V. L. Fainshmidt

ON SOME REGULARLY MONOTONE POLYNOMIALS LEAST DEVIATING FROM ZERO

(Presented by Academician S. N. Bernstein on 24 II 1962)

Let \(\lambda\) be a positive integer. Denote, as was done in \((^2)\), by \(\Ц_{\lambda,m}\) the class of regularly monotone functions of order \(m\) on \([0,1]\) for which the first and last type numbers do not exceed \(\lambda\), while all the remaining type numbers are equal to \(\lambda\). The class \(\Ц_{\lambda,m}\) can be divided into \(2\lambda\) nonintersecting subclasses \(\Ц_{\lambda,m}^{(j)}\) in the following way: a function \(f(x)\in \Ц_{\lambda,m}^{(j)}\) \((j=1,\ldots,\lambda)\), if \(f(x)\in \Ц_{\lambda,m}\), \(f(x)f'(x)\geqslant 0\) for \(x\in[0,1]\), and the first type number of this function is equal to \(\lambda+1-j\); a function \(f(x)\in \Ц_{\lambda,m}^{(j)}\) \((j=\lambda+1,\ldots,2\lambda)\), if \(f(x)\in \Ц_{\lambda,m}\), \(f(x)f'(x)\leqslant 0\) for \(x\in[0,1]\), and the first type number of this function is equal to \(2\lambda+1-j\).

In \((^2)\) there were also introduced into consideration the generalized Euler–Bernstein polynomials \(A_{j,m}(x)\in \Ц_{\lambda,m}^{(j)}\), having degree \(m\) and satisfying the conditions:

\[ A_{j,m}^{(m)}(x)=1,\qquad A_{j,m}^{(k)}\bigl(\alpha_k^{(j)}\bigr)=0 \quad (k=0,1,\ldots,m-1), \]

in which

\[ \alpha_k^{(j)}= \begin{cases} 0 & \text{for } k=0,1,\ldots,\lambda-j,\\ 1 & \text{for } k\equiv \lambda-j+1,\ldots,2\lambda-j \pmod{2\lambda},\\ 0 & \text{for } k\equiv 2\lambda-j+1,\ldots,3\lambda-j \pmod{2\lambda}, \end{cases} \]

if \(1\leqslant j\leqslant \lambda\), and

\[ \alpha_k^{(j)}= \begin{cases} 1 & \text{for } k=0,1,\ldots,2\lambda-j,\\ 0 & \text{for } k\equiv 2\lambda-j+1,\ldots,3\lambda-j \pmod{2\lambda},\\ 1 & \text{for } k\equiv 3\lambda-j+1,\ldots,4\lambda-j \pmod{2\lambda}, \end{cases} \]

if \(\lambda+1\leqslant j\leqslant 2\lambda\).

The introduction of \(2\lambda\) sequences of generalized Euler–Bernstein numbers \(E_m^{(j)}\) \((j=1,\ldots,2\lambda;\ m=0,1,2,\ldots)\), defined for each fixed \(j\) by the recurrence relations

\[ E_0^{(j)}=1, \]

\[ (1+E^{(j)})_m=0 \quad \text{for } m\equiv j,\ldots,j+\lambda-1 \pmod{2\lambda}, \]

\[ E_m^{(j)}=0 \quad \text{for } m\equiv j+\lambda,\ldots,j+2\lambda-1 \pmod{2\lambda}, \]

allows one to write any polynomial \(A_{j,m}(x)\) in the form\(^*\)

\[ A_{j,m}(x)=\frac{\bigl(x+E^{(m+j)}\bigr)_m}{m!}, \]

\(^*\) The symbol \((x+E)_m\) means that the parentheses are to be expanded by Newton’s binomial formula, and then the powers \(E^k\) are to be replaced by the numbers \(E_k\).

where the index \(m+j\) must be replaced by \(m+j-2\lambda\), if \(m+j>2\lambda\). From a theorem of S. N. Bernstein (\(^{1}\), p. 515) there follows the following extremal theorem:

Theorem 1. Of all polynomials \(P_m(x)\in \mathrm{Ц}_{\lambda,m}^{(j)}\) of the form

\[ P_m(x)=\frac{1}{m!}x^m+p_{m-1}x^{m-1}+\cdots+p_0 \]

the polynomial \(A_{j,m}(x)\) deviates least from zero at each point \(x\in[0,1]\). The magnitude of the least deviation on the whole interval \(L_m^{(j)}\) is determined by the equalities

\[ L_m^{(j)}=\left|A_{j,m}\left(1-\alpha_0^{(j)}\right)\right|= \begin{cases} \dfrac{\left|E_m^{(m+j+\lambda)}\right|}{m!}, & \text{for } j=1,\ldots,\lambda,\\[1.2em] \dfrac{\left|E_m^{(m+j)}\right|}{m!}, & \text{for } j=\lambda+1,\ldots,2\lambda. \end{cases} \]

The following propositions hold, making it possible to compare the quantities \(L_m^{(j)}\) for different \(j\).

Theorem 2. If \(1\leq j\leq \lambda\), then \(L_m^{(j)}=L_m^{(\lambda+j)}\).

Theorem 3. If \(1\leq j\leq \lambda\), then \(L_{\lambda p+j}^{(\lambda+1-j)}=L_{\lambda p+j}^{(\lambda)}\).

Theorem 4. Of all polynomials \(P_{\lambda p+j}(x)\in \mathrm{Ц}_{\lambda,\lambda p+j}\) of the form

\[ P_{\lambda p+j}(x)=\frac{1}{(\lambda p+j)!}x^{\lambda p+j} +p_{\lambda p+j-1}x^{\lambda p+j-1}+\cdots+p_0 \]

the polynomials \(A_{\lambda+1-j,\lambda p+j}(x)\), \(A_{2\lambda+1-j,\lambda p+j}(x)\), \(A_{\lambda,\lambda p+j}(x)\), and \(A_{2\lambda,\lambda p+j}(x)\) deviate least from zero on \([0,1]\), and the magnitude of the least deviation is determined by the formula

\[ L_{\lambda p+j} = \frac{\left|E_{\lambda p+j}^{(\lambda p+1)}\right|}{(\lambda p+j)!} = \frac{\left|E_{\lambda p+j}^{(\lambda p+j)}\right|}{(\lambda p+j)!}. \]

Received
7 II 1962

CITED LITERATURE

\(^{1}\) S. N. Bernstein, On Certain Properties of Cyclically Monotone Functions, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1954, p. 493.
\(^{2}\) V. L. Fainshmidt, Izv. AN BSSR, Series of Physical-Technical Sciences, No. 3 (1960).