UDC 517.512.6+517.512.7

MATHEMATICS

D. L. BERMAN

ON SOME IDENTITIES OF THE GENERAL THEORY OF ORTHOGONAL POLYNOMIALS AND THEIR APPLICATIONS

(Presented by Academician S. N. Bernstein, 4 I 1966)

1°. Let $\{\omega_k(x)\}_{k=0}^{\infty}$ be a system of polynomials orthonormal on the interval $[-1,1]$ with respect to the weight $g(x)$, and let $a_n$ be the leading coefficient of $\omega_n(x)$. It is known that the recurrence formula holds

\[ \omega_k(x)=(\alpha_k x+\beta_k)\omega_{k-1}(x)-\gamma_k\omega_{k-2}(x),\qquad k=2,3,\ldots, \tag{1} \]

where $\alpha_k,\beta_k,\gamma_k$ are constants, with $\alpha_k=a_k/a_{k-1}$, $\gamma_k=\alpha_k/\alpha_{k-1}$.

Of importance in the general theory of orthogonal polynomials is the Christoffel–Darboux formula

\[ \sum_{k=0}^{n}\omega_k(x)\omega_k(t)= \frac{a_n}{a_{n+1}}\, \frac{\omega_{n+1}(t)\omega_n(x)-\omega_n(t)\omega_{n+1}(x)}{t-x}. \tag{2} \]

In the present note we study the sum $\sum_{k=0}^{n}\omega_k'(x)\omega_k'(t)$, where the prime, as usual, denotes the first derivative. We shall show that this sum is also of interest.

2°. Theorem 1. The identity holds

\[ \sum_{k=0}^{n}\omega_k'(x)\omega_k'(t) = \frac{a_n}{a_{n+1}} \left\{ \frac{\omega_{n+1}'(t)\omega_n'(x)-\omega_n'(t)\omega_{n+1}'(x)}{t-x} +\right. \]

\[ \left. +(t-x)^{-3}\left[ \bigl(\omega_{n+1}'(t)\omega_n(x)-\omega_n'(t)\omega_{n+1}(x) +\omega_{n+1}'(x)\omega_n(t)\right. \right. \]

\[ \left. \left. -\omega_n'(x)\omega_{n+1}(t)\bigr)(t-x) +2\bigl(\omega_{n+1}(x)\omega_n(t)-\omega_{n+1}(t)\omega_n(x)\bigr) \right]\right\}. \tag{3} \]

We shall outline the proof. Differentiate equality (1) and then multiply it by $\omega_{k-1}(t)$. In the resulting equality replace $x$ by $t$ and $t$ by $x$, and subtract the second equality from the first. The subsequent reasoning is the same as in the derivation of the Christoffel–Darboux formula. The only difference is that on the right-hand side of the equality there appear sums of the form $\sum_{k=0}^{n}\omega_k'(x)\omega_k(t)$ and $\sum_{k=0}^{n}\omega_k(x)\omega_k'(t)$, which are found by differentiating formula (2), respectively with respect to $x$ and to $t$.

Theorem 2. The identity holds

\[ \sum_{k=0}^{n}\bigl(\omega_k'(x)\bigr)^2 = \frac{a_n}{a_{n+1}} \left[ \frac{1}{2}\bigl(\omega_{n+1}''(x)\omega_n'(x)-\omega_n''(x)\omega_{n+1}'(x)\bigr) +\right. \]

\[ \left. +\frac{1}{6}\bigl(\omega_{n+1}'''(x)\omega_n(x)-\omega_n'''(x)\omega_{n+1}(x)\bigr) \right]. \]

Proof. This theorem is easily derived from Theorem 1. It suffices in equality (3) to put $t=x$ and then compute its right-hand side by L’Hôpital’s rule.

Corollary 1. For any orthonormal system of polynomials \(\{\omega_k(x)\}_{k=0}^{\infty}\) the inequality holds
\[ \omega'''_{n+1}(x)\omega_n(x)+3\omega''_{n+1}(x)\omega'_n(x) \ge \omega'''_n(x)\omega_{n+1}(x)+3\omega''_n(x)\omega'_{n+1}(x), \]
\[ n=0,1,2,\ldots,\quad -\infty<x<\infty. \]

Theorem 3. For any orthonormal system of polynomials \(\{\omega_k(x)\}_{k=0}^{\infty}\) with weight \(g(x)\), the equality holds
\[ \sum_{k=0}^{n}\int_{-1}^{1}(\omega'_k(x))^2 g(x)\,dx = \frac{a_n}{2a_{n+1}} \int_{-1}^{1} \bigl[\omega''_{n+1}(x)\omega'_n(x)-\omega''_n(x)\omega'_{n+1}(x)\bigr]g(x)\,dx. \]

Proof. Since \(\omega'''_{n+1}(x)\), \(\omega'''_n(x)\) are polynomials of degrees respectively \([n-2]\) and \((n-3)\), by orthogonality,
\[ \int_{-1}^{1}\omega'''_{n+1}(x)\omega_n(x)g(x)\,dx = \int_{-1}^{1}\omega'''_n(x)\omega_{n+1}(x)g(x)\,dx =0. \]
Therefore Theorem 3 follows from Theorem 2.

Corollary 2. For any orthonormal system of polynomials \(\{\omega_k(x)\}_{k=0}^{\infty}\) with weight \(g(x)\), the inequality holds
\[ \int_{-1}^{1}\omega''_{n+1}(x)\omega'_n(x)g(x)\,dx \ge \int_{-1}^{1}\omega''_n(x)\omega'_{n+1}(x)g(x)\,dx, \quad n=0,1,2,\ldots . \]

Theorem 4. For the orthonormal system of Legendre polynomials \(\{p_k(x)\}_{k=0}^{\infty}\), the equality holds
\[ \sum_{k=0}^{n}\int_{-1}^{1}(p'_k(x))^2\,dx = \frac{n(n+1)^2(n+2)}{4}, \quad n=0,1,2,\ldots . \]

We indicate the proof. From the orthogonality of the Legendre polynomials and the formula of integration by parts it follows that
\[ \int_{-1}^{1} \bigl[p''_{n+1}(x)p'_n(x)-p''_n(x)p'_{n+1}(x)\bigr]\,dx = \bigl[p_n(x)p''_{n+1}(x)-p_{n+1}(x)p''_n(x)\bigr]_{-1}^{1}. \]

It is not difficult to calculate that the right-hand side of this equality is equal to
\[ \frac{n(n+1)(n+2)}{2}\sqrt{(2n+1)(2n+3)}. \]

Therefore Theorem 4 follows from Theorem 3.

\(3^\circ\). With the help of Theorem 4 one can obtain an analogue of A. A. Markov’s inequality for the space \(L_2\) of all functions square-summable on the interval \([-1,1]\) with norm
\[ \|f\|_{L_2}=\left(\int_{-1}^{1} f^2\,dx\right)^{1/2}. \]

Theorem 5. For any polynomial \(R_n\) of degree \(n\), the inequality holds
\[ \|R'_n\|\le \frac{n+1}{2}\sqrt{n(n+2)}\,\|R_n\|_{L_2}, \quad n=0,1,2,\ldots \tag{4} \]

Proof. Consider the operator
\[ U_n(f,x)=\int_{-1}^{1} f(t)\sum_{j=0}^{n}p'_j(x)p_j(t)\,dt, \quad f\in L_2. \]

It is clear that
\[ U_n(R_n,x)=R'_n(x). \]

Therefore, from the Cauchy—Bunyakovsky inequality and the orthogonality of the Legendre polynomials it follows that

\[ \|R_n'\|_{L_2}\leq \left(\sum_{k=0}^{n}\int_{-1}^{1}(p_k'(x))^2\,dx\right)^{1/2}\|R_n\|. \tag{5} \]

By Theorem 4 and this inequality we have (4).

Remark 1. For \(n=1\) and \(R_1(x)=p_1(x)\), equality holds in (4).

Remark 2. If the norm is defined by the formula

\[ \|f\|=\left(\int_{-1}^{1} f^2 g\,dx\right)^{1/2}, \]

then the preceding arguments lead to the inequality

\[ \|R_n^{(k)}\|_{L_2}\leq \left(\sum_{j=0}^{n}\int_{-1}^{1}(\omega_j^{(k)}(x))^2\,g(x)\,dx\right)^{1/2}\|R_n\|, \]

which is a generalization of inequality (5) to the general case of orthogonal polynomials.

\(4^\circ\). Let \(\bar{\Omega}_n^{(k)}(L_2)\) be the set of all linear operations \(V(f)\) from \(L_2\) into \(L_2\) having the property that \(V(f,x)=f^{(k)}(x)\), if \(f\) is a polynomial of degree \(\leq n\), and let \(\bar{\Omega}_{n,n}^{(k)}(L_2)\) be the set of all linear operations \(V(f)\) from \(L_2\) into \(L_3\) satisfying the conditions: 1) for any \(f\in L_2\), \(V(f)\) is a polynomial of degree \(\leq n\); 2) if \(f\) is a polynomial of degree \(\leq n\), then \(V(f,x)=f^{(k)}(x)\). Obviously,
\(\bar{\Omega}_{n,n}^{(k)}(L_2)\subset \bar{\Omega}_n^{(k)}(L_2)\). It can be shown that
\(\bar{\Omega}_n^{(k)}(L_2)\ne \bar{\Omega}_{n,n}^{(k)}(L_2)\). Put

\[ \bar{\rho}_{n,n}^{(k)}=\rho_{n,n}^{(k)}(L_2)= \inf_{V\in\bar{\Omega}_{n,n}^{(k)}(L_2)}\|V\|,\qquad \bar{\rho}_{n}^{(k)}=\rho_{n}^{(k)}(L_2)= \inf_{V\in\bar{\Omega}_{n}^{(k)}(L_2)}\|V\|. \]

In the space \(C\) of all continuous \(2\pi\)-periodic functions with norm
\(\|f\|=\max_{0\leq x<2\pi}|f(x)|\), the equality \((1)\)* holds

\[ \lim_{n\to\infty}\left(\frac{\rho_{n,n}^{(k)}}{\rho_n^{(k)}}:\frac{4}{\pi^2}\ln n\right)=1. \]

In the space \(C\) of all continuous functions on the interval \([-1,1]\), with norm
\(\|f\|=\max_{-1\leq x\leq 1}|f(x)|\), \(\bar{\rho}_n^{(k)}=\bar{\rho}_{n,n}^{(k)}=T_n^{(k)}(1)\), \(k=1,2,\ldots,n\), \(T_n(x)=\cos n\arccos x\) \((^2)\). A. N. Kolmogorov drew my attention to the fact that in the space \(L_2\) of all \(2\pi\)-periodic square-integrable functions with norm
\[ \|f\|=\left(\int_{0}^{2\pi} f^2\,dx\right)^{1/2} \]
the equalities \(\rho_{n,n}^{(k)}=\rho_n^{(k)}=n^k\), \(k=0,1,2,\ldots\), hold. We shall now prove a theorem concerning the space \(L_2\).

Theorem 6. For any \(0\leq k\leq n\),

\[ \|p_n^{(k)}\|\leq \bar{\rho}_n^{(k)}(L_2)\leq \bar{\rho}_{n,n}^{(k)}(L_2)\leq \|\Pi_n\|, \tag{6} \]

\[ \Pi_n(x)=\left[\sum_{j=0}^{n}(p_j^{(k)}(x))^2\right]^{1/2}, \]

\[ \text{* For the periodic case the bar over \(\rho\) is omitted.} \]

where \(\{p_j(x)\}_{j=0}^{\infty}\) are the orthonormal Legendre polynomials. In particular,

\[ \bar{\rho}_n^{(n)}(L_2)=\rho_{n,n}^{(n)}(L_2)=(2n-1)!!\sqrt{2n+1}. \tag{7} \]

Proof. It is obvious that the operation

\[ \bar V(f,x)=\int_{-1}^{1} f(t)\sum_{j=0}^{n} p_j^{(k)}(x)\,p_j(t)\,dt \]

is of the class \(\bar{\Omega}_{n,n}^{(k)}(L_2)\), and \(\|\bar V\|\leq \|\Pi_n\|\). Therefore

\[ \bar{\rho}_n^{(n)}(L_2)\leq \bar{\rho}_{n,n}^{(n)}(L_2)\leq \|\Pi_n\|. \tag{8} \]

On the other hand, for any \(V\in \bar{\Omega}_n^{(k)}(L_2)\), \(V(p_n)=p_n^{(k)}\), and since \(\|p_n\|=1\), it follows that \(\bar{\rho}_n^{(k)}(L_2)\geq \|p_n^{(k)}\|\). Hence, together with (8), (6) follows.

If \(k=n\), then, by virtue of (6), we have

\[ \bar{\rho}_n^{(n)}(L_2)=\bar{\rho}_{n,n}^{(n)}(L_2)=p_n^{(n)}(x)\sqrt{2}, \]

and this is equivalent to the equalities (7).

Leningrad Institute of Soviet Trade
named after Fr. Engels

Received
9 XII 1965

REFERENCES

\(^{1}\) D. L. Berman, DAN, 138, No. 4 (1961).
\(^{2}\) D. L. Berman, DAN, 140, No. 3 (1961).