Abstract Generated abstract
The paper studies how the derivative relation for Jacobi orthogonal polynomials changes when the Jacobi weight is multiplied by a positive Lipschitz factor. For polynomials orthonormal with respect to \((1-x)^\alpha(1+x)^\beta q(x)\), with \(\alpha,\beta \geq -1/2\), it proves that the derivative is asymptotically a constant of order \(n\) times the corresponding polynomial orthonormal with respect to the modified weight multiplied by \(1-x^2\), with logarithmic error on compact subintervals and a larger uniform error on the full interval. The argument uses expansion in the modified orthonormal system, the Christoffel-Darboux formula, Korous-type estimates, and asymptotics of leading coefficients. Consequences include estimates for higher derivatives, almost-everywhere convergence criteria for differentiated orthogonal series, and quasi-orthonormality of normalized derivative systems.
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.