Reports of the Academy of Sciences of the USSR

1. Volume 160, No. 5

MATHEMATICS

B. L. GOLINSKII

ON LIMIT RELATIONS AND ASYMPTOTIC FORMULAS FOR POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE

(Presented by Academician S. N. Bernstein on 4 VIII 1964)

1. Let

\[ P_n(z)=\varkappa_n z^n+\ldots;\qquad \varkappa_n>0,\quad n=0,1,2,\ldots, \]

be polynomials orthonormal on the unit circle with respect to the measure \(d\sigma(\theta)\), i.e.

\[ \frac{1}{2\pi}\int_0^{2\pi} P_n(e^{i\theta})\,\overline{P_m(e^{i\theta})}\,d\sigma(\theta)=\delta_{nm}, \]

where \(\sigma(\theta)\) is a bounded nondecreasing function with an infinite set of points of increase. Let

\[ \int_0^{2\pi}\ln p(\theta)\,d\theta>-\infty, \tag{1} \]

where \(p(\theta)\) is the derivative \(\sigma'(\theta)\), which exists almost everywhere. As is known, in this case the function

\[ \pi(z,p)=\exp\left\{-\frac{1}{4\pi}\int_0^{2\pi} \ln p(\theta)\,\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta\right\} \]

is regular in \(|z|<1\), and

\[ \lim_{n\to\infty} P_n^*(z)=\pi(z,p) \qquad \left(P_n^*(z)=z^n\overline{P_n\!\left(\frac{1}{z}\right)}\right), \]

\[ \lim_{n\to\infty}\varkappa_n=\varkappa=\pi(0,p), \tag{2} \]

where the limit relation (2) holds uniformly for \(|z|\le r<1\). S. N. Bernstein \((^1)\) and G. Szegő \((^2)\) were the first to pose and solve the problem of the uniform limit relation

\[ \lim_{n\to\infty} P_n^*(e^{i\theta})=\pi(e^{i\theta},p),\qquad \theta\in[0,2\pi], \tag{3} \]

where

\[ \pi(e^{i\theta})=\lim_{r\to 1-0}\pi(re^{i\theta}), \]

and found an estimate for the difference

\[ \rho_n(\theta)=|P_n^*(e^{i\theta})-\pi(e^{i\theta})| \]

in the case when \(\sigma(\theta)\) is an absolutely continuous function: \(\sigma(\theta)\in aC^*\), and the weight \(p(\theta)\) is a positive \(2\pi\)-periodic continuous function satisfying a Dini–Lipschitz condition of order \(>1\).

\[ \text{* } d\sigma(\theta)=p(\theta)\,d\theta,\quad p(\theta)\text{ is a nonnegative summable function.} \]

Ya. L. Geronimus \((^3)\) generalized the formulation of this problem, extending it also to the case of an interior interval \([\alpha,\beta]\subset[0,2\pi]\). In the present note new cases are given for the existence of the limiting relation (3) at each point \(\theta\in[\alpha,\beta]\), almost everywhere on \([0,2\pi]\) or on \([\alpha,\beta]\), uniformly on \([0,2\pi]\) or on \([\alpha,\beta]\).

We denote, as usual, the integral modulus of continuity of a \(2\pi\)-periodic function \(f(\theta)\in \mathscr L_q(0,2\pi)\), \(1\le q<\infty\), by

\[ \Omega_q(\delta,f)=\sup_{|h|\le\delta}\|f(\theta+h)-f(\theta)\|_{\mathscr L_q(0,2\pi)}, \]

and the modulus of continuity of a \(2\pi\)-periodic continuous function \(g(\theta)\in C(0,2\pi)\) by

\[ \Omega(\delta,g)=\max_{|h|\le\delta}\|g(\theta+h)-g(\theta)\|_{C(0,2\pi)}. \]

If \(f(\theta)\in \mathscr L_q(\alpha,\beta)\), \(g(\theta)\in C(\alpha,\beta)\), then

\[ \omega_q(\delta,f)=\sup_{|h|\le\delta\le\delta_0}\|f(\theta+h)-f(\theta)\|_{\mathscr L_q(\alpha',\beta')}, \]

\[ \omega(\delta,g)=\max_{|h|\le\delta\le\delta_0}\|g(\theta+h)-g(\theta)\|_{C(\alpha',\beta')}, \]

\[ [\alpha',\beta']\subset[\alpha,\beta],\qquad \delta_0=\min(\alpha'-\alpha,\beta-\beta'). \]

2. Theorem 1. Suppose that on \([0,2\pi]\)

\[ \sigma(\theta)\in aC, \]

\[ 0<B_0\le p(\theta)\in C(0,2\pi); \tag{4} \]

\[ \frac{\Omega(t,p)}{t}\in\mathscr L_1. \tag{5} \]

Then, starting from some \(n\ge N_0\), we have

\[ \rho_n(\theta)\le B_1\int_0^{\delta_n}\frac{\Omega(t,p)}{t}\,dt +B_2\frac{1}{\delta_n}\Omega\!\left(\frac1n,p\right)^*, \tag{6} \]

where \(\delta_n\) is chosen so that

\[ \frac{1}{\delta_n}\Omega\!\left(\frac1n,p\right)=o(1)\quad\text{as }n\to\infty. \]

If, instead of conditions (4) and (5), the conditions

\[ p(\theta)\le B_2,\qquad \frac{1}{p(\theta)}\in\mathscr L_2, \tag{7} \]

and, on \([\alpha,\beta]\),

\[ 0<B_0\le p(\theta)\in C(\alpha,\beta); \tag{8} \]

\[ \frac{\omega(t,p)}{t}\in\mathscr L_1, \tag{9} \]

hold, then, starting from some \(n\ge N_1\), we have for \(\theta\in[\alpha',\beta']\)

\[ \rho_n(\theta)\le B_3\int_0^{\varepsilon_n}\frac{\omega(t,p)}{t}\,dt +\frac{B_4}{\varepsilon_n}\Omega_2\!\left(\frac1n,\frac1p\right) +B_5\Omega_1^{1/2}\!\left(\frac1n,\frac1p\right), \tag{10} \]

and \(\varepsilon_n\) is chosen so that

\[ \frac{1}{\varepsilon_n}\Omega^2\!\left(\frac1n,\frac1p\right)=o(1)\quad\text{as }n\to\infty. \]

If \(\Omega(\delta,p)=O\left\{\left(\ln\frac1\delta\right)^{-(1+\varepsilon)}\right\}\), \(\varepsilon>0\), then, putting in (6) \(\delta_n=(\ln n)^{-1}\), we obtain the estimate of S. N. Bernstein and G. Szegő.

3. Theorem 2. Suppose that condition (1) is satisfied, and on \([\alpha,\beta]\)

\[ \sigma(\theta)\in aC,\qquad 0<B_0\le p(\theta)\in C(\alpha,\beta); \tag{11} \]

\[ {}^*\, B_0,B_1,B_2,\ldots \text{ are various constants.} \]

\[ \frac{\omega(t,p)}{t}\in \mathscr L_2; \tag{12} \]

\[ (Z_1):\qquad \delta\int_\delta^{\delta_0}\frac{\omega(t,p)}{t^2}\,dt =O\{\omega(\delta,p)\}^{*}. \tag{13} \]

Then, uniformly for \(\theta\in[\alpha',\beta']\), (3) holds.

We note that conditions (12) and (13) are satisfied if

\[ \omega(\delta,p)=O\left\{\delta^\nu\left(\ln\frac{B}{\delta_0}\right)^{-\mu}\right\} \tag{14} \]

with
\[ 0\leq \mu<(1-\nu)\ln\frac{B}{\delta_0}-1,\qquad B>\delta_0,\qquad \tfrac12\leq \nu<1. \]

Theorem 3. Let \(\rho^{-1}(\theta)\in \mathscr L_1\), and suppose that on \([\alpha,\beta]\) condition (9), condition \((Z_1)\) (or the equivalent conditions), and, instead of condition (12), the condition

\[ (Z)\qquad \int_0^\delta \frac{\omega(t,p)}{t}\,dt =O\{\omega(\delta,p)\}^{**} \tag{15} \]

hold. Then, uniformly for \(\theta\in[\alpha',\beta']\), (3) holds.

We note that conditions \((Z)\) and \((Z_1)\) are satisfied if \(\omega(\delta,p)\) satisfies inequality (14) with
\[ 0\leq \mu<(1-\nu)\ln\frac{B}{\delta_0}-1,\quad B>\delta_0,\quad 0<\nu<1. \]

Remark 1. Under the hypotheses of Theorem 3 we have, uniformly for \(\theta\in[\alpha',\beta']\) and for all \(n\geq N_2\), the lower estimate
\[ \rho_n(\theta)\geq B_4\omega\left(\frac1n\right). \]
If
\[ B_5\delta^\alpha\leq \omega(\delta,p)\leq B_6\delta^\beta,\qquad 0\leq \alpha<\beta<1, \]
then
\[ \rho_n(\theta)\geq B_7 n^{-\beta(1-\alpha)/(1-\beta)}. \]

Theorem 4. Let \(\rho^{-1}(\theta)\in \mathscr L_1\), and suppose that on \([\alpha,\beta]\) condition (9) holds and

\[ \frac{\omega(t,p)}{t}\ln\frac1t\in \mathscr L_1. \]

Then, uniformly for \(\theta\in[\alpha',\beta']\), (3) holds.

We prove Theorems 2–4 by the method of Ya. L. Geronimus (3), applying a local analogue of a theorem of A. Zygmund, which makes it possible, in terms of the modulus of continuity of a given function \(f(\theta)\), to estimate from above the modulus of continuity of the conjugate function \(\widetilde f(\theta)\) (5):

\[ \omega(\delta,\widetilde f)= O\left\{ \int_0^\delta \frac{\omega(t,f)}{t}\,dt +\delta\int_\delta^{\delta_0}\frac{\omega(t,f)}{t^2}\,dt \right\} \tag{16} \]

under the condition that

\[ f(\theta)\in \mathscr L_1,\qquad f(\theta)\in C(\alpha,\beta),\qquad \frac{\omega(t,f)}{t}\in \mathscr L_1. \]

Remark 2. Let

\[ B_9\delta^\nu\lambda(\delta)\leq \omega(\delta,p)\leq B_8\delta^\nu\lambda(\delta), \tag{17} \]

where \(0<\nu<1\); \(\lambda(\sigma)\) is an almost nondecreasing function on \([0,\delta_0]\), i.e.

\[ \lambda(\delta_2)\leq B_{10}\lambda(\delta_1) \qquad (\delta_2>\delta_1,\; B_{10}\geq 1). \tag{18} \]

Starting from inequality (16), we obtain

\[ \omega(\delta,\pi)= O\left\{ \int_0^\delta \frac{\omega(t,p)}{t}\,dt \right\}, \]

and in Theorem 2, instead of condition \((Z_1)\), one may take condition (17).

\[ \text{* Condition }(Z_1)\text{ is equivalent to conditions }(S_1),(L_1),(P_1),(B_1)\text{ in the notation of }(4). \]
\[ \text{** Condition }(Z)\text{ is equivalent to conditions }(S),(L),(P),(B)\text{ in the notation of }(4). \]

1. Theorem 5. Suppose that on \([0,2\pi]\) \(\sigma(\theta)\in aCu\),

\[ 0<B_0\leq p(\theta)\leq B_1,\qquad \Omega_2^2(t,p)\frac1t\ln\frac1t\in\mathcal L_1 \tag{19} \]

or

\[ p(\theta)\in\mathcal L_q,\qquad \frac1{p(\theta)}\in\mathcal L_{q'},\qquad \frac1q+\frac1{q'}=1,\qquad 1<q\leq 2, \tag{20} \]

\[ \Omega_q(t,p)\frac1t\ln\frac1t\in\mathcal L_1. \]

Then for almost all \(\theta\in[0,2\pi]\), (3) holds.

The proof is based on a theorem of G. Rademacher and D. E. Menshov \({}^{6}\) and on estimates of Ya. L. Geronimus \(\bigl({}^{3}\), Table I\(\bigr)\).

Theorem 6. Suppose that on \([0,2\pi]\) \(\sigma(\theta)\in aC\), \(\dfrac1{p(\theta)}\in\mathcal L_2\), and on \([\alpha,\beta]\)

\[ |P_n(e^{i\theta})|\leq B_{11},\qquad n=0,1,2,\ldots \tag{21} \]

If on \([\alpha,\beta]\) one of the conditions

\[ p(\theta)\leq B_1,\qquad \omega_2^2(t,p)\frac1t\ln\frac1t\in\mathcal L_1; \tag{22} \]

\[ p(\theta)\in\mathcal L_q,\qquad \frac1{p(\theta)}\in\mathcal L_{q'},\qquad \omega_q(t,p)\frac1t\ln\frac1t\in\mathcal L_1; \tag{23} \]

\[ p(\theta)\leq B_1,\qquad \frac{\omega_q^q(t,p)}{t}\in\mathcal L_1,\qquad 1<q\leq 2; \tag{24} \]

\[ \frac{\omega_1(t,p)}{t}\in\mathcal L_1, \tag{25} \]

is satisfied, then for almost all \(\theta\in[\alpha',\beta']\), (3) holds.

The proof of Theorem 6 is based on the following lemma:

Lemma. Suppose that a weight \(0\leq\varphi(\theta)\leq B_{12}\) satisfies condition (1), and the corresponding system of orthonormal polynomials \(\{\Phi_n(e^{i\theta})\}_0^\infty\) is uniformly bounded on \([0,2\pi]\). Then at a point \(e^{i\theta_0}\) of the unit circle we have

\[ \lim_{n\to\infty}\Phi_n^*(e^{i\theta_0})=\pi(e^{i\theta_0},\varphi). \]

If the new weight \(\psi(\theta)=\varphi(\theta)\) for \(\theta\in[\alpha,\beta]\) and \(\psi(\theta)\in\mathcal L_1\), \(\psi^{-1}(\theta)\in\mathcal L_2\), then for the corresponding system of orthonormal polynomials \(\{\Psi_n(e^{i\theta})\}_0^\infty\) we have

\[ \lim_{n\to\infty}\Psi_n^*(e^{i\theta})=\pi(e^{i\theta},\psi),\qquad \theta\in[\alpha',\beta']. \]

Kharkov Aviation Institute

Received
1 VI 1964

CITED LITERATURE

\({}^{1}\) S. N. Bernstein, Collected Works, 2, 1954.
\({}^{2}\) G. Szegő, Orthogonal Polynomials, Moscow, 1962.
\({}^{3}\) Ya. L. Geronimus, Polynomials Orthogonal on the Circle and on an Interval, Moscow, 1958.
\({}^{4}\) N. K. Bari, S. B. Stechkin, Trans. Moscow Math. Soc., 5 (1956).
\({}^{5}\) B. L. Golinskii, Mathematical Collection, 51, 93, 4 (1960).
\({}^{6}\) G. Alexits, Problems of Convergence of Orthogonal Series, Moscow, 1963.