Abstract
Full Text
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)
- 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). Set:}^{(r)
[
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
\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(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.
- 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.
- 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_i0) 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
\varphi_{\sigma,q}(\lambda)e^{-iu\lambda}\,d\lambda,
]
[
\widetilde{f}{\sigma,q}(x)
=\int}^{\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(u)),
(\widetilde{f}{\sigma,q}(x)=f) is free of the points (c_i).}(x)), if the interval (I_{\sigma,q
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(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}.
]
- 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}^{(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(x)-}^{(k)
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(x)\right|}^{(k)
\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}^{(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
- A. F. Timan, Izv. AN SSSR, Ser. Mat., 24, No. 3 (1960).
- A. L. Garkavi, Izv. AN SSSR, Ser. Mat., 24, No. 1 (1960).
- N. I. Akhiezer, Lectures on Approximation Theory, Moscow–Leningrad, 1947.
- A. F. Timan, DAN, 64, No. 2 (1949).
- E. A. Bredikhina, Matem. sbornik, 50 (92), 3 (1960).