Mathematics

G. K. LEBED’

INEQUALITIES FOR POLYNOMIALS AND THEIR DERIVATIVES

(Presented by Academician M. A. Lavrent’ev, 8 VI 1957)

In this note we give several theorems that provide estimates for the derivatives of a trigonometric polynomial

[
T_n(\theta)=\sum_{k=0}^{n}(a_k\cos k\theta+b_k\sin k\theta)
\tag{1}
]

or of an algebraic polynomial

[
P_n(x)=\sum_{k=0}^{n}a_k x^k .
\tag{2}
]

Theorem 1. If a trigonometric polynomial (1) of degree not exceeding (n) satisfies, for all (\theta), the inequality (|T_n(\theta)|\le t_n(\theta)), where (t_n(\theta)) is a nonnegative function having a continuous derivative (t_n^{(r)}(\theta)) of order (r) ((r=0,1)) with modulus of continuity

[
\omega_r(h)=\sup_{|\theta_1-\theta_2|\le h}\left|t_n^{(r)}(\theta_1)-t_n^{(r)}(\theta_2)\right|
\quad (\theta_1,\theta_2\in[-\pi,\pi]),
]

then

[
\left|T_n^{(k)}(\theta)\right|\le (An)^k
\left{t_n(\theta)+k\frac{\omega_r(1/n)}{n^2}\right}
\quad (n,k=1,2,\ldots,\ r=0,1),
\tag{3}
]

where (A) is a constant independent of (T_n) and (k).

In particular, when (t_n(\theta)=\mathrm{const}), (k=1), and (T_n(\theta)) is an even trigonometric polynomial, inequality (3) becomes the well-known inequality for algebraic polynomials ((¹), pp. 72–75; (²), p. 27), but with a cruder constant. If (t_n(\theta)) has a bounded second discontinuous derivative, then (\omega_1(t)=O(t)), and therefore the remainder term (k\,\frac{\omega_r(1/n)}{n^r}) on the right-hand side of (3) will have order (O(n^{-2})).

Let us note in this connection that if we improve the properties of the function (t_n(\theta)), for example, require that it have a bounded third derivative, this circumstance does not lead to an improvement in the order of the remainder term. It still remains equal to (O(n^{-2})), as is easily verified from the example (T_n(\theta)=\sin^2 n\theta\le n^2\sin^2\theta\le t_n(\theta)).

Definition. We shall say that a function (\omega(t)) satisfies condition (A_\alpha^\beta) if the following properties hold for it:

1) (\omega(t)>0,\ t>0);

2) (\omega(t_1)\le c\omega(t_2)\ \left(0\le t_1\le t_2\le \dfrac{2}{n^\alpha},\ 0\le \alpha\right));

3) (\dfrac{\omega(t_2)}{t_2^\beta}\le c\,\dfrac{\omega(t_1)}{t_1^\beta}\ \left(0<t_1\le t_2\le \dfrac{2}{n^\alpha},\ 0\le \beta\right)),

where (c) is a constant independent of (t_1) and (t_2).

Theorem 2. If, for the trigonometric polynomial (T_n(\theta)), the inequality

[
|T_n(\theta)| \leq \omega\left(\frac{|\sin\theta|}{n^\alpha}+\frac{1}{n^{1+\alpha}}\right)
\qquad (0\leq \alpha),
]

holds, where (\omega(t)) satisfies condition (A_\alpha^\beta), then

[
\left|T_n^{(k)}(\theta)\right|
\leq (An)^k\omega\left(\frac{|\sin\theta|}{n^\alpha}+\frac{1}{n^{1+\alpha}}\right)
\qquad (n,k=1,2,\ldots),
]

where (A) is a constant independent of (T_n(\theta)) and (k).

Theorem 3. If the function (\omega(t)) satisfies condition (A_\alpha^\beta), where (\alpha=1,\ 0\leq\beta\leq 1), then for any algebraic polynomial (P_n(x)) and any (r,p,p') satisfying the inequalities (1\leq p\leq p'\leq\infty), one has

[
\left|
\frac{P_n(x)\delta^{r-1/p'}(x,n)}
{\omega[\delta(x,n)/n]}
\right|{L}(a,b)
\leq
A\left(\frac{2n}{b-a}\right)^{1/p-1/p'}
\left|
\frac{P_n(x)\delta^{r-1/p}(x,n)}
{\omega[\delta(x,n)/n]}
\right|_{L_p(a,b)},
]

where

[
\delta(u,\nu)=\frac{2\sqrt{(b-u)(u-a)}}{b-a}+\frac{1}{\nu};
\qquad
A=A(r)
]

is a constant depending only on (r), and

[
|\varphi|_{L_p(a,b)}
=
\left(\int_a^b |\varphi(x)|^p dx\right)^{1/p}.
]

For nonperiodic functions this inequality is an analogue of the inequality of S. M. Nikol’skii ((^3)) for trigonometric polynomials or entire functions of finite degree.

In particular, if in it we put (\omega(t)\equiv 1,\ r=1/p,\ p'=\infty), then we obtain the inequality

[
|P_n(x)|
\leq
\frac{A}{\delta^{1/p}(x,n)}
\left(\frac{2n}{b-a}\right)^{1/p}
|P_n|_{L_p(a,b)},
\qquad
1\leq p\leq\infty,
]

which is a strengthening of Jackson’s inequality ((^4))

[
|P_n(x)|\leq K(a,b)n^{2/p}|P_n|_{L_p(a,b)}.
]

Theorem 4. Under the assumptions on the function (\omega(t)) imposed in Theorem 3, for every algebraic polynomial (P_n(x)) of degree not exceeding (n) the inequality

[
\left|
\frac{P_n^{(k)}\delta^{k+r}(x,n)}
{\omega[\delta(x,n)/n]}
\right|{L_p(a,b)}
\leq
\left(A\frac{2a}{b-a}\right)^k
\left|
\frac{P_n(x)\delta^r(x,n)}
{\omega[\delta(x,n)/n]}
\right|,
\tag{4}
]

[
(n,k=1,2,\ldots;\quad 1\leq p\leq\infty),
]

holds, where (A=A(r)) is a constant depending only on (r) ((r) arbitrary).

In particular, if in (4) we put (k=1,\ \omega(t)\equiv 1,\ r=-\rho) and (p=\infty), then we obtain the inequality of V. K. Dzyadyk ((^5))*

[
\left|
\frac{P_n'(x)}{\delta^{\rho-1}(x,n)}
\right|
\leq
An\max_{a\leq x\leq b}
\left|
\frac{P_n(x)}{\delta^\rho(x,n)}
\right|.
]

* This inequality was obtained by us by another method independently of Dzyadyk and was reported at the seminar on the theory of approximation of functions at the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR at the end of 1956.

If in (4) we put (\omega(t)\equiv 1) and (r=0), then we obtain the inequality

[
\bigl|P_n^{(k)}(x)\delta^k(x,n)\bigr|{L_p(a,b)}
\leq
\left(\frac{An}{b-a}\right)^k \bigl|P_n(x)\bigr|,
]

which is obviously stronger than the corresponding inequality of N. K. Bari ({}^{(6)})

[
\bigl|P_n^{(k)}\bigr|{L_p(a,b)} \leq c n^{2k}\bigl|P_n\bigr|.
]

In proving the theorems stated above, we used the fact that a trigonometric polynomial (T_n(\theta)) of order (n) can be represented (in infinitely many ways) in the form of the integral

[
T_n(\theta)=\frac{1}{\pi}\int_0^{2\pi} K_m(u)T_n(\theta+u)\,du \qquad (m\leq n),
]

where (K_m(u)) has the form

[
K_m(u)=\frac{1}{2}+\sum_{k=1}^{n}\cos ku+\sum_{k=n+1}^{m}(a_k\cos ku+b_k\sin ku),
]

and the coefficients (a_k) and (b_k) may be arbitrary. Then

[
T'_n(\theta)=\frac{1}{\pi}\int_0^{2\pi} K'_m(u)T_n(\theta+u)\,du.
]

To obtain our inequalities we used this representation of the derivative (T'_n(\theta)), each time choosing the coefficients (a_k) and (b_k) in the appropriate way.

In these investigations the following lemma is also used:

Lemma. Let (P_n(x)) be an algebraic polynomial of degree (\leq n), and suppose that on the interval

[
\left[-1+\frac{c^2}{n^2},\;1-\frac{c^2}{n^2}\right]
]

the inequality

[
|P_n(x)|\leq
\left(\sqrt{1-x^2}+\frac{1}{n}\right)^s
\omega\left(\frac{\sqrt{1-x^2}}{n}+\frac{1}{n^2}\right)
]

holds. Then, if (\omega) satisfies the conditions of Theorem 3, the inequality

[
|P_n(x)|\leq
A\left(\sqrt{1-x^2}+\frac{1}{n}\right)^s
\omega\left(\frac{\sqrt{1-x^2}}{n}+\frac{1}{n^2}\right),
]

where (A) is some constant (\geq 1), independent of (n) and (x), holds on the entire interval ([-1,1]).

This note arose as a result of work on a Candidate dissertation carried out under the supervision of Prof. S. M. Nikol’skii.

Received
29 V 1957

References

({}^{1}) A. A. Markov, Selected Works on the Theory of Continued Fractions and the Theory of Functions Deviating Least from Zero, 1948.
({}^{2}) S. N. Bernstein, Collected Works, 1, Constructive Theory of Functions, 1905–1930, Publ. Acad. Sci. USSR, 1952.
({}^{3}) S. M. Nikol’skii, Trudy Mat. Inst. im. V. A. Steklova, Acad. Sci. USSR 38, 244 (1951).
({}^{4}) D. Jackson, Trans. Am. Math. Soc. 40, 225 (1936).
({}^{5}) V. K. Dzyadyk, Izv. Acad. Sci. USSR, Math. Ser., 20, No. 5, 623 (1956).
({}^{6}) N. K. Bari, Izv. Acad. Sci. USSR, Math. Ser., 18, No. 2, 159 (1954).