MATHEMATICS

I. I. IBRAGIMOV

EXTREMAL PROBLEMS IN THE CLASS OF TRIGONOMETRIC POLYNOMIALS

(Presented by Academician S. N. Bernstein on 17 III 1958)

Many problems in the theory of approximation of periodic functions by means of trigonometric polynomials are connected with extremal properties of these polynomials.

An arbitrary trigonometric polynomial of order \((n_1, n_2,\ldots,n_k)\) has the form

\[ T_{n_1,\ldots,n_k}(x_1,x_2,\ldots,x_k) = \sum_{\nu_1=-n_1}^{n_1}\cdots \sum_{\nu_k=-n_k}^{n_k} C_{\nu_1,\ldots,\nu_k} e^{i\nu_1x_1+i\nu_2x_2+\cdots+i\nu_kx_k}. \]

We introduce the notation:

\[ \|T_{n_1,\ldots,n_k}\|_p = \left( \int_0^{2\pi}\cdots\int_0^{2\pi} |T_{n_1,\ldots,n_k}(t_1,\ldots,t_k)|^p \,dt_1\cdots dt_k \right)^{1/p}, \]

where \(p\ge 1\) is an arbitrary number.

Among the numerous investigations in the class of trigonometric polynomials we note the well-known inequality of S. N. Bernstein \((^{1}),\) p. 26, and its generalization, obtained by N. K. Bari \((^{3})\) in the space \(L_p(a,b)\), where \(0<a<b<2\pi\). A remarkable inequality was obtained by S. M. Nikol’skii \((^{4})\) for trigonometric polynomials of many variables:

\[ \|T_{n_1,\ldots,n_k}\|_{p'} \le 2^k \left(\prod_{j=1}^{k} n_j\right)^{1/p-1/p'} \|T_{n_1,\ldots,n_k}\|_{p} \qquad (1\le p\le p'). \tag{1} \]

In this direction a number of general problems were first considered by S. N. Bernstein (see, for example, \((^{1})\), pp. 97—100; \((^{2})\), pp. 426—432 and 605—606).

We assume that the function \(K(x_1,x_2,\ldots,x_k)\) is integrable inside the \(k\)-dimensional cube \((G_k)=(0\le x_j\le 2\pi,\ 1\le j\le k)\), and consider the functional

\[ J_k(T_{n_1,\ldots,n_k}) = \frac{1}{(2\pi)^k} \int_0^{2\pi}\cdots\int_0^{2\pi} T_{n_1,\ldots,n_k}(t_1,\ldots,t_k) K(t_1,\ldots,t_k)\,dt_1\cdots dt_k. \]

With respect to this functional the following proposition holds.

Theorem 1. If \(T_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\) is an arbitrary trigonometric polynomial of order \((n_1,n_2,\ldots,n_k)\), and if \(K(x_1,\ldots,x_k)\) is an integrable function inside the \(k\)-dimensional cube \((G_k)=(0\le x_j\le 2\pi,\ 1\le j\le k)\), with Fourier coefficients \(b_{\nu_1,\nu_2,\ldots,\nu_k}\) \((\nu_j=0,\pm1,\ldots;\ 1\le j\le k)\), then for the functional \(J_k(T_{n_1,\ldots,n_k})\) the following inequality holds:

\[ \left|J_k\left(T_{n_1,\ldots,n_k}\right)\right| \leq (2\pi)^{-k/p}\left\|T_{n_1,\ldots,n_k}\right\|_p \left(\sum_{-n_1}^{n_1}\cdots \sum_{-n_k}^{n_k} |b_{\nu_1,\ldots,\nu_k}|^p\right)^{1/p}. \]

for any \(p\) satisfying the condition \(1\leq p\leq 2\).

We note several consequences of Theorem 1 for a specifically given \(K(x_1,\ldots,x_k)\).

1. For an arbitrary trigonometric polynomial \(T_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\), the inequality holds
   \[ \max_{\substack{0\leq x_j\leq 2\pi\\(1\leq j\leq k)}} \left|T_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\right| \leq \prod_{j=1}^{k}\left(\frac{2n_j+1}{2\pi}\right)^{1/p} \left\|T_{n_1,\ldots,n_k}\right\|_p . \]

2. For the derivatives of the trigonometric polynomial \(T_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\), for \(1\leq p\leq 2\) we have:
   \[ \max_{\substack{0\leq x_j\leq 2\pi\\(1\leq j\leq k)}} \left| \frac{\partial^{m_1+\cdots+m_k}T_{n_1,\ldots,n_k}} {\partial x_1^{m_1}\cdots \partial x_k^{m_k}} \right| \leq \pi^{-k/p} \prod_{j=1}^{k} \frac{n_j^{m_j+1/p}}{(pm_j+1)^{1/p}} \cdot \left\|T_{n_1,\ldots,n_k}\right\|_p(1+o(1)). \]

3. Let real or complex numbers \(\lambda_{s_1,\ldots,s_k}\) be given
   \((s_j=0,\pm1,\pm2,\ldots,\pm n_j;\ 1\leq j\leq k)\), and let a trigonometric polynomial
   \(T_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\) have coefficients
   \(C_{\nu_1,\ldots,\nu_k}\)
   \((\nu_j=0,\pm1,\pm2,\ldots,\pm n_j;\ 1\leq j\leq k)\). Then for the trigonometric sum
   \[ f_\lambda(x_1,\ldots,x_k) = \sum_{s_1=-n_1}^{n_1}\cdots \sum_{s_k=-n_k}^{n_k} \lambda_{s_1,\ldots,s_k}C_{s_1,\ldots,s_k} e^{i(s_1x_1+\cdots+s_kx_k)} \]
   for \(1\leq p\leq 2\) the inequality holds
   \[ \max_{\substack{0\leq x_j\leq 2\pi\\(1\leq j\leq k)}} \left|f_\lambda(x_1,\ldots,x_k)\right| \leq (2\pi)^{-k/p}\left\|T_{n_1,\ldots,n_k}\right\|_p \left(\sum_{-n_k}^{n_k}\cdots\sum_{-n_1}^{n_1} |\lambda_{s_1,\ldots,s_k}|^p\right)^{1/p}. \]

4. If the polynomial \(\widetilde T(x)\) is conjugate to the polynomial \(T(x)\), then for \(1\leq p\leq 2\) we have
   \[ \max_{0\leq x\leq 2\pi}|\widetilde T(x)| \leq 2\left(\frac{n}{2\pi}\right)^{1/p}\|T\|_p, \]
   \[ \max_{0\leq x\leq 2\pi}|\widetilde T'(x)| \leq 2\left(\frac{n}{2\pi(p+1)}\right)^{1/p} n\|T\|_p(1+o(1)). \]

Theorem 2. If \(1\leq p\leq 2\), \(p'\geq p\), then for an arbitrary trigonometric polynomial
\(T_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\) the inequality holds
\[ \left\|T_{n_1,\ldots,n_k}\right\|_{p'} \leq \left(\prod_{j=1}^{k}\frac{2n_j+1}{2\pi}\right)^{1/p-1/p'} \left\|T_{n_1,\ldots,n_k}\right\|_p . \tag{2} \]

We observe that inequality (2) sharpens the inequality of S. M. Nikol’skii (1). Similar assertions hold for algebraic polynomials of the form
\[ Q(x_1,\ldots,x_k) = \sum_{\nu_1=0}^{n_1}\cdots \sum_{\nu_k=0}^{n_k} a_{\nu_1,\ldots,\nu_k}x_1^{\nu_1}\cdots x_k^{\nu_k}. \]

Theorem 3. For the algebraic polynomial \(Q(x_1,\ldots,x_k)\) in the \(k\)-dimensional cube
\((W_k)=(a\leq x_j\leq b,\ 1\leq j\leq k)\), for \(1\leq p\leq 2\) the inequalities hold

\[ \max_{\substack{a \le x_j \le b\\ (1 \le j \le k)}} \left|\frac{\partial Q(x_1,\ldots,x_k)}{\partial x_j}\right| \le \left(\frac{2}{\pi}\right)^{k/p} \frac{n_j^{1+1/p}}{\sqrt{(x_j-a)(b-x_j)}} \left(\prod_{\substack{l=1\\ l\ne j}}^{k} n_l^{1/p}\right) L_{k,p}(Q), \]

\[ \max_{\substack{a \le x_j \le b\\ (1 \le j \le k)}} |Q(x_1,x_2,\ldots,x_k)| \le \prod_{j=1}^{k} \left(\frac{2n_j+1}{\pi}\right)^{1/p} L_{k,p}(Q), \]

where

\[ L_{k,p}(Q)= \left( \int_a^b \cdots \int_a^b |Q(x_1,\ldots,x_k)|^p \frac{dx_1\,dx_2\cdots dx_k} {\prod_{j=1}^{k}(x_j-a)(b-x_j)} \right)^{1/p}. \]

Institute of Physics and Mathematics
of the Academy of Sciences of the Azerbaijan SSR

Received
17 III 1958

REFERENCES

1. S. N. Bernstein, Collected Works, 1, Publishing House of the Academy of Sciences of the USSR, 1952.
2. S. N. Bernstein, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1954.
3. N. K. Bari, Izv. Akad. Nauk SSSR, Mathematical Series, 18, No. 2, 159 (1954).
4. S. M. Nikol’skii, Trudy Mat. Inst. im. V. A. Steklova, 38 (1951).