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)})=0;
]

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

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

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