Reports of the Academy of Sciences of the USSR
1962. Volume 145, No. 1

MATHEMATICS

E. A. BREDIKHINA

ON THE SIMULTANEOUS APPROXIMATION OF ALMOST-PERIODIC FUNCTIONS AND THEIR DERIVATIVES

(Presented by Academician V. I. Smirnov on 16 II 1962)

1. Let \(W^{(r)}\) be the class of all functions bounded on \((-\infty,\infty)\) that have a derivative of order \(r\) bounded on the entire real axis; let \(P^{(r)}\) and \(P^{(r)}_{2\pi}\) be, respectively, the classes of all periodic and all \(2\pi\)-periodic functions from \(W^{(r)}\). We shall say that a uniformly almost-periodic function \(f(x)\) belongs to the class \(A_s\) if \(N_f(x,x+1)=O(1)\), where \(N_f(x,x+1)\) is the number of Fourier exponents of the function \(f(x)\) on the interval \((x,x+1)\). (The structural characteristic of the class \(A_s\) is given in \((^5)\).) Denote by \(W_s^{(r)}\) the intersection of the classes \(W^{(r)}\) and \(A_s\). Obviously, \(W_s^{(r)} \supset P_{2\pi}^{(r)}\). Set:

\[ C_{\sigma,r}(f)=\operatorname{Inf}_{g_\sigma(x)}\max_{0\le k\le r} \frac{\operatorname{Sup}_x |f^{(k)}(x)-g_\sigma^{(k)}(x)|}{E_\sigma(f^{(k)})}, \]

where \(g_\sigma(x)\) is an entire function of degree \(\le \sigma\) and

\[ E_\sigma(f)=\operatorname{Inf}_{g_\sigma(x)}\operatorname{Sup}_x |f(x)-g_\sigma(x)|; \]

\[ C_{n,r}^*(f)=\operatorname{Inf}_{T_n(x)}\max_{0\le k\le r} \frac{\operatorname{Sup}_x |f^{(k)}(x)-T_n^{(k)}(x)|}{E_n^*(f^{(k)})}, \]

where

\[ T_n(x)=\sum_{\nu=0}^{n} a_\nu\cos \nu x+b_\nu\sin \nu x \]

and

\[ E_n^*(f)=\operatorname{Inf}_{T_n(x)}\operatorname{Sup}_x |f(x)-T_n(x)|. \]

Let

\[ C_{\sigma,r}(W^{(r)})=\operatorname{Sup}_{f\in W^{(r)}} C_{\sigma,r}(f); \]

analogously the quantities \(C_{\sigma,r}(W_s^{(r)})\), \(C_{\sigma,r}(P^{(r)})\), \(C_{n,r}^*(P_{2\pi}^{(r)})\) are defined.

A. F. Timan \((^1)\) showed that, as \(r\to\infty\), uniformly with respect to all \(\sigma>0\), the following asymptotic equality holds:

\[ C_{\sigma,r}(W^{(r)})=\frac{4}{\pi^2}\ln(r+1)+O(\ln\ln\ln r). \tag{1} \]

A. L. Garkavi \((^2)\) obtained the asymptotic formula

\[ C_{n,r}^*(P_{2\pi}^{(r)})=\frac{4}{\pi^2}\ln(p+1)+O(\ln\ln\ln p), \tag{2} \]

where \(p=\min\{n,r\}\).

It follows from the theorems given below that the result of A. L. Garkavi can be extended to the class \(W_s^{(r)}\) of almost-periodic functions, while estimate (1) for the quantity \(C_{\sigma,r}\) for the class of all uniformly almost-periodic functions belonging to \(W^{(r)}\) cannot be improved, since it cannot be improved even for the class \(P^{(r)}\) contained in it.

1. We formulate the main results of the note.

Theorem 1. Whatever the positive real \(\sigma>0\) and the natural number \(r\), as \(p\to\infty\) the asymptotic equality

\[ C_{\sigma,r}\left(W_s^{(r)}\right)=\frac{4}{\pi^2}\ln(p+1)+O(\ln\ln\ln p),\qquad \text{where } p=\min\{\sigma,r\}. \tag{3} \]

holds.

Theorem 2. As \(r\to\infty\), uniformly with respect to all \(\sigma>0\), the asymptotic equality

\[ C_{\sigma,r}\left(P^{(r)}\right)=\frac{4}{\pi^2}\ln(r+1)+O(\ln\ln\ln r). \tag{4} \]

holds.

1. Let us recall some known facts and give three lemmas on which the proof of Theorem 1 is based.

The integral operator of N. I. Akhiezer—B. M. Levitan \({}^{(3)}\)

\[ f_{\sigma,q}(x)=\int_{-\infty}^{\infty} f(x+u)\Psi_{\sigma,\sigma(1+1/q)}(u)\,du, \tag{5} \]

where

\[ \Psi_{\sigma,\sigma(1+1/q)}(u)=\frac{q}{\pi\sigma}\, \frac{\cos\sigma u-\cos\sigma(1+1/q)u}{u^2} \]

(\(q>0\)), assigns to every continuous and bounded function \(f(x)\) on \((-\infty,\infty)\) an entire function \(f_{\sigma,q}(x)\) of degree \(\leqslant \sigma(1+1/q)\), and moreover \(f_{\sigma,q}(x)=f(x)\) if \(f(x)\) is a bounded entire function on \((-\infty,\infty)\) of degree \(\leqslant \sigma\).

Let \(L(q)\) be the norm of the operator (5) in the space of all bounded functions on the real axis. A. F. Timan \({}^{(4)}\) established that, uniformly with respect to all \(q>0\), the asymptotic equality

\[ L(q)=\frac{4}{\pi^2}\ln(q+1)+O(1) \tag{6} \]

holds.

Let the interval \(I_{\sigma,q}=(\sigma,\sigma(1+1/q))\) contain \(n\) (\(n\geqslant0\)) points \(c_i\) \((c_i<c_{i+1},\ i=1,2,\ldots,n)\); let \(\varepsilon>0\) be chosen so that the intervals \((c_i-\varepsilon,c_i+\varepsilon)\) do not intersect and belong to the interval \(I_{\sigma,q}\). Consider the function, continuous and linear on the intervals \((c_i-\varepsilon,c_i)\), \((c_i,c_i+\varepsilon)\) \((i=1,2,\ldots,n)\),

\[ \varphi_{\sigma,q}(\lambda)=\varphi_{\sigma,q}(\lambda,c_1,c_2,\ldots,c_n,\varepsilon)= \]

\[ = \begin{cases} 1, & |\lambda|\leqslant\sigma,\\[4pt] q+1-\dfrac{q}{\sigma}|\lambda|, & \sigma<|\lambda|<\sigma\left(1+\dfrac{1}{q}\right),\quad |\lambda|\notin(c_i-\varepsilon,c_i+\varepsilon),\\[8pt] 0, & |\lambda|=c_i,\\[4pt] 0, & |\lambda|\geqslant\sigma\left(1+\dfrac{1}{q}\right). \end{cases} \]

Set

\[ \widetilde{\Psi}_{\sigma,\sigma(1+1/q)}(u) =\frac{1}{2\pi}\int_{-\infty}^{\infty} \varphi_{\sigma,q}(\lambda)e^{-iu\lambda}\,d\lambda, \]

\[ \widetilde{f}_{\sigma,q}(x) =\int_{-\infty}^{\infty} f(x+u)\widetilde{\Psi}_{\sigma,\sigma(1+1/q)}(u)\,du; \tag{5'} \]

it is easy to see that
\(\widetilde{\Psi}_{\sigma,\sigma(1+1/q)}(u)=\Psi_{\sigma,\sigma(1+1/q)}(u)\),
\(\widetilde{f}_{\sigma,q}(x)=f_{\sigma,q}(x)\), if the interval \(I_{\sigma,q}\) is free of the points \(c_i\).

Let \(\widetilde{L}(q)\) be the norm of the operator (5′), and \(\widetilde{\widetilde{L}}(q)\) the norm of the operator

\[ \int_{-\infty}^{\infty} f(x+u)\left[\widetilde{\Psi}_{\sigma,\sigma(1+1/q)}(u)-\Psi_{\sigma(1-1/q),\sigma}(u)\right]\,du \]

in the space of all functions bounded on the real axis.

Lemma 1. If \(f(x)\) is continuous and bounded on \((-\infty,\infty)\), then \(\tilde f_{\sigma,q}(x)\) is an entire function of degree \(\leq \sigma(1+1/q)\). If \(f(x)\) is an entire function of degree \(\leq \sigma\) bounded on \((-\infty,\infty)\), then \(\tilde f_{\sigma,q}(x)=f(x)\).

Lemma 2. The inequalities hold

\[ \tilde L(q)\leq \bar L(q)+2n, \tag{7} \]

\[ \tilde L(q)\leq 2(n+1). \tag{8} \]

Lemma 3. If

\[ f(x)\sim \sum_{k=-\infty}^{\infty} A_k e^{i\lambda_k x} \quad (\lambda_0=0,\ \lambda_{-k}=-\lambda_k,\ \lambda_k<\lambda_{k+1} \ \text{for } k\geq 0,\ \lim_{k\to\infty}\lambda_k=\infty) \]

is a uniformly almost-periodic function, then

\[ \tilde f_{\sigma,q}(x)= \sum_{|\lambda_k|<\sigma(1+1/q)} \varphi_{\sigma,q}(\lambda_k) A_k e^{i\lambda_k x}. \]

1. We shall give the proof of Theorem 1. Let \(I_\sigma\) be the common part of the intervals \(I_{\sigma,r}\) and \((\sigma,\sigma+1)\); let \(c_1,c_2,\ldots,c_n\) \((0\leq n\leq N_f(\sigma,\sigma+1))\) be the Fourier exponents of the function \(f(x)\) belonging to the interval \(I_\sigma\). We shall show that for every function \(f(x)\in W_s^{(r)}\)

\[ \max_{0\leq k\leq r} \frac{\operatorname{Sup}_{x}\left|f^{(k)}(x)-G_\sigma^{(k)}(\tilde f_{\sigma,r},x)\right|} {E_\sigma(f^{(k)})} \leq \frac{4}{\pi^2}\ln(p+1)+O(1), \tag{9} \]

where \(p=\min\{\sigma,r\}\); \(G_\sigma(\tilde f_{\sigma,r},x)\) is an entire function of degree \(\leq \sigma\), realizing the best approximation to the function \(\tilde f_{\sigma,r}(x)\).

The equalities

\[ \tilde f_{\sigma,r}^{(k)}(x)= \int_{-\infty}^{\infty} f^{(k)}(x+u)\,\tilde\Psi_{\sigma,\sigma(1+1/r)}(u)\,du \quad (k=0,1,\ldots,r) \tag{10} \]

are valid.

Denote by \(g_\sigma(f^{(k)},x)\) an entire function of degree \(\leq \sigma\), realizing the best approximation to the function \(f^{(k)}(x)\). By Lemma 1,

\[ g_\sigma(f^{(k)},x)= \int_{-\infty}^{\infty} g_\sigma(f^{(k)},x+u)\, \tilde\Psi_{\sigma,\sigma(1+1/q)}(u)\,du \quad (k=0,1,\ldots,r) \tag{11} \]

for any \(q>0\).

Let \(\sigma\leq r\). In consequence of (10) and Lemma 3,

\[ \tilde f_{\sigma,r}^{(k)}(x)= \int_{-\infty}^{\infty} f^{(k)}(x+u)\,\tilde\Psi_{\sigma,\sigma+1}(u)\,du = S_\sigma^{(k)}(f,x), \tag{12} \]

where

\[ S_\sigma(f,x)=\sum_{|\lambda_k|\leq \sigma} A_k e^{i\lambda_k x}, \]

and therefore

\[ G_\sigma^{(k)}(\tilde f_{\sigma,r},x)=\tilde f_{\sigma,r}^{(k)}(x) \quad (k=0,1,\ldots,r). \tag{13} \]

Putting \(q=\sigma\) in equality (11), we obtain from (11), (12), and (13)

\[ \operatorname{Sup}\left|f^{(k)}(x)-G_\sigma^{(k)}(\tilde f_{\sigma,r},x)\right| \leq \{\tilde L(\sigma)+1\}E_\sigma(f^{(k)}) \quad (k=0,1,\ldots,r). \]

From the last inequality, by virtue of (7), it follows that for \(\sigma \leqslant r\)

\[ \max_{0\leqslant k\leqslant r} \frac{\operatorname{Sup}_{x}\left|f^{(k)}(x)-G_{\sigma}^{(k)}(\tilde f_{\sigma,r},x)\right|} {E_{\sigma}(f^{(k)})} \leqslant L(\sigma)+2N_f(\sigma,\sigma+1)+1. \tag{14} \]

Let \(\sigma>r\). It is easy to see that

\[ \int_{-\infty}^{\infty} [f(x+u)-g_{\sigma}(f,x+u)] [\tilde\Psi_{\sigma,\sigma(1+1/r)}(u)-\Psi_{\sigma(1-1/r),\sigma}(u)]\,du = \]

\[ =\tilde f_{\sigma,r}(x)-\Phi_{\sigma}(x), \]

where \(\Phi_{\sigma}(x)\) is some entire function of degree \(\leqslant \sigma\); therefore, from inequality (8) one obtains the estimate

\[ E_{\sigma}(\tilde f_{\sigma,r})\leqslant 2[1+N_f(\sigma,\sigma+1)]E_{\sigma}(f). \tag{15} \]

For any function \(f(x)\in W^{(r)}\) (see \((^1,^3)\)),

\[ E_{\sigma}(f)\leqslant \frac{\pi}{2\sigma^{k}}E_{\sigma}(f^{(k)}) \qquad (k=0,1,\ldots,r). \tag{16} \]

From S. N. Bernstein’s inequality \((^3)\) and inequalities (15) and (16) it follows that

\[ \operatorname{Sup}_{x}\left|\tilde f_{\sigma,r}^{(k)}(x)- G_{\sigma}^{(k)}(\tilde f_{\sigma,r},x)\right|\leqslant \]

\[ \leqslant \pi\left(1+\frac{1}{r}\right)^k [1+N_f(\sigma,\sigma+1)]E_{\sigma}(f^{(k)}) \qquad (k=0,1,\ldots,r). \]

Taking \(q=r\) in equality (11), we obtain from (10), (11), and (7)

\[ \operatorname{Sup}_{x'}\left|\tilde f^{(k)}(x)- \tilde f_{\sigma,r}^{(k)}(x)\right| \leqslant [L(r)+2N_f(\sigma,\sigma+1)+1]E_{\sigma}(f^{(k)}) \]

\[ (k=0,1,\ldots,r). \]

From the last two inequalities it follows that, for \(\sigma>r\),

\[ \max_{0\leqslant k\leqslant r} \frac{\operatorname{Sup}_{x}\left|f^{(k)}(x)-G_{\sigma}^{(k)}(\tilde f_{\sigma,r},x)\right|} {E_{\sigma}(f^{(k)})} \leqslant L(r)+(2+\pi e)N_f(\sigma,\sigma+1)+\pi e+1. \tag{17} \]

Estimate (9) is a consequence of inequalities (14), (17), and the asymptotic equality (6). From (9) we obtain the inequality

\[ C_{\sigma,r}(W_s^{(r)})\leqslant \frac{4}{\pi^2}\ln(p+1)+O(1), \qquad \text{where } p=\min\{\sigma,r\}, \]

which, by virtue of (2), leads to the asymptotic equality (3).

5. We do not present here the proof of Theorem 2, based on the effective construction of a function \(g_{\sigma}(x)\in P^{(r)}\) for which

\[ C_{\sigma,r}(g_{\sigma})\geqslant \frac{4}{\pi^2}\ln(r+1)+O(\ln\ln\ln r). \]

Kuibyshev Aviation
Institute

Received
9 II 1962

CITED LITERATURE

1. A. F. Timan, Izv. AN SSSR, Ser. Mat., 24, No. 3 (1960).
2. A. L. Garkavi, Izv. AN SSSR, Ser. Mat., 24, No. 1 (1960).
3. N. I. Akhiezer, Lectures on Approximation Theory, Moscow–Leningrad, 1947.
4. A. F. Timan, DAN, 64, No. 2 (1949).
5. E. A. Bredikhina, Matem. sbornik, 50 (92), 3 (1960).