Mathematics

Ya. L. Geronimus

ON SOME ESTIMATES IN THE THEORY OF TOEPLITZ FORMS AND ORTHOGONAL POLYNOMIALS

(Presented by Academician V. I. Smirnov on 23 V 1957)

1. Consider the Toeplitz forms
   \[ T_n=\sum_{i,k=0}^{n} c_{i-k}x_i\overline{x}_k,\qquad c_{-n}=\overline{c}_n,\qquad \Delta_n=\left|c_{i-k}\right|_0^n \quad (n=0,1,2,\ldots); \tag{1} \]
   they are positive definite if \(\{\Delta_n\}_0^\infty>0\). Introduce the notation
   \[ h_n=\frac{\Delta_{n+1}}{\Delta_n}\qquad (n=0,1,2,\ldots). \tag{2} \]
   It is not difficult to prove that \(0<h_{n+1}\leq h_n\) \((^6)\); therefore there exists the limit
   \[ \lim_{n\to\infty} h_n=h\geq 0. \]
   We shall consider the quantity
   \[ \mu_n=h_n-h=h_n-\lim_{n\to\infty}h_n\qquad (n=0,1,2,\ldots) \tag{3} \]
   and indicate some estimates for it.

If one introduces the parameters \((({}^1), \text{pp. }36\text{--}38)\)
\[ a_n=\frac{(-1)^n}{\Delta_n}\left|c_{i-k+1}\right|_0^n,\qquad h_n=h_0\prod_{k=0}^{n-1}\{1-|a_k|^2\}\qquad (n=0,1,2,\ldots), \tag{4} \]
then it is clear that the conditions \(\{\Delta_n\}_0^\infty>0\) are equivalent to the conditions \(\{|a_n|\}_0^\infty<1\), and the condition \(h>0\) is equivalent to the convergence of the series \(\sum_{k=0}^{\infty}|a_k|^2\); consequently, when these conditions are fulfilled, we have the estimate of the quantity \(\mu_n\) in terms of the parameters:
\[ \frac{\mu_n}{h_n}=\left|1-\prod_{k=n}^{\infty}\{1-|a_k|^2\}\right|\sim \sum_{k=n}^{\infty}|a_k|^2,\qquad \mu_n=O\left\{\sum_{k=n}^{\infty}|a_k|^2\right\}. \tag{5} \]

1. When the conditions \(\{\Delta_n\}_0^\infty>0\) are fulfilled, we have the representation
   \[ \frac{1}{2\pi}\int_0^{2\pi} e^{-ik\theta}\,d\sigma(\theta)=c_k\qquad (k=0,1,2,\ldots), \tag{6} \]
   where \(\sigma(\theta)\) is a bounded nondecreasing function with an infinite set of points of increase; if one introduces the polynomials \(\{P_n(z)\}\), orthonormal on the circle \(z=e^{i\theta}\), \(0\leq\theta\leq 2\pi\), with respect to the mass distribution \(d\sigma(\theta)\), then the quantity \(h_n\) also has the meaning
   \[ h_n=\frac{1}{\alpha_n^2},\qquad P_n(z)=\alpha_n z^n+\cdots\qquad (n=0,1,2,\ldots), \tag{7} \]

and, thus,

\[ \mu_n=\frac{1}{\alpha_n^2}-\frac{1}{\alpha^2},\qquad \lim_{n\to\infty}\alpha_n=\alpha\leqslant\infty, \tag{8} \]

The conditions

\[ h=\frac{1}{\alpha}>0,\qquad \sum_{k=0}^{\infty}|a_k|^2<\infty,\qquad \lg\sigma'(\theta)\in L_1 \tag{9} \]

are equivalent to one another by virtue of the relation ((7); (1), p. 38)

\[ h=\lim_{n\to\infty}\frac{\Delta_{n+1}}{\Delta_n} =h_0\prod_{k=0}^{\infty}\{1-|a_k|^2\} =\exp\left\{\frac{1}{2\pi}\int_{0}^{2\pi}\lg\sigma'(\theta)\,d\theta\right\}; \tag{10} \]

in what follows we shall assume that these conditions are satisfied; they, in turn, are equivalent to the condition ((1), p. 21)

\[ \lim_{n\to\infty}P_n^*(z)=\pi(z) =\exp\left\{-\frac{1}{4\pi}\int_{0}^{2\pi} \frac{e^{i\theta}+z}{e^{i\theta}-z}\lg\sigma'(\theta)\,d\theta\right\}, \]

\[ |z|<1,\qquad P_n^*(z)=z^n\overline{P_n}\left(\frac{1}{z}\right), \tag{11} \]

and for \(|z|\leqslant r<1\) we have uniformly

\[ |P_n^*(z)-\pi(z)|=o\bigl(\sqrt{\mu_n}\bigr), \tag{12} \]

i.e., the error of the asymptotic formula inside the disk \(|z|<1\) is estimated by means of \(\mu_n\). We have the growth estimate for the orthonormal polynomials:

\[ |P_n(e^{i\theta})|=\max_{0\leqslant\theta<2\pi}|P_n(e^{i\theta})| \leqslant |\pi(re^{i\theta})|\bigl(C_1+C_2\sqrt{n\mu_n}\bigr),\qquad r=1-\frac{1}{2n}. \tag{13} \]

The case \(\mu_n=O(1/n)\) is especially interesting: then we may put \(C_2=0\) in (13). If, moreover, the condition

\[ \sigma(\theta_2)-\sigma(\theta_1)\geqslant m(\theta_2-\theta_1),\qquad 0\leqslant\theta_1<\theta_2\leqslant2\pi, \tag{14} \]

is satisfied, then in the closed disk \(|z|\leqslant1\) the estimate

\[ |P_n(z)|\leqslant C_3+C_4\sqrt{n\mu_n},\qquad |z|\leqslant1 \tag{15} \]

holds. If both conditions are satisfied, the entire orthonormal system is uniformly bounded in the closed disk \((^2)\).

If \(\mu_n=o(1/n)\), then, under (14), at every point \(z_0=e^{i\theta_0}\) at which there exists the radial boundary value \(\pi(e^{i\theta_0})\), we have

\[ |P_n^*(z_0)-\pi(z_0)| \leqslant \lambda_n,\qquad \lambda_n=C_1^3\sqrt{n\mu_n}+C_2|\pi(z_0)-\pi(rz_0)|,\qquad r=1-\mu_n^{1/3}n^{-2/3}. \tag{16} \]

Under the more restrictive condition

\[ \sum_{n=1}^{\infty}\sqrt{\frac{\mu_n}{n}}<\infty \]

the function \(\sigma(\theta)\) is absolutely continuous on the whole interval \([0,2\pi]\), the function \(\pi(z)\) is continuous in the closed disk, and the asymptotic formula (16) holds with error estimate \((^5)\)

\[ \lambda_n=C\sum_{k=n+1}^{\infty}\sqrt{\frac{\mu_k}{k}}. \tag{17} \]

Let us indicate still other estimates expressed in terms of the quantity \(\mu_n\):

\[ \frac{1}{2\pi}\int_0^{2\pi}\left|\frac{P_n^+(e^{i\theta})}{\pi(e^{i\theta})}-1\right|^2\,d\theta \le C_1\mu_n;\qquad \frac{1}{2\pi}\int_0^{2\pi}|P_n(e^{i\theta})|^2\,d\sigma_1(\theta)\le C_2\mu_n; \tag{18} \]

\(\sigma_1(\theta)\) is the sum of the jump function and the singular component of the function \(\sigma(\theta)\).

1. As we have shown, many estimates are connected with the quantity \(\mu_n\); therefore it is very important to find estimates for \(\mu_n\). From estimate (5) it is clear that \(\mu_n\)

Table 1

can tend to zero arbitrarily slowly, since the parameters \(\{a_n\}\) may be chosen quite arbitrarily, provided only that the conditions

\[ |a_n|<1,\qquad n=0,1,2,\ldots,\qquad \sum_{n=0}^{\infty}|a_n|^2<\infty \tag{19} \]

are satisfied.

Assuming that the function \(\sigma(\theta)\) is absolutely continuous on the whole interval \([0,2\pi]\), and introducing the notation

\[ \omega_r(\delta;f)=\sup_{|h|\le\delta}\|f(\theta+h)-f(\theta)\|_r,\qquad f\in L_r, \]

\[ I(\delta)=\sup_{|h|\le\delta}\left\{\frac{1}{2\pi}\int_0^{2\pi}\frac{|p(\theta+h)-p(\theta)|}{p(\theta)}\,d\theta\right\}, \]

we indicate in Table 1 some estimates for the quantity \(\mu_n\), expressed in terms of the structural characteristics of the weight \(p(\theta)=\sigma'(\theta)\).

Kharkov Aviation Institute

Received
21 V 1957

REFERENCES

1. Ya. L. Geronimus, Communications of the Kharkov Mathematical Society, ser. 4, 19, 35 (1948).
2. Ya. L. Geronimus, DAN, 83, No. 1 (1952).
3. Ya. L. Geronimus, DAN, 88, No. 2 (1953).
4. Ya. L. Geronimus, DAN, 106, No. 2 (1956).
5. Ya. L. Geronimus, DAN, 88, No. 4 (1953).
6. G. Szegő, Math. Zs., 6, 167 (1920); 9, 167 (1921).
7. S. Verblunsky, Proc. Lond. Math. Soc., 40, 290 (1935).