Abstract Generated abstract
The paper studies how the growth of polynomials orthonormal on the unit circle with respect to a mass distribution depends on local and global properties of that distribution, especially the zeros of its density. It derives upper bounds in terms of the modulus of increase of the distribution and lower bounds in terms of local symmetric averages near a point, showing that bounded subsequences near a point require a positive generalized symmetric derivative there. These estimates are then applied to densities whose zeros are of logarithmic or algebraic order, yielding corresponding logarithmic or power growth bounds for the orthonormal polynomials and converse lower-growth implications. The paper concludes by relating observed growth rates at a point to possible global types of zeros of the density.
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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 \(\sigma^{(1)}(\theta_0)>0\).
- 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
\[
\varlimsup_{n\to\infty}\frac{\lg M_n}{\lg n}=\infty.
\tag{15}
\]
- 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:
| Order of growth at the given point $\theta_0$ | Nature of the zeros of the function $\sigma'(\theta)$ on the interval $[0,2\pi]$ |
|---|---|
| $\displaystyle \varlimsup_{n\to\infty}\left\{\frac{\lg|\varphi_n(e^{i\theta_0})|}{\lg\lg n}\right\}=\infty$ | There are zeros above logarithmic order |
| $\displaystyle \varlimsup_{n\to\infty}\left\{\frac{\lg|\varphi_n(e^{i\theta_0})|}{\lg n}\right\}=\infty$ | There are zeros above algebraic order |
| $\displaystyle \varlimsup_{n\to\infty}\left\{\frac{\lg|\varphi_n(e^{i\theta_0})|}{n}\right\}>0$ | $\sigma'(\theta)=0,\ \theta\in E,\ \operatorname{mes} E>0$ |
| $\displaystyle \varlimsup_{n\to\infty}\left\{\frac{\lg|\varphi_n(e^{i\theta_0})|}{\sqrt n}\right\}=\infty$ | $\lg\sigma'(\theta)\notin L_1(0,2\pi)$ |
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).