V. L. FAINSHMIDT

ON A CLASS OF REGULARLY MONOTONE POLYNOMIALS

(Presented by Academician S. N. Bernstein on 22 X 1959)

Following S. N. Bernstein \((^1)\), we shall call a function \(f(x)\) regularly monotone of order \(m\) on \([0,1]\) if it and its first \(m\) derivatives do not change sign on the interval \([0,1]\). Every regularly monotone function is characterized by a sequence of type numbers \(\lambda_1,\lambda_2,\ldots,\lambda_s\), which are defined as follows. For definiteness, we shall assume that \(f(x)f'(x)\ge 0\) for \(x\in[0,1]\). Then \(\lambda_1\) has the property that \(f^{(i-1)}(x)f^{(i)}(x)\ge 0\) for \(i\le \lambda_1\) and \(f^{(\lambda_1)}(x)f^{(\lambda_1+1)}(x)\le 0\). The type number \(\lambda_2\) is determined from the condition \(f^{(i-1)}(x)f^{(i)}(x)\le 0\) for \(\lambda_1<i\le \lambda_1+\lambda_2\) and \(f^{(\lambda_1+\lambda_2)}(x)f^{(\lambda_1+\lambda_2+1)}(x)\ge 0\). The numbers \(\lambda_3,\lambda_4,\ldots,\lambda_s\) are defined analogously, and one must have \(\lambda_1+\lambda_2+\cdots+\lambda_s=m\).

Consider the class \(Ц_{2,m}\) of polynomials regularly monotone of order \(m\) on \([0,1]\), for which the first and last type numbers are equal to 1 or 2, while all the remaining type numbers are equal to 2. If the polynomial \(P_n(x)\in Ц_{2,m}\), then, obviously, for it and its first two derivatives the following possibilities are admissible:

\[ \begin{aligned} 1)\quad & P_n(x)P_n'(x)\ge 0, \qquad && P_n'(x)P_n''(x)\le 0;\\ 2)\quad & P_n(x)P_n'(x)\le 0, \qquad && P_n'(x)P_n''(x)\le 0;\\ 3)\quad & P_n(x)P_n'(x)\le 0, \qquad && P_n'(x)P_n''(x)\ge 0;\\ 4)\quad & P_n(x)P_n'(x)\ge 0, \qquad && P_n'(x)P_n''(x)\ge 0. \end{aligned} \tag{A} \]

We divide the class \(Ц_{2,m}\) into four subclasses \(Ц_{2,m}^{(i)}\) \((i=1,2,3,4)\); moreover, we shall say that \(P_n(x)\in Ц_{2,m}^{(i)}\) if \(P_n(x)\in Ц_{2,m}\) and satisfies the \(i\)-th of conditions (A). It is clear that
\[ Ц_{2,m}=Ц_{2,m}^{(1)}+Ц_{2,m}^{(2)}+Ц_{2,m}^{(3)}+Ц_{2,m}^{(4)}. \]

The most interesting in the class \(Ц_{2,m}\) are the polynomials \(A_{i,m}(x)\in Ц_{2,m}^{(1)}\) \((i=1,2,3,4)\), satisfying the conditions

\[ A_{1,m}^{(4k)}(0)=A_{1,m}^{(4k+1)}(1)=A_{1,m}^{(4k+2)}(1)=A_{1,m}^{(4k+3)}(0)=0; \]

\[ A_{2,m}^{(4k)}(1)=A_{2,m}^{(4k+1)}(1)=A_{2,m}^{(4k+2)}(0)=A_{2,m}^{(4k+3)}(0)=0; \]

\[ A_{3,m}^{(4k)}(1)=A_{3,m}^{(4k+1)}(0)=A_{3,m}^{(4k+2)}(0)=A_{3,m}^{(4k+3)}(1)=0; \]

\[ A_{4,m}^{(4k)}(0)=A_{4,m}^{(4k+1)}(0)=A_{4,m}^{(4k+2)}(1)=A_{4,m}^{(4k+3)}(1)=0 \]

and normalized by the condition
\[ A_{i,m}^{(m)}(x)=1. \]

These polynomials, as is not difficult to show, are connected with one another by the relations

\[ A_{i,m}^{(4k+j)}(x)=A_{i+j,m-4k-j}(x), \]

\[ A_{i,m}(x)=(-1)^m A_{i+2,m}(1-x), \]

where in the last formulas \(A_{k,p}(x)\equiv A_{k-4,p}(x)\) if \(k>4\).

The polynomials \(A_{i,m}(x)\) are constructed by the method indicated by S. N. Bernstein in the paper \((^2)\). For the construction, we introduce the numbers \(E_m^{(i)}\) \((i=1,2,3,4)\), analogous to the Euler–Bernstein numbers and defined by the equalities

\[ E_0^{(i)}=1 \qquad (i=1,2,3,4); \]

\[ E_{4k}^{(1)}=E_{4k+1}^{(1)}=0;\quad (1+E^{(1)})_{4k+2}=(1+E^{(1)})_{4k+3}=0; \]

\[ E_{4k+1}^{(2)}=E_{4k+2}^{(2)}=0;\quad (1+E^{(2)})_{4k+3}=(1+E^{(2)})_{4k}=0; \]

\[ E_{4k+2}^{(3)}=E_{4k+3}^{(3)}=0;\quad (1+E^{(3)})_{4k}=(1+E^{(3)})_{4k+1}=0; \]

\[ E_{4k+3}^{(4)}=E_{4k}^{(4)}=0;\quad (1+E^{(4)})_{4k+1}=(1+E^{(4)})_{4k+2}=0, \]

in which the expression \((1+E)_m\) means that the brackets are to be expanded according to Newton’s binomial formula and the powers \(E^r\) replaced by the numbers \(E_r\).

For the numbers \(E_m^{(i)}\) the equalities

\[ E_m^{(i)}=(-1)^m(1+E^{(i+2)})_m,\qquad E_{4k}^{(2)}=E_{4k}^{(3)},\qquad E_{4k+2}^{(1)}=-E_{4k+2}^{(4)}, \]

hold, with \(E_r^{(k)}=E_r^{(k-4)}\) for \(k>4\).

With the aid of these numbers the polynomials \(A_{i,m}(x)\) can be written in the form

\[ A_{i,4k+j}(x)=\frac{(x+E^{(i+j)})_{4k+j}}{(4k+j)!}. \]

The following extremal theorems hold:

Theorem 1. Of all polynomials \(P_m(x)\in L_{2,m-s}^{(i)}\) of the form

\[ P_m(x)=\sum_{k=m-s}^{m}\sigma_k x^k+\sum_{k=0}^{m-s-1}p_k x^k, \]

where \(\sigma_k\) \((k=m-s,\ldots,m)\) are fixed, the polynomial least deviating from \(0\) on \([0,1]\) is

\[ P_m^*(x)=\sum_{r=0}^{s}(m-r)!\,a_{m-r}A_{i,m-r}(x), \]

in which

\[ a_{m-r}= \begin{cases} \sigma_{m-r}, & \text{for } r\equiv m+i+3,\ m+i \pmod 4,\\[4pt] \displaystyle \sum_{k=0}^{r} C_{m-k}^{\,r-k}\sigma_{m-k}, & \text{for } r\equiv m+i+1,\ m+i+2 \pmod 4. \end{cases} \]

Moreover, the least deviation is determined by the formula

\[ L_m^{(i)}= \left| \sum_{r=0}^{s}(-1)^{\alpha r}a_{m-r}E_{m-r}^{(j)} \right|, \]

where

\[ \alpha=\frac{i^2-i+2}{2},\qquad j\equiv m-r-\frac{3+(-1)^i}{2}\pmod 4. \]

Remark 1. The coefficients \(a_{m-r}\) are found as the solution of the system

\[ \sigma_{m-r}=\sum_{k=0}^{r} C_{m-k}^{\,r-k}E_{r-k}^{(m+i-k)}a_{m-k} \qquad (r=0,1,\ldots,s). \]

Remark 2. For the set of polynomials under consideration to be nonempty, it is necessary and sufficient that the coefficients \(\sigma_k\) be such that the polynomial

\[ \sum_{k=m-s}^{m} \frac{k!}{(k-m+s)!}\,\sigma_k x^{k-m+s} \]

does not change sign on \([0,1]\).

Remark 3. From the theorem just formulated, in particular, for \(s=0\) there follows the extremal assertion of S. N. Bernstein from paper (3) (p. 548).

Theorem 2. Among all polynomials \(P_m(x)\in Ц_{2,s+1}^{(i)}\) of the form

\[ P_m(x)=\sum_{k=s+1}^{m}\rho_k x^k+\sigma_s x^s+\sum_{k=0}^{s-1}\rho_k x^k, \]

where \(\sigma_s\) is fixed, the polynomial

\[ P_m^*(x)=s!\sigma_s A_{i,s}(x) \]

deviates least from \(0\) on \([0,1]\), and the least deviation is determined by the formula

\[ L_m^{(i)}=|\sigma_s E_s^{(i)}|, \]

where

\[ j\equiv s-1 \pmod 4 \quad \text{for } i=1,3;\qquad j\equiv s-2 \pmod 4 \quad \text{for } i=2,4. \]

Theorem 3. Among all polynomials \(P_m(x)\in Ц_{2,m-1}^{(i)}\) of the form

\[ P_m(x)=\rho_m x^m+\sigma_{m-1}x^{m-1}+\sum_{k=0}^{m-2}\rho_k x^k, \]

where \(\sigma_{m-1}\) is fixed, the polynomial

\[ P_m^*(x)= \begin{cases} (m-1)!\,\sigma_{m-1}\bigl[-A_{i,m}(x)+A_{i,m-1}(x)\bigr], & \text{for } m+i\equiv 1,2 \pmod 4,\\[4pt] -(m-1)!\,\sigma_{m-1}A_{i,m}(x), & \text{for } m+i\equiv 0,3 \pmod 4, \end{cases} \]

deviates least from \(0\) on \([0,1]\), and the magnitude of the least deviation is determined by the formula

\[ L_m^{(i)}= \begin{cases} \left|\sigma_{m-1}\left[(-1)^\beta \dfrac{E_m^{(j+1)}}{m}+E_m^{(j)}\right]\right|, & \text{for } m+i\equiv 1,2 \pmod 4,\\[8pt] \left|\sigma_{m-1}\dfrac{E_m^{(j+1)}}{m}\right|, & \text{for } m+i\equiv 0,3 \pmod 4, \end{cases} \]

where

\[ \beta=\frac{i^2-i}{2},\qquad j\equiv m-\frac{5+(-1)^i}{2}\pmod 4. \]

Remark. Theorem 3 is obviously not a special case of Theorem 2 corresponding to \(s=m-1\), since in Theorem 3 the extremal polynomial is sought in the class \(Ц_{2,m-1}^{(i)}\), which is broader than the class \(Ц_{2,m}^{(i)}\).

Received
20 X 1959

CITED LITERATURE

\(^1\) S. N. Bernstein, Collected Works, 1, No. 32, Publishing House of the Academy of Sciences of the USSR, 1952.
\(^2\) S. N. Bernstein, Collected Works, 2, No. 100, Publishing House of the Academy of Sciences of the USSR, 1954.
\(^3\) S. N. Bernstein, Collected Works, 2, No. 106, Publishing House of the Academy of Sciences of the USSR, 1954.