} c_x e^{iy\theta}$ need not converge to $F(\theta)$ . There remains to be shown that the determinant $D_m(f_m)$ does in fact approximate $D_m(f)$ as $m \to \infty$ . To do this we make use of (7.19), (7.20) and assumption (g). Since $|F(\theta)| \le 1$ and $|v_m(e^{i\theta})| \ge \delta > 0$ for all $m > N_0$ , it is clear from (7.20) that (8.27) $$m|F_m(\theta) - F(\theta)| = \left| \frac{F(\theta)}{v_m(e^{i\theta})} \right| \cdot m|Q_{m+1}(e^{i\theta})|,$$ which by (f) tends to zero uniformly in $\theta$ . Furthermore we can deduce that for fixed $\theta$ either $|F_m(\theta)| \leq |F(\theta)| \leq 1$ or $$|F_m(\theta)| \leq |F(\theta)| \left\{ 1 + \left| \frac{Q_{m+1}(e^{i\theta})}{v_m(e^{i\theta})} \right| \right\},$$ so that $M \leq 1 + o(1/m)$ . From (7.19) we now arrive at (8.29) $$|\zeta_m| \leq \left[1 + o\left(\frac{1}{m}\right)\right]^{m-1} \cdot m \cdot \max_{\theta \in I_1} |F_m(\theta) - F(\theta)|,$$ which by (8.27) tends to zero as $m \to \infty$ . Alternatively (7.23) can be used on exponentiating the product together with (8.4) and the facts that (8.30) $$|e^z-1| \le |z|e^{|z|}$$ and $D_m(|f|) \le 1$ . COROLLARY. (i) Conditions (d), (e) and (f) may be replaced by the stronger but simpler conditions that on $[0, 2\pi]$ - (1) $u'(e^{i\theta})$ exists and is continuous - (8.31) (2) $v'(e^{i\theta})$ exists and satisfies a Lipschitz condition of order $\alpha$ with $0 < \alpha < 1$ . These conditions give rise to uniformity on the unit circle. The first assumption implies that u(z) and u'(z) are uniformly continuous and bounded for all $|z| \leq 1$ , and hence (d) and (e) are satisfied. The second assumption, together with the fact that $v(e^{i\theta})$ is non-zero, shows that (8.2) holds and also that (8.32) $$v(e^{i\theta})$$ and $\frac{v'(e^{i\theta})}{v(e^{i\theta})}$ are in Lip $\alpha$ (0 < $\alpha$ < 1). From Lemma 1 it can be seen that $u_p$ , $v_p$ , $pk_{-p}$ and $pk_p$ are the Fourier coefficients of their respective functions. This allows the application of two theorems (generalised to complex Fourier series) from the general theory of approximations [15], to the functions $v'(e^{i\theta})/v(e^{i\theta})$ and $v(e^{i\theta})$ , giving that (8.33) $$|k_p| = O\left(\frac{1}{p^{1+\alpha}}\right) \text{ and } |Q_{p+1}(e^{i\theta})| = O\left(\frac{\ln p}{p^{1+\alpha}}\right)$$ $(p > 0).$ Whence assumption (f) is now also satisfied. Moreover it follows that (8.2) holds for all $|z| \le 1$ and so (8.34) $$\ln F(\theta) = \sum_{-\infty}^{\infty} k_p e^{ip\theta} \text{ with } \frac{u_0}{v_0} = e^{k_0},$$ and furthermore $F(\theta) = \sum_{-\infty}^{\infty} c_p e^{ip\theta}$ . (ii) We note that the existence and finiteness of $v'(e^{i\theta})$ does not necessarily imply that $mQ_{m+1}(e^{i\theta}) \to 0$ or vice versa.