MATHEMATICS

Ya. L. GERONIMUS

ON THE DEPENDENCE BETWEEN THE ORDER OF GROWTH OF ORTHONORMAL POLYNOMIALS AND THE CHARACTER OF THE MASS DISTRIBUTION

(Presented by Academician S. N. Bernstein on 9 IV 1962)

In paper (^1) we considered certain conditions for boundedness of a system of polynomials \(\{\varphi_n(z)\}_0^\infty\), orthonormal on the unit circle \(z=e^{i\theta}\), \(0\leq \theta \leq 2\pi\), with respect to the mass distribution \(d\sigma(\theta)\); in the present note we shall consider the question of the order of growth of these polynomials as a function of the character of the zeros of the function \(\sigma'(\theta)\).

1. Theorem 1. If \(a(\delta)\) is the modulus of increase of the function \(\sigma(\theta)\) on the interval \([0,2\pi]\) (see (4))

\[ a(\delta)=\inf_{\theta}\int_{\theta}^{\theta+\delta} d\sigma(\theta), \qquad \theta,\theta+\delta\in[0,2\pi],\ \delta>0, \tag{1} \]

then the inequality

\[ K_n(\theta)\,a\!\left(\frac1n\right)\leq 8\pi,\qquad K_n(\theta)=\sum_{s=0}^{n}|\varphi_s(e^{i\theta})|^2,\ \theta\in[0,2\pi]\quad (n=1,2,\ldots) \tag{2} \]

holds. Hence, in addition to the obvious inequality

\[ |\varphi_n(e^{i\theta})|\sqrt{a\!\left(\frac1n\right)}\leq 2\sqrt{2\pi},\qquad \theta\in[0,2\pi]\quad (n=1,2,\ldots), \tag{3} \]

there follows, under the additional condition \(\lg \sigma'(\theta)\in L_1\), the stronger result (see (^2), § 3.5)

\[ \lim_{n\to\infty}\left\{|\varphi_n(e^{i\theta})|\sqrt{a\!\left(\frac1n\right)}\right\}=0,\qquad \theta\in[0,2\pi]. \tag{4} \]

Theorem 2. If \(s\geq 1\) is an integer and if we introduce the notation

\[ M_n=\max_{\theta\in e_n}|\varphi_n(e^{i\theta})|, \qquad \theta\in e_n=\left[\theta_0+\frac{1}{2(n-s)},\ \theta_0+\frac{1}{2(n-s)}\right],\quad n>s, \]

\[ \mu(h)=\frac{\sigma(\theta_0+h)-\sigma(\theta_0-h)}{2h},\qquad h>0, \tag{5} \]

then the inequality* holds

\[ M_n^2\mu_1\!\left(\frac1n\right)\geq 1,\qquad \mu_1(x)=x^{2s-1}\left\{C_1+C_2\int_x^{\pi}\frac{\mu(t)\,dt}{t^{2s}}\right\}. \tag{6} \]

* \(C, C_1, C_2,\ldots\) are constants independent of \(n\).

Hence, as a special case, there follows a theorem supplementing Theorem 1 of our note \((^{1})\):

Theorem 3. Suppose that at the point \(\theta_0\) there exists a generalized symmetric derivative
\[ \sigma^{(1)}(\theta_0)=\lim_{h\to 0}\mu(h); \]
if there exists a subsequence \(\{\varphi_{n_i}(z)\}\), uniformly bounded in a neighborhood of the point \(\theta_0\),
\[ |\varphi_{n_i}(e^{i\theta})|\leq C,\qquad \theta_0-\varepsilon\leq \theta\leq \theta_0+\varepsilon,\quad \varepsilon>0,\quad n_i>\frac{1}{2\varepsilon}+1, \tag{7} \]
then \(\sigma^{(1)}(\theta_0)>0\).

1. In Theorems 1 and 2 there occur certain structural characteristics of the function \(\sigma(\theta)\); the quantity \(\mu(h)\) is a purely local characteristic, since \(h\) may be taken arbitrarily small; the quantity \(a(\delta)\), i.e. the modulus of increase on the whole interval \([0,2\pi]\), is not a local characteristic; if, however, in Theorem 1 one considers the modulus of increase only on some interior interval, then in (2) one must replace \(\frac{1}{n}\) by \(\frac{1}{n^2}\).

Theorems 1 and 2 make it possible to establish a certain dependence between the order of growth of the orthonormal system and the character of the zeros of the function \(\sigma'(\theta)\).

Theorem 4. 1) Suppose that all zeros \(\{\theta_i\}_1^m\) \((m<\infty)\) of the function \(\sigma'(\theta)\) on the interval \([0,2\pi]\) are not higher than of logarithmic order, i.e. there exist constants \(C,\gamma>0\) such that in a neighborhood of each of the zeros we have
\[ \sigma'(\theta)\geq C|\lg|\theta-\theta_i||^{-\gamma},\qquad \sigma'(\theta_i)=0\quad (i=1,2,\ldots,m); \tag{8} \]
then on the whole interval \([0,2\pi]\) the inequality
\[ |\varphi_n(e^{i\theta})|\leq C_1(\lg n)^{\gamma/2}\qquad (n=2,3,\ldots) \tag{9} \]
holds.

2) If in a neighborhood of some point \(\theta_0\) we have
\[ \mu(h)\leq C_2\left(\lg\frac{1}{h}\right)^{-\gamma},\qquad C_2,\gamma>0, \tag{10} \]
then \(M_n\geq C_3(\lg n)^{\gamma/2}\), \(n=2,3,\ldots\); in particular, if in this neighborhood the function \(\sigma(\theta)\) is differentiable and \(\sigma'(\theta)\) has at the point \(\theta_0\) a zero higher than of logarithmic order, then
\[ \varlimsup_{n\to\infty}\left\{\frac{\lg M_n}{\lg\lg n}\right\}=\infty. \tag{11} \]

Theorem 5. 1) Suppose that all zeros of the function \(\{\theta_i\}_1^m\) on the interval \([0,2\pi]\) are not higher than of algebraic order, i.e. there exist constants \(C,\gamma>0\) such that in a neighborhood of each of the zeros we have
\[ \sigma'(\theta)\geq C|\theta-\theta_i|^\gamma,\qquad \sigma'(\theta_i)=0\quad (i=1,2,\ldots,m); \tag{12} \]
then on the whole interval \([0,2\pi]\) the inequality
\[ |\varphi_n(e^{i\theta})|\leq C_1 n^{\gamma/2}\qquad (n=1,2,\ldots) \tag{13} \]
holds.

2) If in a neighborhood of some point \(\theta_0\) we have
\[ \mu(h)\leq C_2h^\gamma, \tag{14} \]
then \(M_n\geq C_3 n^{\gamma/2}\), \(n=2,3,\ldots\); in particular, if in this neighborhood the function \(\sigma(\theta)\) is differentiable and \(\sigma'(\theta)\) has at the point \(\theta_0\) a zero higher than of algebraic order, then
\[ \varlimsup_{n\to\infty}\frac{\lg M_n}{\lg n}=\infty. \tag{15} \]

1. Theorems 4 and 5, as well as the estimates obtained earlier by us (see (²), § 8.2; (³)), make it possible to draw some conclusions about the nature of the zeros of the function $\sigma'(\theta)$ on the whole interval $[0,2\pi]$, if the order of growth of the orthonormal system at one point is known; these results are summarized in the table:

Kharkov Aviation Institute

Received
28 III 1962

CITED LITERATURE

¹ Ya. L. Geronimus, DAN, 142, No. 3 (1962).
² Ya. L. Geronimus, Polynomials Orthogonal on the Circle and on an Interval, 1958.
³ Ya. L. Geronimus, Matem. sborn., 23 (65), 1, 77 (1948).
⁴ J. Shohat, Ann. di Mat., 18, 201 (1939).