MATHEMATICS

R. G. MAMEDOV

LOCAL SATURATION OF A FAMILY OF LINEAR POSITIVE OPERATORS

(Presented by Academician V. I. Smirnov, 15 I 1964)

I. Let \(\mathcal{L}_\lambda(f;x)\) be a family of linear positive operators (l.p.o.), whose domain of definition \(H\), in particular, contains also the set of all functions \(f(x)\) having on \([a,b]\) an absolutely continuous \((2m-1)\)-st derivative \(f^{(2m-1)}(x)\). Define the linear operator

\[ M_\lambda(f;x)=\frac{1}{\tau_\lambda^{[2m]}} \left[ \mathcal{L}_\lambda(f;x) - \sum_{k=0}^{2m-1}\frac{f^{(k)}(x)}{k!}\tau_\lambda^{[k]} \right], \]

where \(\tau_\lambda^{[k]}=\mathcal{L}_\lambda[(t-x)^k;x]\) and \(m\geq 1\) is a given integer.

Denote by \(H(n,x)\) the set of all functions \(f(x)\), defined on \([a,b]\) and having a finite derivative of order \(n\), \(f^{(n)}(x)\), at the point \(x\in [a,b]\).

Let the operator \(M_\lambda(f;x)\) map every function \(f(x)\in H\) into a function continuous on \([a,b]\). Suppose, moreover, that the conditions

\[ |M_\lambda(f;x)|\leq M<+\infty, \tag{1} \]

\[ \lim_{\lambda\to\infty} M_\lambda(f;x)=\frac{f^{(2m)}(x)}{(2m)!} \tag{2} \]

are satisfied for every function \(f\in H(2m,x)\) on the set of all points \(x\) of the interval \([a,b]\) where the finite \(f^{(2m)}(x)\) exists. It is known \((^{3,4})\) that, in order that the asymptotic equality (2) hold for all \(f\in H(2m,x)\), it is necessary and sufficient that the condition

\[ \lim_{\lambda\to\infty}\frac{\tau_\lambda^{[2m+2j]}}{\tau_\lambda^{[2m]}}=0 \]

hold for at least one value \(j=1,2,\ldots\).

In what follows we also assume that for the l.p.o. the condition

\[ \lim_{\lambda\to\infty}\int_a^b \left[M_\lambda(f;x)\varphi(x)-M_\lambda(\varphi;x)f(x)\right]\,dx=0 \tag{3} \]

is satisfied for all \(f(x)\in H\) and \(\varphi(x)\in H\).

Theorem 1. If \(M_\lambda(f;x)=o(1)\) as \(\lambda\to\infty\) uniformly on \([a,b]\), then \(f(x)\) is an algebraic polynomial of degree \(\leq 2m-1\).

Proof. Denote by \(C^{(2m)}(a,b)\) the totality of all functions \(f(t)\), defined on the interval \([a,b]\), having on this interval a continuous derivative of order \(2m\) and equal to zero outside \([a,b]\). Since \(\lim_{\lambda\to\infty} M_\lambda(f;x)=0\) uniformly on \([a,b]\), we have

\[ \lim_{\lambda\to\infty}\int_a^b M_\lambda(f;x)\varphi(x)\,dx=0 \tag{4} \]

for every function \(\varphi\in C^{(2m)}(a,b)\).

We note that, by virtue of (3), the relation

\[ \lim_{\lambda\to\infty}\int_a^b M_\lambda(f;x)\varphi(x)\,dx = \lim_{\lambda\to\infty}\int_a^b M_\lambda(\varphi;x)f(x)\,dx . \tag{5} \]

holds.

On the other hand, the equality

\[ \lim_{\lambda\to\infty} M_\lambda(\varphi;x) = \frac{\varphi^{(2m)}(x)}{(2m)!} \]

is valid on \([a,b]\) for \(\varphi\in C^{(2m)}(a,b)\). Consequently, from (1), (4), and (5) we find

\[ \int_a^b \varphi^{(2m)}(x) f(x)\,dx = 0 \]

for every function \(\varphi\in C^{(2m)}(a,b)\). Hence, on the basis of the fundamental lemma of the calculus of variations, it follows that \(f(x)\) is an algebraic polynomial of degree \(2m-1\).

Theorem 2. In order that, as \(\lambda\to\infty\), the relation

\[ M_\lambda(f;x)=O(1) \tag{6} \]

hold uniformly on \([a,b]\), it is necessary and sufficient that \(f^{(2m-1)}(x)\in \operatorname{Lip} 1\).

Necessity. Since (6) holds uniformly on \([a,b]\), by the weak compactness of the functionals of the space \(L_1(a,b)\), there exist a subsequence \(\{\lambda_k\}\) and a function \(\psi(x)\in L_\infty(a,b)\) such that

\[ \lim_{k\to\infty}\int_a^b M_{\lambda_k}(f;x)\varphi(x)\,dx = \int_a^b \psi(x)\varphi(x)\,dx \tag{7} \]

for every \(\varphi\in C^{(2m)}(a,b)\). The left-hand side of relation (7), by virtue of (1), (2), and (3), is equal to

\[ \lim_{k\to\infty}\int_a^b M_{\lambda_k}(f;x)\varphi(x)\,dx = \lim_{k\to\infty}\int_a^b M_{\lambda_k}(\varphi;x)f(x)\,dx = \frac{1}{(2m)!}\int_a^b \varphi^{(2m)}(x)f(x)\,dx . \tag{8} \]

The right-hand side of equality (7), in turn, can be represented in the form

\[ \int_a^b \psi(x)\varphi(x)\,dx = \int_a^b \varphi^{(2m)}(x)F_{2m}(x)\,dx , \tag{9} \]

where \(F_{2m}(x)\) is the \(2m\)-th indefinite integral of \(\psi(x)\).

Consequently, from (7), (8), and (9) we find

\[ \int_a^b \varphi^{(2m)}(x)\,[f(x)-(2m)!F_{2m}(x)]\,dx = 0 \]

for every \(\varphi\in C^{(2m)}(a,b)\). Hence we conclude that \(f(x)-(2m)!F_{2m}(x)\) is an algebraic polynomial of degree \(2m-1\), i.e. \(f^{(2m-1)}(x)\in \operatorname{Lip} 1\).

Sufficiency. If \(f^{(2m-1)}(x)\in \operatorname{Lip} 1\), then \(f^{(2m)}(x)\) exists almost everywhere and is bounded on \([a,b]\). Hence, by virtue of (1),

\[ \|M_\lambda(f;x)\|_{L_\infty}=O(1) \]

as \(\lambda\to\infty\). Owing to the continuity of \(M_\lambda(f;x)\), from the last relation we find that (6) holds uniformly on \([a,b]\).

Theorems 1 and 2 determine the orders and classes of local saturation (see, for this, (5)) of the family \(\mathcal{L}_\lambda(f;x)\).

II. By analogous reasoning one determines the classes and orders of local saturation of the family of l.p.o. \(W_\lambda(f;x)\), defined on the set of \(2\pi\)-periodic functions. We shall show this for \(m=1\); the case \(m>1\) is treated similarly. We assume that for the family \(W_\lambda(f;x)\) the conditions stated at the beginning are also satisfied, in particular conditions (1), (2), and (3), in which \(M_\lambda(f;x)\) must be replaced by the operator

\[ N_\lambda(f;x)=\frac{1}{\mu_\lambda^{[2]}}\,[W_\lambda(f;x)-f(x)-f'(x)\mu_\lambda^{[1]}], \]

where

\[ \mu_\lambda^{[k]}=W_\lambda\left[2^k\sin^k\frac{t-x}{2};x\right]. \]

Theorem 3. 1. If, as \(\lambda\to\infty\),

\[ N_\lambda(f;x)=o(1) \tag{10} \]

uniformly, then \(f(x)\) is a constant.

1. In order that, as \(\lambda\to\infty\), the relation

\[ N_\lambda(f;x)=O(1) \tag{11} \]

hold uniformly, it is necessary and sufficient that \(f'(x)\in \operatorname{Lip}1\).

Proof. 1. Since (10) holds uniformly on \([-\pi,\pi]\), we have

\[ \lim_{\lambda\to\infty}\int_{-\pi}^{\pi} N_\lambda(f;x)e^{-ikx}\,dx=0. \]

Using condition (3) and the asymptotic equality (2), we find

\[ \lim_{\lambda\to\infty}\int_{-\pi}^{\pi} N_\lambda(f;x)e^{-ikx}\,dx = \lim_{\lambda\to\infty}\int_{-\pi}^{\pi} N_\lambda(e^{-ikt};x)f(x)\,dx = -\pi k^2 C_k(f)=0 \]

for all \(k\), where \(C_k(f)\) is the \(k\)-th Fourier coefficient of \(f(x)\). Consequently, \(C_k(f)=0\) for \(k=\pm1,\pm2,\ldots\), i.e. \(f(x)=\mathrm{const}\).

1. Let us prove necessity. Taking into account that (11) holds uniformly on \([-\pi,\pi]\), by the weak compactness of the space \(L_1(-\pi,\pi)\) one can find a function \(\psi(x)\in L_\infty(-\pi,\pi)\) and a subsequence \(\lambda_k\) such that the relation

\[ \lim_{k\to\infty}\int_{-\pi}^{\pi} N_{\lambda_k}(f;x)e^{-ikx}\,dx = \int_{-\pi}^{\pi}\psi(x)e^{-ikx}\,dx \]

is valid.

The left-hand side is equal to \(-\pi k^2 C_k(f)\). Consequently,

\[ -\pi k^2 C_k(f)=\int_{-\pi}^{\pi}\psi(x)e^{-ikx}\,dx \]

for all \(k\). Hence it follows that \(f''(x)\in L_\infty(-\pi,\pi)\), i.e. \(f'(x)\in \operatorname{Lip}1\).

The proof of sufficiency is obvious.

As an example, consider the l.p.o. of P. P. Korovkin \((^1)\)

\[ A_n(f;x)=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x+t)V_n(t)\,dt, \tag{12} \]

where

\[ V_n(t)=\frac{1}{2}\sum_{k=1}^{n}\rho_k^{(n)}\cos kt,\qquad \rho_1^{(n)}=\cos\frac{\pi}{n+2},\ldots \]

For the l.p.o. (12) all the conditions of Theorem 3 are satisfied, and moreover

\[ \mu_n^{[1]}=0,\qquad \mu_n^{[2]}=2[1-\rho_1^{(n)}] = 2\left(1-\cos\frac{\pi}{n+2}\right)\simeq \frac{\pi^2}{n^2}\quad (n\to\infty). \]

Thus, from Theorem 3 we obtain

Corollary. 1. If, as \(n\to\infty\), \(A_n(f;x)-f(x)=o(n^{-2})\) uniformly, then \(f(x)=\mathrm{const}\).

1. In order that, as \(n\to\infty\), the relation
   \(A_n(f;x)-f(x)=O(n^{-2})\) hold uniformly, it is necessary and sufficient that \(f'(x)\in \mathrm{Lip}\,1\).

The last result is customarily written as follows:

\[ \operatorname{Sat}[A_n]_C=\bigl[\{f\mid f'\in \mathrm{Lip}\,1\},\ n^{-2},\ \text{constant}\bigr]. \]

For another l.p.o. of P. P. Korovkin \({}^{(2)}\),

\[ A_r(f;x)=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x+t)u_r(t)\,dt, \]

\[ u_r(t)=\frac{1}{2}+\sum_{k=1}^{\infty} r^{k^2}\cos kt, \]

we have

\[ \operatorname{Sat}[A_n]_C=\bigl[\{f\mid f'\in \mathrm{Lip}\,1\},\ 1-r,\ \text{constant}\bigr]. \]

On the basis of Theorem 3 one can also verify that the class and order of saturation of the well-known l.p.o. of Vallée-Poussin and Jackson (see \({}^{(6)}\)) are determined in the form

\[ \operatorname{Sat}[B_n]_C=\bigl[\{f\mid f'\in \mathrm{Lip}\,1\},\ n^{-1},\ \text{constant}\bigr], \]

\[ \operatorname{Sat}[D_n]_C=\bigl[\{f\mid f'\in \mathrm{Lip}\,1\},\ n^{-2},\ \text{constant}\bigr], \]

respectively. These results were obtained earlier by A. Kh. Turetskii \({}^{(7)}\).

We note that assertions analogous to Theorems 1–3 are also valid in the metric of the space \(L_p\) \((p>1)\). Moreover, if approximation of functions by means of l.p.o. on the unbounded interval \((-\infty,\infty)\) is considered, then in proving the corresponding assertions one should use the Fourier transform of functions instead of Fourier coefficients.

Azerbaijan Polytechnic
Institute

Received
13 I 1964

CITED LITERATURE

\({}^{1}\) P. P. Korovkin, UMN, 13, no. 6 (84), 99 (1958).
\({}^{2}\) P. P. Korovkin, DAN, 127, no. 3, 513 (1959).
\({}^{3}\) R. G. Mamedov, DAN, 146, no. 5, 1013 (1962).
\({}^{4}\) R. G. Mamedov, Izv. AN AzerbSSR, Ser. Phys., Math. and Techn., 4, 3 (1962).
\({}^{5}\) R. G. Mamedov, DAN, 144, no. 2, 272 (1962).
\({}^{6}\) I. P. Natanson, Constructive Theory of Functions, Moscow–Leningrad, 1949.
\({}^{7}\) A. Kh. Turetskii, DAN, 126, no. 6, 1207 (1959).