UDC 517.512.6

MATHEMATICS

G. A. SOGOMONOVA

CERTAIN PROPERTIES OF AN EXTREMAL SEQUENCE OF REGULARLY MONOTONE POLYNOMIALS

(Presented by Academician V. I. Smirnov on 8 VI 1966)

1°. We shall consider the class \(Ц_n^{(\bar\lambda_s)}\) of regularly monotone polynomials \(P_n(x)\), \(P_n^{\,n}(x) \equiv 1\), of order \(n\) on \([0,1]\) with type numbers \(\mu_1,\mu_2,\ldots,\mu_m\), where
\[ \sum_{j=1}^{m}\mu_j=n,\qquad \mu_{ps+l}=\lambda_l,\quad l=1,\ldots,s,\quad \mu_m\le \lambda_{l_0}, \]
\(m=p_0s+l_0\). Let, for definiteness, the first type number \(\mu_1=\lambda_1\) correspond to a permanence. From the generalized theorem of S. N. Bernstein ((\(^{2}\), p. 515)) it follows that in the class \(Ц_n^{(\bar\lambda_s)}\) the polynomial \(P_n(x)\) (hereafter—the extremal polynomial) deviates least from zero on \([0,1]\), and together with its derivatives satisfies the conditions \(P_n^{(\nu)}(\alpha_\nu)=0\), where \(\alpha_\nu=0\), if
\[ \begin{aligned} \nu-qb_s&=\overline{b_{2k},\,b_{2k+1}^{-1}},\qquad &&s=2p,\quad &&k=0,1,\ldots,(s-2)/2,\\ \nu-2qb_s&=\overline{b_{2k},\,b_{2k+1}^{-1}},\qquad &&s=2p+1,\quad &&k=0,1,\ldots,(s-1)/2, \tag{A}\\ \nu-(2q+1)b_s&=\overline{b_{2k+1},\,b_{2k+2}^{-1}},\qquad &&s=2p+1,\quad &&k=0,1,\ldots,(s-3)/2, \end{aligned} \]
and \(\alpha_\nu=1\), if
\[ \begin{aligned} \nu-qb_s&=\overline{b_{2k+1},\,b_{2k+2}^{-1}},\qquad &&s=2p,\quad &&k=0,1,\ldots,(s-2)/2,\\ \nu-2qb_s&=\overline{b_{2k+1},\,b_{2k+2}^{-1}},\qquad &&s=2p+1,\quad &&k=0,1,\ldots,(s-3)/2, \tag{B}\\ \nu-(2q+1)b_s&=\overline{b_{2k},\,b_{2k+1}^{-1}},\qquad &&s=2p+1,\quad &&k=0,1,\ldots,(s-1)/2, \end{aligned} \]
where
\[ b_p=\sum_{j=1}^{p}\mu_j,\qquad b_0=0. \]
(We note that for the class under consideration
\[ b_{qs+i}=qb_s+b_i,\quad i=0,\ldots,s-1,\quad q=0,1,2,\ldots.) \]

In what follows we shall write \(\nu\in A\), if \(\nu\) satisfies conditions (A), and \(\nu\in B\), if \(\nu\) satisfies conditions (B). Let \(r\) be an integer nonnegative number. Consider the sequence \(\{P_{n,r}(x)\}_{n=0}^{\infty}\), where
\[ P_{n,r}(x)\equiv P_{n+r}^{(r)}(x), \]
and the polynomial \(P_{n+r}(x)\in Ц_{n+r}^{(\bar\lambda_s)}\) is extremal in this class. Represent \(r\) in the form \(r=qb_s+b_i+j\) \((i=0,\ldots,s-1,\ j=0,\ldots,\lambda_{i+1}-1)\). It is not difficult to see that any polynomial \(P_{n,r}(x)\) of the sequence under consideration is a regularly monotone polynomial of order \(n\) on \([0,1]\) with type numbers
\[ \lambda_{i+1}-j,\lambda_{i+2},\ldots,\lambda_s,\lambda_1,\lambda_2,\ldots,\lambda_s,\lambda_1,\ldots \tag{1} \]
The class of regularly monotone polynomials of order \(n\) on \([0,1]\) with type numbers (1) will be denoted by \(Ц_{n,r}^{(\bar\lambda_s)}\).

Theorem 1. Of all polynomials \(y_{n,r}(x)\in Ц_{n,r}^{(\bar\lambda_s)}\), \(y_{n,r}^{(n)}(x)\equiv 1\), the polynomial \(P_{n,r}(x)\) deviates least from zero on \([0,1]\), for which the conditions
\[ P_{n,r}^{(\nu)}\bigl(\alpha_\nu^{(r)}\bigr)=0,\qquad \nu=0,\ldots,n-1, \]
are satisfied, where
\[ \alpha_\nu^{(r)}= \begin{cases} 0, & \text{if } \nu+r\in A,\\ 1, & \text{if } \nu+r\in B. \end{cases} \tag{2} \]
Moreover, the magnitude of the least deviation is
\[ L_n=\bigl|P_{n,r}(1-\alpha_0^{(r)})\bigr|. \]

Theorem 2. The sequence \(\{P_{n,r}(x)\}_{n=0}^{\infty}\) of extremal polynomials of the class \(Ц_{n,r}^{(\bar\lambda_s)}\) is a sequence of generalized Appell polynomials for the class \(A_k^{(\bar\lambda_s)}\) (see (3)). Here \(k=b_s\), if \(s\) is even, and \(k=2b_s\), if \(s\) is odd.

Theorem 3. Every polynomial \(P_{n,r}(x)\) of the extremal sequence \(\{P_{n,r}(x)\}_{n=0}^{\infty}\) of the class \(Ц_{n,r}^{(\bar\lambda_s)}\) can be represented in the form
\[ P_{n,r}(x)=\frac{1}{n!}\sum_{\nu=0}^{n} C_n^\nu E_{n-\nu}^{\,n,r}x^\nu, \]
where the numbers \(E_\nu^{n,r}\) are successively determined from the system of equations
\[ \begin{aligned} E_0^{n,r}&=1,\\ E_{n-\nu}^{n,r}&=0, && \text{if } \nu+r\in A,\\ (1+E^{n,r})_{n-\nu}&=0, && \text{if } \nu+r\in B. \end{aligned} \]

\(2^\circ\). Denote by \(Ц_{n-l,r}^{(\bar\lambda_s)}\), \(l<n-1\), the class of regularly monotone polynomials \(y_{n,r}(x)\) of order \(n-l\) on \([0,1]\) with type numbers (1).

Theorem 4. Of all polynomials \(y_{n,r}(x)\in Ц_{n-l,r}^{(\bar\lambda_s)}\) of the form
\[ y_{n,r}(x)=\sum_{k=0}^{n}\sigma_k x^k, \]
where the coefficients \(\sigma_k\), \(k=n-l,\ldots,n\), are fixed, the polynomial
\[ y_{n,r}^*(x)=\sum_{k=n-l}^{n} a_k P_{k,r}(x), \]
deviates least from zero on \([0,1]\), with
\[ a_k= \begin{cases} \sigma_k k!, & \text{if } k+r\in A,\\[6pt] \displaystyle \sum_{m=k}^{n}\frac{m!}{(m-k)!}\,\sigma_m, & \text{if } k+r\in B. \end{cases} \]

Denote by \(\mathfrak P\) the subclass of polynomials
\[ y_{n,r}(x)=\sum_{k=n-l}^{n}\sigma_k x^k+\sum_{k=0}^{n-l-1} P_k x^k\in Ц_{n-l,r}^{(\bar\lambda_s)} \]
with \(m\), \(0\le m\le l\), fixed coefficients \(\sigma_{k_1},\sigma_{k_2},\ldots,\sigma_{k_m}\),
\[ n-l\le k_1<k_2<\cdots<k_m\le n. \]

Theorem 5. If there exists a polynomial \(y_{n,r}^*(x)\in\mathfrak P\) such that for any polynomial \(y_{n,r}(x)\in\mathfrak P\) and any \(x\in[0,1]\) the inequality
\[ \bigl|y_{n,r}^{*(\,n-l)}\bigr|\le \bigl|y_{n,r}^{(n-l)}(x)\bigr| \]
is satisfied, then it is the polynomial least deviating from zero on \([0,1]\).

In particular, the following holds.

Theorem 6. Among all polynomials \(y_{n,r}(x)\in Ц_{n-2,r}^{(\bar\lambda_s)}\) with fixed leading coefficient \(\sigma_n\) such that
\[ y_{n,r}^{(n-2)}(x)\sigma_n\le 0,\qquad x\in[0,1], \]
the polynomial
\[ y_{n,r}^*(x)= \begin{cases} n!\,\sigma_n\bigl[P_{n,r}(x)-\tfrac12 P_{n-1,r}(x)\bigr], & \text{if } n-1+r\in A,\\ n!\,\sigma_n\bigl[P_{n,r}(x)+\tfrac12 P_{n-1,r}(x)\bigr], & \text{if } n-1+r\in B. \end{cases} \]
deviates least from zero on \([0,1]\).

3°. Consider the sequence \(\{P_{n,r}(x)\}_{n=0}^{\infty}\) of extremal polynomials of the class \(Ц_{n,r}^{(\bar\lambda_s)}\), satisfying the conditions of Theorem 1. Let \(L\) be a linear differential operator generated by the differential expression \(l(f)=f^{(\widetilde k)}\) and by the separated boundary conditions
\(f^{(\nu)}(\alpha_{\nu}^{(r)})=0,\ \nu=0,\ldots,\widetilde k-1\). If \(\bar\lambda_s\) and \(r\) are such that the corresponding boundary-value problem has a unique eigenvalue \(\rho_1\) of smallest modulus (see (4)), then the following holds.

Theorem 7. If for \(L\) one of the following conditions is fulfilled: 1) \(L=L^*\), 2) \(L\ne L^*\), but all eigenvalues of the operator \(L\) are simple zeros of the characteristic determinant \(\Delta(\rho)\), then the asymptotic equality holds
\[ \lim_{n\to\infty}\frac{P_{n,r}(x)}{P_{n,r}(1-\alpha_0^{(r)})} = \frac{\varphi_1(x)}{\varphi_1(1-\alpha_0^{(r)})}, \]
where \(\varphi_1(x)\) is the eigenfunction of the operator \(L\) corresponding to the eigenvalue \(\rho_1\) of smallest modulus.

From this theorem there immediately follows a refinement of one result of Tagamlitskii (see \({}^{(5)}\), p. 199):

Theorem 8. Let \(f(x)\) be a regularly monotone function with an infinite order of monotonicity and with type numbers (1). Then it can be represented in the form
\[ f(x)=\sum_{k=0}^{\infty} c_k P_{k,r}(x)+A\,\frac{\varphi_1(x)}{\varphi_1(1-\alpha_0^{(r)})}, \]
where \(c_k\) and \(A\) are certain nonnegative constants.

Leningrad Mechanical
Institute

Received
26 V 1966

References

\({}^{1}\) S. N. Bernstein, Collected Works, 1, No. 32, Publishing House of the Academy of Sciences of the USSR, 1954.
\({}^{2}\) S. N. Bernstein, Collected Works, 2, No. 100, Publishing House of the Academy of Sciences of the USSR, 1954.
\({}^{3}\) V. B. Ozhegov, On certain properties of generalized Appell polynomials, Candidate’s dissertation, L., 1954.
\({}^{4}\) M. A. Naimark, Linear Differential Operators, Moscow, 1954.
\({}^{5}\) Ya. Tagamlitskii, Izv. Matem. inst. Bŭlg. AN, 3, no. 2, p. 187 (1959).