N. I. Chernykh

ON THE APPROXIMATION OF FUNCTIONS BY POLYNOMIALS WITH CONSTRAINTS

(Presented by Academician S. N. Bernstein on 2 XII 1964)

The problem of approximating functions by algebraic polynomials and trigonometric polynomials whose coefficients are connected by a linear relation was first studied by V. A. Markov \((^5)\). The paper \((^5)\) initiated numerous investigations in this direction (see \((^{3,4})\)).

The present paper is devoted to Jackson-type estimates for best approximations of a continuous function in the case of homogeneous linear constraints of a certain special form. For the same constraints, the paper indicates the exact order of the upper bound of best approximations over a class of functions analytic and bounded in a prescribed domain. For approximations without constraints, the latter problem was solved by N. A. Akhiezer (\((^1)\), pp. 230—235).

§ 1. Let \(X\) be a linear normed space and \(\{x_k\}_0^\infty \subset X\). By \(X_n=\{p_n\}\) \((n=0,1,\ldots)\) we denote the subspace of \(X\) spanned by the system \(\{x_k\}_0^n\). Let \(\Psi\) be a homogeneous additive functional defined on the union of all \(X_n\), and let its norm on \(X_n\) be equal to \(\|\Psi\|_n\). For any element \(f\in X\) put

\[ E_n(f)=\inf_{p_n\in X_n}\|f-p_n\|=\|f-p_n(f)\|,\qquad E_n(f;\Psi)=\inf_{\Psi(p_n)=0}\|f-p_n\|, \]

where the norm is the same as in the space \(X\). In this section we agree to regard \(0/0=0\). In these notations the following inequalities are valid*:

\[ {}^{1}\!/_{3}\{E_n(f)+|\Psi(p_n(f))|(\|\Psi\|_n)^{-1}\}\leq E_n(f;\Psi)\leq \]
\[ \leq E_n(f)+|\Psi(p_n(f))|(\|\Psi\|_n)^{-1} \qquad (n=0,1,\ldots); \tag{1.1} \]

\[ E_n(f;\Psi)\leq |\Psi(p_0(f))|(\|\Psi\|_n)^{-1}+ \]
\[ +2\left\{E_n(f)+(\|\Psi\|_n)^{-1}\sum_{k=0}^{n-1}E_k(f)\|\Psi\|_{k+1}\right\} \qquad (n=0,1,\ldots). \]

§ 2. Let \(X=C_{[-1,1]}\); \(X_n\subset X\) is the subspace of algebraic polynomials \(p_n(x)\) of degree \(n\). We shall denote the functional \(\Psi\) by \(\Psi_{\alpha,a}\) \((\alpha>-1,\ a>1)\), if it can be represented in the form

\[ \Psi(p_n)=\int_{-a}^{a} p_n(x)(a^2-x^2)^\alpha \varphi(x)\,dx =\Psi_{\alpha,a}(p_n), \tag{2.1} \]

where \(\varphi(x)\) is a function summable on the interval \([-a,a]\), continuous at the points \(x=\pm a\), and \(\varphi^2(a)+\varphi^2(-a)\neq 0\). For \(a>1\) put \(g(a)=a+(a^2-1)^{1/2}>1\).

* A special case of inequality (1.1) was indicated earlier \((^6)\).

Let \(\omega(\delta; f)\) be the modulus of continuity of the function \(f(x)\in C_{[-1,1]}\). From the estimate (1,2), inequality (3) of the work [7], and D. Jackson’s inequality

\[ E_n(f)\leqslant \frac{c_r}{n^r}\,\omega\!\left(\frac1n;\, f^{(r)}\right)\qquad (n>r) \tag{2,2} \]

there follows the following

Theorem 1. Let \(r\geqslant 0\) be an integer, \(\alpha>-1\), \(a>1\). If \(f^{(r)}(x)\in C_{[-1,1]}\), then for \(n>r\)

\[ E_n(f;\Psi_{\alpha,a})\leqslant \frac{K_1}{n^r}\,\omega\!\left(\frac1n;\, f^{(r)}\right) +K_2\|f\|_{C[-1,1]}g^{-n/2}(a), \tag{2,3} \]

where the constants \(K_1\) and \(K_2\) depend only on \(r\) and \(\Psi_{\alpha,a}\).

Let \(\Gamma_b\) be the ellipse with foci at the points \(x=\pm 1\), passing through the point \(b>1\); let \(G(b)\) be the finite domain in the \(z\)-plane with boundary \(\Gamma_b\). We shall say that a function \(f(z)\) analytic in the domain \(G(b)\) belongs to \(MB(b)\) if it is real on the segment \([-1,1]\) and \(|f(z)|\leqslant M\) for \(z\in G(b)\).

Theorem 2. Let \(\alpha>-1\), \(a>1\), \(b>1\). For every functional \(\Psi_{\alpha,a}\) of the form (2,1) there exists a constant \(K_3=K_3(b,\Psi_{\alpha,a})\) such that, if the function \(f(z)\in MB(b)\), then

\[ E_{n-1}(f;\Psi_{\alpha,a})\leqslant \begin{cases} K_3Mg^{-n}(b), & \text{if } b<a,\\ K_3Mn^{\alpha+1}g^{-n}(a), & \text{if } b\geqslant a, \end{cases} \tag{2,4} \]

\[ (n=1,2,\ldots). \]
For every number \(b>1\) in the class \(MB(b)\), inequality (2,4) is exact in order as \(n\to\infty\).

Let \(h(x)\) \((-a\leqslant x\leqslant a)\) be a piecewise-constant function of bounded variation such that the endpoints of the segment \([-a,a]\) are not limit points for the discontinuities of \(h(x)\), and
\[ [h(a)-h(a-0)]^2+[h(-a+0)-h(-a)]^2\ne 0. \]
Put

\[ D_a(p_n)=\int_{-a}^{a} p_n(x)\,dh(x)\qquad (n=0,1,\ldots). \tag{2,5} \]

For \(a>1\), inequality (2,3) is preserved if in it \(\Psi_{\alpha,a}\) is replaced by \(D_a\). Therefore, for the best approximations \(E_n(f;D_a)\) we shall formulate only an analogue of Theorem 2.

Theorem 3. Let \(a>1\), \(b>1\), and let \(D_a\) be a functional of the form (2,5). Then there exists a constant \(K_4=K_4(b,D_a)\) such that, if \(f(z)\in MB(b)\), then

\[ E_n(f;D_a)\leqslant \begin{cases} K_4Mg^{-n}(b), & \text{if } b<a,\\ K_4M\ln(n+2)\,g^{-n}(a), & \text{if } b=a,\\ K_4Mg^{-n}(a), & \text{if } b>a, \end{cases} \tag{2,6} \]

\[ (n=0,1,\ldots). \]
For every number \(b>1\) in the class \(MB(b)\), inequality (2,6) is exact in order as \(n\to\infty\).

Let us note that Theorem 3 contains an important special case, when
\[ D_a(p_n)\equiv p_n(a),\qquad a>1. \]

§ 3. Let \(0<a<b\leqslant \pi\); \(X=C_{[-a,a]}\); \(X_{2n}\subset X\) be the subspace of trigonometric polynomials \(t_n(u)\) of order \(n\). The best approximations \(E'_{2n}(f)\) and \(E_{2n}(f;\Psi)\) (see § 1) in this case will be denoted by \(E_n(f,a)\) and \(E_n(f,a,\Psi)\), respectively. Put

\[ \Psi(t_n)=\Psi_{a,b}(t_n)=\int_{-b}^{b} t_n(u)(b^2-u^2)^\alpha\varphi(u)\,du, \tag{3,1} \]

where \(\varphi(u)\) is a function summable on the segment \([-b,b]\), continuous at the points \(u=\pm b\), and
\[ \varphi^2(b)+\varphi^2(-b)\ne 0. \]

In estimating the quantity \(E_n(f,a,\Psi_{\alpha,\pi})\) for analytic functions, it is useful to distinguish two cases according as \(\varphi(u)\) does or does not satisfy the equality

\[ \varphi(\pi)+\varphi(-\pi)=0. \tag{3,2} \]

If this equality holds, we shall also require that in neighborhoods of the points \(u=\pm\pi\) the inequalities

\[ |\varphi(u)-\varphi(\pm\pi)|\leq K(\pi\mp u) \tag{3,3} \]

hold, respectively.

Let \(q_1(w)=(1-\cos a)^{-1}(2\cos w-1-\cos a)\) and \(g_1(w)=g(q_1(w))\). For any fixed number \(\lambda>0\), denote by \(D(a,\lambda)\) the finite domain lying in the strip \(-\pi\leq \operatorname{Re} w\leq \pi\) and bounded by the level line \(|g_1(w)|=g_1(i\lambda)>1\).

Let the number \(\lambda_b\) be the root of the equation \(g_1(i\lambda_b)=|g_1(b)|\) \((a<b\leq\pi)\). For \(\lambda>\lambda_\pi\), the boundary of the domain \(D(a,\lambda)\) will include segments of the straight lines \(w=\pm\pi+iv\). We shall say that a function \(f(w)\), analytic in the domain \(D(a,\lambda)\), belongs to \(MB(a,\lambda)\) if it is real on the interval \([-a,a]\) and \(|f(w)|\leq M\) for \(w\in D(a,\lambda)\). In this case, when \(\lambda>\lambda_\pi\), we shall assume that the function \(f(w)\), extended with period \(2\pi\), remains analytic in the corresponding domain.

Using a result of N. I. Akhiezer ([1], p. 235), one can prove the following assertion:

Lemma. If \(f(w)\in MB(a,\lambda)\), then

\[ E_n(f,a)\leq HMg_1^{-n}(i\lambda)\quad (n=0,1,\ldots), \tag{3,4} \]

where the constant \(H\) does not depend on \(f(w)\). In the class \(MB(a,\lambda)\), the estimate (3,4) is sharp in order as \(n\to\infty\).

Hence, with the aid of inequality (1,1) and inequality (1)—(2) of paper [7], the following theorem follows:

Theorem 4. Let \(0<a<b\leq\pi\), \(\alpha>-1\), \(\lambda>0\), and let \(\Psi_{\alpha,b}\) be a functional of the form (3,1). Then there exists a constant \(H_1=H_1(a,\lambda,\Psi_{\alpha,b})\) such that, for any function \(f(w)\in MB(a,\lambda)\) and any \(n=1,2,\ldots\), we have:

1) if \(a<b\leq\pi\) and \(\lambda<\lambda_b\), then

\[ E_{n-1}(f,a,\Psi_{\alpha,b})\leq H_1Mg_1^{-n}(i\lambda); \]

2) if \(a<b<\pi\) and \(\lambda\geq\lambda_b\), then

\[ E_{n-1}(f,a,\Psi_{\alpha,b})\leq H_1Mn^{\alpha+1}g_1^{-n}(i\lambda_d); \]

3) if \(b=\pi\), \(\lambda\geq\lambda_\pi\), and the functional \(\Psi_{\alpha,\pi}\) does not satisfy condition (3,2), then

\[ E_{n-1}(f,a,\Psi_{\alpha,\pi})\leq H_1Mn^{(\alpha+1)/2}(\operatorname{ctg} a/4)^{-2n}; \]

4) if \(b=\pi\), \(\lambda\geq\lambda_\pi\), and the functional \(\Psi_{\alpha,\pi}\) satisfies conditions (3,2) and (3,3), then

\[ E_{n-1}(f,a,\Psi_{\alpha,\pi})\leq H_1Mn^{(\alpha+2)/2}(\operatorname{ctg} a/4)^{-2n}. \]

For any number \(\lambda>0\), all these inequalities in the class \(MB(a,\lambda)\) are sharp in order as \(n\to\infty\).

For continuous functions, with the aid of a theorem of S. B. Stechkin (see, for example, [2], p. 113), one can prove the following assertion:

Theorem 5. Let \(0<a<d\leq\pi\) and \(f(x)\in C[-d,d]\). For every functional \(\Psi_{\alpha,b}\) \((\alpha>-1,\ a<b\leq\pi)\) of the form (3,1), there exists a constant \(C_k=C_k(a,\Psi_{\alpha,b},\|f\|_{C[-d,d]})\) and a number \(N_k(a)\) such that for \(n>N_k(a)\) we have:

\[ E_n(f,a,\Psi_{\alpha,b})\leq C_k\omega_k(1/n,f,a), \]

where \(\omega_k(\delta,f,a)\) is the modulus of continuity of the function \(f(x)\) of order \(k=1,2,\ldots\) on the interval \([-a,a]\).

Denote by \(E_n^{(0)}(f,a,m)\) and \(E_n^{(1)}(f,a,m)\) the best approximations of the function \(f(x)\) by trigonometric polynomials \(t_n(x)\) that do not contain \(\cos mx\) or \(\sin mx\), respectively.

Corollary 1. Let \(0<a<d\leqslant \pi\), \(l=0,1\), \(f(x)\in C[-d,d]\).
For any integer \(m\geqslant 0\) there exist a constant
\(C_k^{(l)}=C_k^{(l)}(a,m,\|f\|_{C[-d,d]})\) and an \(N_k(a)\) such that, for \(n>N_k(a)\), we have

\[ E_n^{(l)}(f,a,m)\leqslant C_k^{(l)}\omega_k(1/n,f,a). \]

Corollary 2. Let \(0<a<\pi\), \(\lambda>0\), \(l=0,1\), \(f(w)\in MB(a,\lambda)\).
For every integer \(m\geqslant l\) there exist constants \(H_m^{(l)}=H_m^{(l)}(a,\lambda)\) such that, for \(n>m\), we have

\[ E_n^{(l)}(f,a,m)\leqslant \begin{cases} H_m^{(l)}Mg_1^{-n}(i\lambda), & \text{if } \lambda<\lambda_\pi,\\ H_m^{(l)}Mnl^{1/2}(\operatorname{ctg} a/4)^{-2n}, & \text{if } \lambda\geqslant \lambda_\pi \end{cases} \qquad (l=0,1). \]

For any \(\lambda>0\) these inequalities in the class \(MB(a,\lambda)\) are sharp in order as \(n\to\infty\).

Sverdlovsk Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
23 XI 1964

CITED LITERATURE

1. N. I. Akhiezer, Lectures on Approximation Theory, 1947.
2. N. K. Bari, Scientific Notes of Moscow State University, Mathematics, 8, issue 181, 107 (1956).
3. S. M. Lozinskii, I. P. Natanson, Mathematics in the USSR for 40 Years, 1, 1959, p. 363.
4. B. D. Korpuslichev, Problems of Mathematical Physics and Function Theory, 1, 1964, p. 72.
5. A. A. Markov, On functions least deviating from zero, St. Petersburg, 1892.
6. N. I. Chernykh, Proceedings of the II Scientific Conference of Mathematics Departments of Pedagogical Institutes of the Volga Region, issue 1, 1962, p. 113.
7. N. I. Chernykh, UMN, 19, issue 2 (116), 216 (1964).