UDC 517.512.7

MATHEMATICS

Ya. L. Geronimus

ON SOME LIMITING PROPERTIES OF ORTHOGONAL POLYNOMIALS

(Presented by Academician S. N. Bernstein on 30 III 1965)

1. Let the polynomials \(\{\varphi_n(z)\}_0^\infty\) be orthonormal on the unit circle

\[ \frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi_n(z)\overline{\varphi_m(z)}\,d\sigma(\theta)=\delta_{nm},\qquad z=e^{i\theta}\quad (m,n=0,1,2,\ldots), \tag{1} \]

where the set \(e\) of points of increase of the bounded nondecreasing function \(\sigma(\theta)\) is closed; suppose that on the set \(e_1\Subset e\) there exists \(\sigma'(\theta)>0\), and \(e_2\subset e_1\) is the set of Lebesgue points of the characteristic function of the set \(e_1\); by \(E_2\subset E_1\subset E\) we denote the corresponding sets on the unit circle and by \(d\) their transfinite diameters (capacities).

Theorem. If \(d(E_2)=d(E)>0\) and \(\mu(\theta)\) is the Robin distribution function for the set \(E\), then at every point \(z_0=e^{i\theta_0}\) at which \(\mu'(\theta_0)\), \(\sigma'(\theta_0)>0\) exist, we have

\[ \lim_{n\to\infty}\frac{K_n(z_0,z_0)}{n+1} = 2\pi\,\frac{\mu'(\theta_0)}{\sigma'(\theta_0)},\qquad K_n(z,z_0)=\sum_{s=0}^{n}\varphi_s(z)\overline{\varphi_s(z_0)} \quad (n=0,1,2\ldots). \tag{2} \]

An analogous theorem holds for polynomials orthonormal on a finite interval of the real axis. These results are a generalization of earlier results \((^{1-7})\).

1. We outline the course of the proof of the theorem. Consider the polynomial

\[ \Phi_{n+1}(z,z_0)= \frac{K_n(z,z_0)(z-z_0)}{\alpha_n^2\Phi_n(z_0)},\qquad \Phi_n(z)=\frac{\varphi_n(z)}{\alpha_n}=z^n+\ldots,\quad (n=0,1,2,\ldots); \tag{3} \]

all its roots \(\{z_k^{(n)}=e^{i\theta_k^{(n)}}\}_0^n\) are simple and lie on \(|z|=1\). Let \(\psi_n(\theta,\theta_0)\) be a stepwise nondecreasing function having a positive jump \(1/(n+1)\) at each point \(\{\theta_k^{(n)}\}_0^n\); consider the infinite sequence of functions \(\{\psi_n(\theta,\theta_0)\}_1^\infty\), and suppose that \(\theta_0=\{\theta_0^{(n)}\}_{n=1}^\infty\) is a point of increase of each of them.

We apply the theorem of P. P. Korovkin \((^8)\) on the existence of the limit

\[ \lim_{n\to\infty} M_n^{1/n}=d(E),\qquad M_n=\max|\Phi_n(z)|,\quad z\in E\quad (n=1,2,\ldots); \tag{4} \]

then, slightly modifying Walsh’s theorem \((^9,\S\S\,7.3,\,7.4)\), we prove the existence of the limits

\[ \lim_{n\to\infty}|\Phi_n(z)|^{1/n} = \lim_{n\to\infty}|\Phi_{n+1}(z,z_0)|^{1/(n+1)} = d(E)|\varphi(z)|,\qquad |z|>R>1, \tag{5} \]

where the function \(w=\varphi(z)\) maps the region \(F\) complementing the set \(E\) in the extended plane onto the region \(|w|>1\).

Hence, using Helly’s theorems, we derive convergence at all continuity points of \(\mu(\theta)\):

\[ \lim_{n\to\infty}\psi_n(\theta,\theta_0)=\mu(\theta). \tag{6} \]

Remark. If one does not use (8), then from our earlier results (5) there follows the validity of (4), and consequently also of (2), under the more restrictive condition: \(\sigma'(\theta)>0\) almost everywhere on the set \(E^{(1)}\), where \(E^{(1)}\) is the sum of a finite number of arcs of the unit circle, and \(E=E^{(1)}+E^{(2)}\), where \(E^{(2)}\) is an isolated countable set.

1. Let us now consider a stepwise nondecreasing function \(\sigma_n(\theta,\theta_0)\), having \(n+1\) points of increase at the same points \(\{\theta_k^{(n)}\}_0^n\), and suppose the distributions \(d\sigma_n(\theta,\theta_0)\) and \(d\sigma(\theta)\) have equal moments \(\{c_k\}_0^n\); it is easy to see that the quantity \(2\pi\{K_n(z_0,z_0)\}^{-1}\) is equal to the mass of the distribution \(d\sigma_n(\theta,\theta_0)\) concentrated at the point \(\theta_0^{(n)}=\theta_0\). Let \(\theta_0=\{\theta_0^{(n)}\}_{n=1}^{\infty}\) be a point of increase of all the functions \(\{\sigma_n(\theta,\theta_0)\}_1^{\infty}\); again using Helly’s theorems and the determinacy of the trigonometric moment problem, we prove convergence in the main,

\[ \lim_{n\to\infty}\sigma_n(\theta,\theta_0)=\sigma(\theta). \tag{7} \]

1. Let a small quantity \(\varepsilon_n>0\) be chosen in such a way that both functions \(\sigma(\theta)\), \(\mu(\theta)\) are continuous at the points \(\theta_0\pm\varepsilon_n\), and that between these points there lies only one point of increase \(\theta_0\) of the functions \(\sigma_n(\theta,\theta_0)\) and \(\psi_n(\theta,\theta_0)\); we have

\[ \frac{K_n(z_0,z_0)}{n+1} = 2\pi\, \frac{\psi_n(\theta_0+\varepsilon_n,\theta_0)-\psi_n(\theta_0-\varepsilon_n,\theta_n)} {\sigma_n(\theta_0+\varepsilon_n,\theta_0)-\sigma_n(\theta_0-\varepsilon_n,\theta_0)}; \tag{8} \]

choosing \(n\) sufficiently large and using the convergence conditions (6) and (7), we obtain

\[ \frac{K_n(z_0,z_0)}{n+1} = 2\pi\, \frac{\mu(\theta_0+\varepsilon_n)-\mu(\theta_0-\varepsilon_n)} {\sigma(\theta_0+\varepsilon_n)-\sigma(\theta_0-\varepsilon_n)} +o(1); \tag{9} \]

dividing the numerator and denominator of the right-hand side by \(2\varepsilon_n\) and using the existence of the derivatives, we arrive at (2).

1. Let us consider several examples.

If \(E\) is the unit circle, then we have

\[ \lim_{n\to\infty}\frac{K_n(z_0,z_0)}{n+1} = \frac{1}{\sigma'(\theta_0)},\qquad \sigma'(\theta_0)>0 \tag{10} \]

under the condition \(d(E_2)=1\), or \(\sigma'(\theta)>0\) almost everywhere on the interval \([-\pi,\pi]\); this latter condition is satisfied, in particular, if \(\ln\sigma'(\theta)\in L_1(-\pi,\pi)\); but it also holds for Polachek polynomials \(((^9),397\text{--}400)\), for which \(\ln\sigma'(\theta)\notin L_1(-\pi,\pi)\).

Now let \(E^{(1)}\) be the arc \([\exp\alpha,\exp(2\pi-\alpha)]\), and let the set \(E^{(2)}\) be situated on the complementary arc; in this case we have

\[ \lim_{n\to\infty}\frac{K_n(z_0,z_0)}{n+1} = \frac{\sin\theta_0/2} {\sigma'(\theta_c)\sqrt{\cos^{(2)}\alpha/2-\cos^{(2)}\theta/2}}, \qquad \sigma'(\theta_0)>0,\quad \alpha<\theta_0<2\pi-\alpha, \]

under the condition \(d(E_2)=\cos\alpha/2\), or \(\sigma'(\theta)>0\) almost everywhere on the interval \([\alpha,2\pi-\alpha]\).

Kharkov Aviation Institute

Received
25 III 1965

CITED LITERATURE

\(^{1}\) P. Erdös, P. Turan, Ann. Math., 4, 510 (1940).
\(^{2}\) U. Grenander, M. Rosenblatt, Trans. Am. Math. Soc., No. 1, 112 (1954).
\(^{3}\) G. Szegö, Math. Zs., 12, 61 (1922).
\(^{4}\) N. I. Akhiezer, Zhurn. Inst. matem. AN USSR, No. 3, 75 (1937).
\(^{5}\) Ya. L. Geronimus, Matem. sborn., 23 (65), No. 1, 77 (1948).
\(^{6}\) Ya. L. Geronimus, DAN, 45, No. 4 (1949).
\(^{7}\) Ya. L. Geronimus, Polynomials Orthogonal on a Circle and an Interval, 1958.
\(^{8}\) P. P. Korovkin, Uch. zap. Kaliningradsk. ped. inst., issue 5, 34 (1958).
\(^{9}\) J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 1961.