MATHEMATICS

Ya. L. Geronimus

ON SOME BASIC INEQUALITIES IN THE THEORY OF ORTHOGONAL POLYNOMIALS

(Presented by Academician S. N. Bernstein, 27 V 1961)

1. Let the polynomials \(\{\varphi_n(z)\}_0^\infty\) be orthonormal on the circle \(z=e^{i\theta}\) \((0\leq \theta \leq 2\pi)\) with respect to the distribution \(d\sigma(\theta)\); consider three inequalities

\[ \sigma'(\theta_0)\geq m>0; \tag{I} \]

\[ K_n(z_0)=\sum_{i=0}^{n}|\varphi_i(z_0)|^2\leq C(n+1),\qquad z_0=e^{i\theta_0}; \tag{II} \]

\[ \{|\varphi_n(z_0)|\}_0^\alpha \leq M. \tag{III} \]

Theorem 1. If \(\sigma'(\theta_0)\) exists and if

\[ \int_0^h |dt\{\sigma(\theta_0+t)-\sigma(\theta_0-t)-2t\sigma'(\theta_0)\}|=o(h), \tag{1} \]

then (I) follows from (II).

The proof follows from the inequality \((n+1)\leq K_n(z_0)\sigma_n(\theta_0)\), where \(\sigma_n(\theta)\) is the Fejér sum for the Fourier–Stieltjes series \(\mathfrak{S}(d\sigma)\); from condition (1) there follows the convergence \(\lim_{n\to\infty}\sigma_n(\theta_0)=\sigma'(\theta_0)\).

Theorem 1 is also valid in the more general case when (II) holds only for a subsequence \(\{K_{n_\nu}(z_0)\}\), and also in the case when there exists only the generalized derivative

\[ \sigma^{(1)}(\theta_0)=\lim_{h\to 0}\frac{\sigma(\theta_0+h)-\sigma(\theta_0-h)}{2h}. \tag{2} \]

We see that (II) implies (I); it is also obvious that (III) implies (II), and consequently also (I),—but does (III) follow from (I) or from (II)? This question was posed in its time by V. A. Steklov (1); in our works (2, 3) we obtained some sufficient conditions for the boundedness of the whole orthonormal system on the entire interval of orthogonality or on a part of it, but in doing so we had to impose on the function \(\sigma(\theta)\) certain restrictions not only of a local but also of a global character, and the question remained open.

2. We shall use the relations

\[ \sqrt{1-|a_n|^2}\,\varphi_{n+1}(z)=z\varphi_n(z)-\bar a_n\varphi_n^*(z), \]

\[ \sqrt{1-|a_n|^2}\,\varphi_{n+1}^*(z)=\varphi_n^*(z)-a_n z\varphi_n(z), \]

\[ \varphi_n^*(z)=z^n\overline{\varphi_n\!\left(\frac{1}{z}\right)} \qquad (n=0,1,\ldots), \tag{3} \]

where the parameters \(\{a_n\}_0^\infty\) are subject to the conditions \(\{|a_n|\}_0^\infty<1\); conversely, if the parameters are chosen completely arbitrarily and independently of one another

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one another, subjecting them only to these conditions, then the polynomials \(\{\varphi_n(z)\}_0^\infty\), constructed by the recurrence formula (3), are orthonormal on \(|z|=1\) with respect to a certain distribution \(d\sigma(\theta)\), which is completely determined by these parameters. From (3) there follows the formula

\[ \sqrt{1-|a_n|^2}=\frac{2\Re\eta_n}{1+|\eta_n|^2}\leqslant \frac{2|\eta_n|}{1+|\eta_n|^2},\qquad \eta_n=\frac{z_0\varphi_n(z_0)}{\varphi_{n+1}(z_0)},\qquad z_0=e^{i\theta_0}. \tag{4} \]

Theorem 2. Under condition (II) one has \(\varliminf_{n\to\infty}|\varphi_n(z_0)|<\infty\); if at the same time \(\varlimsup_{n\to\infty}|\varphi_n(z_0)|=\infty\), then \(\lim_{n\to\infty}|a_n|=1\).

If we had \(\lim_{n\to\infty}|\varphi_n(z_0)|=\infty\), then from this would follow the existence of the limits \(\lim_{n\to\infty}|\varphi_n(z_0)|=\infty\) and \(\lim_{n\to\infty} K_n(z_0)/(n+1)=\infty\), which contradicts condition (II).

Suppose that \(\varlimsup_{n\to\infty}|\varphi_n(z_0)|=\infty\); we divide the set of all natural numbers into two sequences \(\{n_i'\}\) and \(\{n_k''\}\) so as to have

\[ |\varphi_{n_i'}(z_0)|\leqslant A\quad (i=1,2,\ldots);\qquad \lim_{k\to\infty}|\varphi_{n_k''}(z_0)|=\infty; \tag{5} \]

both sequences are, obviously, infinite.

We shall consider pairs of numbers \(n\) and \(n+1\) belonging to two different sequences—such pairs will always be found for arbitrarily large values of \(n\), since, if beginning with \(n=n_0\) no such pairs were found, then one of the sequences would have no more than \(n_0\) terms. Choose \(n\) so large that \(|\eta_n|<\varepsilon\) or \(|\eta_n|>1/\varepsilon\), where \(\varepsilon>0\) is arbitrarily small; in both cases we obtain from (2) \(\sqrt{1-|a_n|^2}<2\varepsilon\), which is equivalent to the condition \(\lim_{n\to\infty}|a_n|=1\).

Remark 1. Formula (4) shows that, knowing \(\varphi_n(z_0)\), we may choose \(\varphi_{n+1}(z_0)\ne0\) completely arbitrarily—this determines \(|a_n|\), and by formula (3) also \(a_n\); under condition (II) we shall indicate several examples of such a choice:

1) Put \(\varphi_m(z_0)=C\sqrt{m}\) for \(m=2^k\) \((k=0,1,2,\ldots)\) and \(\varphi_m(z_0)=1\) for all other values of \(m\).

2) Let \(\varphi_m(z_0)=C\sqrt[3]{m}\) for \(m=k^3\) \((k=1,2,3,\ldots)\) and \(\varphi_m(z_0)=1\) for all other values of \(m\); it can be shown that in both cases we have

\[ \lim_{n\to\infty}\left[\prod_{k=0}^{n-1}(1-|a_k|^2)\right]^{1/2}=1, \]

whence it follows that the set of points of increase of the function \(\sigma(\theta)\) is everywhere dense on the interval \([0,2\pi]\).

1. Theorem 3. 1) From the validity of (II) almost everywhere on \([\alpha,\beta]\) there follows the validity of (I) almost everywhere inside \([\alpha,\beta]\).

2) From the validity of (I) almost everywhere on \([\alpha,\beta]\) there follows the validity of (II) inside \([\alpha,\beta]\).

The proof of assertion 1) follows from the convergence \(\lim_{n\to\infty}\sigma_n(\theta)=\sigma'(\theta)\) almost everywhere on \([\alpha,\beta]\); assertion 2) is known.

Theorem 4. From the validity of (I) almost everywhere on \([\alpha,\beta]\) there follows the existence of an infinite subsequence of polynomials \(\{\varphi_{n_\nu}(e^{i\theta})\}\), bounded at every point inside \([\alpha,\beta]\), uniformly bounded on some interior subinterval; under the additional condition \(\varlimsup_{n\to\infty}|a_n|<1\) the entire sequence \(\{\varphi_n(e^{i\theta})\}_0^\infty\) has this property.

Indeed, from the proof of Theorem 2 it follows that, under the condition \(\overline{\lim}_{\nu\to\infty}|a_\nu|<1\), we have \(\overline{\lim}_{\nu\to\infty}|\varphi_\nu(e^{i\theta})|<\infty\) at every point at which (11) is satisfied; the uniform boundedness follows from Szegő’s theorem.

Remark 2. The condition \(\overline{\lim}_{n\to\infty}|a_n|<1\) is satisfied, in particular, if \(\lg \sigma'(\theta)\in L_1\), since this latter condition is equivalent to the condition
\[ \sum_{n=0}^{\infty}|a_n|^2<\infty; \]
but it is also satisfied in a number of other cases—for example, in the case considered by N. I. Akhiezer \((^4)\), since in this case there exists the limit \(\lim_{n\to\infty} a_n=a,\ |a|<1\).

1. From the theorems proved there follow analogous theorems for polynomials \(\{p_n(x)\}_0^\infty\) orthonormal on the interval \([-1,+1]\) with respect to the distribution \(d\psi(x)\); here the condition \(\overline{\lim}_{n\to\infty}|a_n|<1\) is equivalent to the condition \(\underline{\lim}_{n\to\infty}\lambda_n>0\), where \(\lambda_n\) is the coefficient in the recurrence formula
   \[ \sqrt{\lambda_{n+1}}\,p_n(x)=(x-a_n)p_{n-1}(x)-\sqrt{\lambda_n}\,p_{n-2}(x)\qquad (n=1,2,\ldots); \tag{6} \]
   the condition \(\underline{\lim}_{n\to\infty}\lambda_n>0\) is satisfied, in particular, if \(\lg \psi'(x)/\sqrt{1-x^2}\in L_1\), and is also satisfied in a number of other cases, for example, in the case of Pollaczek polynomials, for which \(\lg \psi'(x)/\sqrt{1-x^2}\notin L_1\), but \(\lim_{n\to\infty}\lambda_n=1/4\).

Kharkov Aviation
Institute

Received
22 V 1961

REFERENCES CITED

\(^1\) V. A. Steklov, Izv. Rossiisk. Akad. nauk, 15, 281 (1921).
\(^2\) Ya. L. Geronimus, Izv. AN SSSR, ser. matem., 16, No. 5, 469 (1952).
\(^3\) Ya. L. Geronimus, Polynomials Orthogonal on the Circle and on an Interval, Moscow, 1958.
\(^4\) N. I. Akhiezer, DAN, 130, No. 2 (1960).