Abstract
Full Text
UDC 517.512.7
MATHEMATICS
S. Z. RAFALSON
ON AN ASYMPTOTIC FORMULA IN THE THEORY OF ORTHOGONAL POLYNOMIALS
(Presented by Academician S. N. Bernstein on 8 II 1966)
1. It is known that the derivatives of the orthogonal Jacobi polynomials are again orthogonal Jacobi polynomials, but with different indices. Namely, the equality holds
[
dJ_n^{(\alpha,\beta)}(x)/dx
=
\sqrt{n(n+\alpha+\beta+1)}\,J_{n-1}^{(\alpha+1,\beta+1)}(x),
\qquad n \geqslant 1
\tag{1}
]
(here (J_m^{(\alpha,\beta)}(x)) is the normalized Jacobi polynomial of degree (m) with indices (\alpha) and (\beta)).
It is of interest to determine how equality (1) changes if, instead of the weight ((1-x)^\alpha(1+x)^\beta), one considers the more general weight (q(x)(1-x)^\alpha(1+x)^\beta), where the function (q(x)) satisfies certain conditions. We shall show that in this case equality (1) is replaced by an asymptotic equality.
Theorem 1. Let ({\omega_n(x)}_0^\infty) be a system of polynomials orthonormal with weight
(p(x)=(1-x)^\alpha(1+x)^\beta q(x)), where (\alpha,\beta \geqslant -1/2), and let the function (q(x)) satisfy the following conditions: 1) (q(x)\geqslant m>0), (x\in[-1,1]); 2) (q(x)\in \operatorname{Lip} 1). Let ({\varphi_n(x)}_0^\infty) be a system of polynomials orthonormal with weight ((1-x^2)p(x)).
The asymptotic formula is valid
[
\omega_n'(x)=a_n\varphi_{n-1}(x)+O(\ln n),
\tag{2}
]
where (a_n\asymp n), and the constant occurring in (O(\ln n)) does not depend on (x\in[-1+h,\,1-h]), (0<h<1).
On the entire interval ([-1,1]) the asymptotic equality holds
[
\omega_n'(x)=a_n\varphi_{n-1}(x)+O(n^{\sigma+3/2}\ln n),
\tag{3}
]
where (\sigma=\max{\alpha,\beta}); the constant occurring in (O(n^{\sigma+3/2}\ln n)) does not depend on (x\in[-1,1]).
Proof. We first prove equality (2). Represent (\omega_n'(x)) in the form of a linear combination of the polynomials ({\varphi_k(x)}_0^{n-1}):
[
\omega_n'(x)=a_n\varphi_{n-1}(x)+\sum_{k=0}^{n-2} c_k\varphi_k(x).
\tag{4}
]
It is easy to obtain the following expression for (c_k):
[
c_k
=
-\int_{-1}^{1}
(1-t)^{\alpha+1}(1+t)^{\beta+1}q'(t)\omega_n(t)\varphi_k(t)\,dt,
\qquad 0\leqslant k\leqslant n-2.
\tag{5}
]
The sum (\displaystyle \sum_{k=0}^{n-2} c_k\varphi_k(x)), with the aid of the expressions obtained for (c_k), is rep—
takes the form
[
I=\sum_{k=0}^{n-2} c_k\varphi_k(x)
= -\int_{-1}^{1} (1-t)^{\alpha+1}(1+t)^{\beta+1}q'(t)\omega_n(t)
\sum_{k=0}^{n-2}\varphi_k(t)\varphi_k(x)\,dt.
\tag{6}
]
Applying the Christoffel–Darboux formula and taking into account that almost everywhere (q'(t)=O(1)), we obtain:
[
|I|=O(1)\int_{-1}^{1}(1-t)^{\alpha+1}(1+t)^{\beta+1}|\omega_n(t)|
\left|\frac{\varphi_{n-2}(x)\varphi_{n-1}(t)-\varphi_{n-2}(t)\varphi_{n-1}(x)}{t-x}\right|\,dt.
\tag{7}
]
We split the integral on the right-hand side of equality (7) into three integrals according to the scheme
[
\int_{-1}^{1}
=
\int_{-1}^{x-1/n}
+
\int_{x-1/n}^{x+1/n}
+
\int_{x+1/n}^{1}
=I_1+I_2+I_3.
]
Let us first estimate the integrals (I_1) and (I_3). Applying Korous’s theorem (see ((^3)), p. 169), we obtain:
[
\begin{aligned}
I_1={}&
\int_{-1}^{x-1/n}
(1-t)^{\alpha+1}(1+t)^{\beta+1}
\bigl[O(1)|J_n^{(\alpha,\beta)}(t)|+O(1)|J_{n-1}^{(\alpha,\beta)}(t)|\bigr]|t-x|^{-1}
\
&\times
\Bigl{|\varphi_{n-2}(x)|
\bigl[O(1)|J_{n-1}^{(\alpha+1,\beta+1)}(t)|
+O(1)|J_{n-2}^{(\alpha+1,\beta+1)}(t)|\bigr]
\
&\quad
+|\varphi_{n-1}(x)|
\bigl[O(1)|J_{n-2}^{(\alpha+1,\beta+1)}(t)|
+O(1)|J_{n-3}^{(\alpha+1,\beta+1)}(t)|\bigr]\Bigr}\,dt.
\end{aligned}
\tag{8}
]
It is enough to estimate one of the integrals entering the right-hand side of (8), for example the first of them, since the remaining integrals are estimated analogously. For this, note, first, that for (-1+h\le x\le 1-h), (0<h<1), the equality (J_m^{(\alpha+1,\beta+1)}(x)=O(1)) holds (see ((^3)), pp. 80 and 204), whence
[
\varphi_m(x)=O(1),\qquad -1+h\le x\le 1-h,\qquad 0<h<1,
\tag{9}
]
and, secondly, we use the relations
[
\begin{gathered}
(1-t)^{\alpha/2+1/4}(1+t)^{\beta/2+1/4}
|J_m^{(\alpha,\beta)}(t)|=O(1),\
(1-t)^{\alpha/2+3/4}(1+t)^{\beta/2+3/4}
|J_m^{(\alpha+1,\beta+1)}(t)|=O(1),
\end{gathered}
\tag{10}
]
which are valid since (\alpha,\beta\ge -1/2) (see ((^2)), pp. 57, 69, 70; see also ((^3)), p. 177). Then we obtain
[
\int_{-1}^{x-1/n}
(1-t)^{\alpha+1}(1+t)^{\beta+1}
\frac{|J_n^{(\alpha,\beta)}(t)|\,|\varphi_{n-2}(x)|\,|J_{n-1}^{(\alpha+1,\beta+1)}(t)|}{|t-x|}\,dt
=
]
[
=O(1)\int_{-1}^{x-1/n}\frac{dt}{|t-x|}
=O(\ln n).
\tag{11}
]
Thus, (I_1=O(\ln n)). It is proved analogously that (I_3=O(\ln n)).
Consider (I_2). With the aid of Korous’s theorem and equalities (9) and (10), it is not difficult to obtain that (I_2=O(1)). Thus, (I=O(\ln n)).
Denoting by (\beta_{n,0}) the leading coefficient of the polynomial (\omega_n(x)) ((\beta_{n,0}>0)), and by (\gamma_{n-1,0}) the leading coefficient of the polynomial (\varphi_{n-1}(x)) ((\gamma_{n-1,0}>0)), from equality (4) we have
[
a_n=n\beta_{n,0}/\gamma_{n-1,0}.
\tag{12}
]
Under the conditions of the theorem the following equalities hold
[
\beta_{n,0} \simeq \pi^{-1/2}\cdot 2^n \exp\left{-\frac{1}{2\pi}\int_{-1}^{1}\ln p(x)\frac{dx}{\sqrt{1-x^2}}\right},
\tag{13}
]
[
\gamma_{n-1,0} \simeq \pi^{-1/2}\cdot 2^{n-1}\exp\left{-\frac{1}{2\pi}\int_{-1}^{1}\ln\bigl[p(x)(1-x^2)\bigr]\frac{dx}{\sqrt{1-x^2}}\right}
\tag{14}
]
(see (³), p. 317).
From equalities (12), (13), and (14) it follows easily that (\alpha_n \simeq n). Equality (2) is proved. Equality (3) is proved similarly; one need only take into account the equality (|J_n^{(\alpha+1,\beta+1)}(x)|_{C([-1,1])}=O(n^{\sigma+3/2})).
Corollary. The following equalities hold
[
\omega_n^{(k)}(x)=\gamma_n^{(k)}\delta_{n-k}(x)+O(n^{k-1}\ln n)
\tag{15}
]
(here (-1+h\le x\le 1-h,\ 0<h<1); the constant entering into (O(n^{k-1}\ln n)) does not depend on (x\in[-1+h,1-h]); (\gamma_n^{(k)}\simeq n^k); (\delta_{n-k}(x)) is a polynomial of degree (n-k) from the sequence of polynomials orthonormal with weight (p(x)(1-x^2)^k));
[
\omega_n^{(k)}(x)=\gamma_n^{(k)}\delta_{n-k}(x)+O(n^{\sigma+3/2+2(k-1)}\ln n)
\tag{16}
]
(here (-1\le x\le 1); the constant entering into (O(n^{\sigma+3/2+2(k-1)}\ln n)) does not depend on (x\in[-1,1])).
- For a system of polynomials ({\omega_k(x)}) satisfying the conditions of Theorem 1, the following holds.
Theorem 2. If
[
\sum_{k=1}^{\infty} c_k^2 k^{2j}\ln^2 k<\infty,
]
then the series
[
\sum_{k=1}^{\infty} c_k\omega_k^{(j)}(x)
]
converges almost everywhere on the interval ([-1,1]), (j=1,2,\ldots).
Corollary 1. If
[
\sum_{k=1}^{\infty} c_k^2<\infty,
]
then almost everywhere on the interval ([-1,1]) the equality
[
s_n(x)=\sum_{k=1}^{n}c_k\omega_k'(x)=o_x(n\ln n)
]
holds.
Corollary 2. If (\lambda_n\uparrow\infty), then from the condition
[
\sum_{n=1}^{\infty}\left[c_n^2\Big/\lambda_n\sum_{k=1}^{n}c_k^2\right]<\infty
]
it follows that almost everywhere on the interval ([-1,1])
[
s_n(x)=o_x\left[\left(\lambda_n(\ln n)^2n^2\sum_{k=1}^{n}c_k^2\right)^{1/2}\right].
]
- Definition. A system of functions ({A_n(x)}_1^\infty) will be called quasi-orthonormal with weight (r(x)) on the interval ([a,b]) if, for any real numbers (c_1,c_2,\ldots,c_n), the equality
[
\int_a^b r(x)\left[\sum_{k=1}^{n}c_kA_k(x)\right]^2dx
=
O(1)\sum_{k=1}^{n}c_k^2,
]
holds, where (O(1)) depends neither on (n) nor on the numbers (c_1,c_2,\ldots,c_n).
It is clear that a system of functions orthonormal with weight (r(x)) on ([a,b]) is quasi-orthonormal. It is also easy to see that if ({A_n(x)}_1^\infty) —
orthonormal with weight (r(x)) on ([a,b]), then the system of functions
[
\left{\sum_{k=0}^{j}\alpha_n^{(k)} A_{n-k}(x)\right}_{n=j+1}^{\infty},
]
where (\alpha_n^{(0)}=O(1),\ \alpha_n^{(1)}=O(1),\ldots,\alpha_n^{(j)}=O(1)), is a quasi-orthonormal system of functions with weight (r(x)) on the interval ([a,b]).
Theorem 3. Let ({\omega_n(x)}_{0}^{\infty}) be a system of polynomials satisfying the conditions of Theorem 1, and let the constants (\alpha_k) ((k=1,2,\ldots)) be taken from equality (2).
The system of polynomials ({\omega_n'(x)/\alpha_n}_{1}^{\infty}) is quasi-orthonormal with weight (p(x)(1-x^2)) on the interval ([-1+h,1-h]), (0<h<1).
For quasi-orthonormal systems the Menshov–Rademacher theorem, the theorem on the equivalence of (A)-summability and ((C,\gamma))-summability, (\gamma>0), etc., are valid.
I express my deep gratitude to Prof. V. S. Videnskii for his help in preparing the article for publication.
Leningrad Financial-Economic Institute
named after N. A. Voznesensky
Received
3 II 1966
REFERENCES
- G. Aleksich, Problems of Convergence of Orthogonal Series, IL, 1963.
- S. N. Bernstein, Collected Works, 2, 1954.
- G. Szegő, Orthogonal Polynomials, Moscow, 1962.