Abstract
Full Text
MATHEMATICS
E. A. BREDIKHINA
ON BEST APPROXIMATIONS OF ALMOST-PERIODIC FUNCTIONS BY ENTIRE FUNCTIONS OF FINITE DEGREE
(Presented by Academician V. I. Smirnov, 20 V 1957)
1. Denote by (B_\lambda) the class of entire functions of degree (\leqslant \lambda), bounded on the real axis. Let the function (f(z)) be defined and bounded on the real axis. Put
[
E_\lambda(f)=\inf_{F(z)\in B_\lambda}\left{\sup_x |f(x)-F(x)|\right}.
]
We shall say that an almost-periodic function (f(x)) belongs to the class (\Pi) if the Fourier series of this function has the form
[
\sum_{k=-\infty}^{\infty} A_k e^{i\lambda_k x}
\quad
(\lambda_0=0;\ \lambda_k>0,\ \lambda_{k+1}/\lambda_k=q_k\geqslant \theta>1 \text{ for } k>0;\ \lambda_k=-\lambda_{-k}).
]
Put
[
R_\lambda(f)=\sup_x\left|f(x)-\sum_{|\lambda_k|\leqslant \lambda} A_k e^{i\lambda_k x}\right|,
\qquad
\alpha_\lambda(f)=\sum_{|\lambda_k|>\lambda}|A_k|.
]
2. Theorem 1. If (f(x)\in \Pi), then (R_\lambda(f)\leqslant C(\theta)E_\lambda(f)), where (C(\theta)) is a constant depending only on (\theta).
The proof of the theorem is based on the following two lemmas.
Lemma 1. If (F(z)\in B_\lambda),
[
\Psi_{a,b}(u)=\frac{2}{\pi(b-a)}
\frac{\sin \frac{b+a}{2}u \, \sin \frac{b-a}{2}u}{u^2},
]
where (\lambda<a<b), then
[
F_{a,b}(x)=\int_{-\infty}^{\infty} F(x+u)\Psi_{a,b}(u)\,du=F(x).
]
Proof. By virtue of the inequality (((1), \text{ p. } 76))
[
\int_{-\infty}^{\infty} |\Psi_{a,b}(u)|\,du
\leqslant
\frac{4}{\pi}+\frac{2}{\pi}\ln\frac{b+a}{b-a}
\tag{1}
]
(F_{a,b}(x)) exists for any function (F(z)\in B_\lambda).
On the basis of the Wiener–Paley theorem (((2), \text{ p. } 151))
[
F(x)=F(0)+\frac{x}{\sqrt{2\pi}}\int_{-\lambda}^{\lambda} e^{itx}\varphi(t)\,dt,
\tag{2}
]
where (\varphi(t)\in L_2(-\lambda,\lambda)); therefore
[
F_{a,b}(x)=F(0)\int_{-\infty}^{\infty}\Psi_{a,b}(u)\,du
+\frac{x}{\sqrt{2\pi}}I_1(x)
+\frac{1}{\sqrt{2\pi}}I_2(x),
]
where
[
I_1(x)=\int_{-\lambda}^{\lambda}\varphi(t)\int_{-\infty}^{\infty} e^{it(x+u)}\Psi_{a,b}(u)\,du\,dt,
]
[
I_2(x)=\int_{-\lambda}^{\lambda}\varphi(t)e^{itx}\int_{-\infty}^{\infty} e^{itu}u\Psi_{a,b}\times(u)\,du\,dt.
]
Taking into account that
[
\int_{-\infty}^{\infty}\Psi_{a,b}(u)\,du=1,\qquad
\int_{-\infty}^{\infty} e^{it(x+u)}\Psi_{a,b}(u)\,du=
]
[
=e^{itx}\quad \text{for } |t|<a\quad ((1),\ \text{pp. }76,77)
]
and
[
\int_{-\infty}^{\infty} e^{itu}u\Psi_{a,b}(u)\,du=0,
]
we obtain
[
F_{a,b}(x)=F(x).
]
Lemma 2. If (F(z)\in B_\lambda,\ \lambda<|\Lambda|), then
[
\lim_{T\to\infty}\frac1T\int_0^T F(x)e^{-i\Lambda x}\,dx=0.
]
Proof follows from (2) and the Riemann—Lebesgue theorem (((3),\ \text{p. }19)).
Proof of Theorem 1 ((6)). If (F(z)\in B_{\lambda_k}) and (\lambda'k=\dfrac{\lambda_k+\lambda2), then, by Lemma 1,}
[
\int_{-\infty}^{\infty}F(x+u)\Psi_{\lambda'k,\lambda(u)\,du=F(x).}
\tag{3}
]
If (f(x)\in L), then (((1),\ \text{p. }76))
[
\int_{-\infty}^{\infty}f(x+u)\Psi_{\lambda'k,\lambda(u)\,du}
=
\sum_{|\lambda_\nu|\le \lambda_k} A_\nu e^{i\lambda_\nu x}.
\tag{4}
]
In the class (B_{\lambda_k}) there exists a function (\widetilde F(z)) such that (((4),\ \text{p. }371))
[
|f(x)-\widetilde F(x)|\le E_{\lambda_k}(f).
\tag{5}
]
From inequalities (5) and (1) it follows that
[
\left|
\int_{-\infty}^{\infty}\widetilde F(x+u)\Psi_{\lambda'k,\lambda(u)\,du}
-
\int_{-\infty}^{\infty}f(x+u)\Psi_{\lambda'k,\lambda(u)\,du}
\right|
\le
]
[
\le
E_{\lambda_k}(f)\left(\frac4\pi+\frac2\pi\ln\frac{3\theta+1}{\theta-1}\right).
\tag{6}
]
From (5) and (6), taking into account (3) and (4), we obtain
[
R_{\lambda'k}(f)\le
\left[
1+\frac2\pi\left(2+\ln\frac{3\theta+1}{\theta-1}\right)
\right]E(f).
\tag{7}
]
Let (\lambda_{k-1}<\lambda<\lambda_k); then
[
R_\lambda(f)\le |A_k|+|A_{-k}|+R_{\lambda_k}(f).
\tag{8}
]
By Lemma 2,
[
A_k=\lim_{T\to\infty}\frac1T\int_0^T [f(x)-F(x)]e^{-i\lambda_k x}\,dx,
]
where (F(x))—the...
an arbitrary function belonging to the class (B_\lambda); hence (|A_k|\leq E_\lambda(f)), (|A_{-k}|\leq E_\lambda(f)), and from inequalities (8) and (7) it follows that for any (\lambda)
[
R_\lambda(f)\leq C(\theta)E_\lambda(f), \quad \text{where}\quad
C(\theta)=3+\frac{2}{\pi}\left(2+\ln \frac{3\theta+1}{\theta-1}\right).
]
Corollary. If (f(x)\in \mathcal L), then the order equalities
(E_\lambda(f)\sim R_\lambda(f)\sim \alpha_\lambda(f)) hold.
Proof. From Theorem 1 of paper ((^8)) there follows the inequality
(\alpha_\lambda(f)\leq C_1(\theta)R_\lambda(f)), where (C_1(\theta)) is a constant depending only on (\theta).
The corollary of Theorem 1 is a generalization of Theorem 3 of paper ((^8)).
Theorem 2. Let the function (f(z)) be defined and bounded on the real axis. If there exists a sequence of nonnegative (nonpositive) numbers
({x_l}) (\left(l=0,1,2,\ldots;\ |x_l|<|x_{l+1}|;\ \lim_{l\to\infty}\frac{l}{|x_l|}>\frac{\lambda}{\pi}\right)) such that for some function (F_0(z)\in B_\lambda)
[
\operatorname{Re}{f(x_l)-F_0(x_l)}=(-1)^l L_l,
\tag{9}
]
where (L_l\geq L>0), then (E_\lambda(f)\geq L).
Proof. Suppose that there exists a function (\Phi(z)\in B_\lambda) for which
[
(-1)^l\Phi(x_l)\geq 1 \quad (l=0,1,2,\ldots).
\tag{10}
]
Let (n(t)) be the number of zeros of (\Phi(z)) in the domain (|\arg z|<\varepsilon,\ |z|<t), if the terms of the sequence ({x_l}) are nonnegative, and let (n(t)) be the number of zeros of (\Phi(z)) in the domain (|\pi-\arg z|<\varepsilon,\ |z|<t), if the terms of the sequence ({x_l}) are nonpositive. It is known ((^5)), Ch. V, § 4, Theorem 11, that for arbitrarily small (\varepsilon)
[
\lim_{t\to\infty}\frac{n(t)}{t}=\frac{d}{2\pi},
]
where (d) is the length of the indicator diagram of the function (\Phi(z)).
On the basis of Pólya’s theorem (((^5),) Ch. I, § 20) (d\leq 2\lambda), therefore
[
\lim_{t\to\infty}\frac{n(t)}{t}\leq \frac{\lambda}{\pi}.
\tag{11}
]
There exists a subsequence ({x_{l_k}}) ((k=0,1,2,\ldots)) of the sequence ({x_l}) such that
[
\lim_{k\to\infty}\frac{l_k}{|x_{l_k}|}>\frac{\lambda}{\pi}.
\tag{12}
]
Taking (10) and (12) into account, we obtain
[
\lim_{k\to\infty}\frac{n(|x_{l_k}|)}{|x_{l_k}|}\geq
\lim_{k\to\infty}\frac{l_k}{|x_{l_k}|}>
\frac{\lambda}{\pi},
]
which contradicts (11). Thus the inequalities (10) are impossible at all points (x_l); consequently, the set of these points is a set of uniqueness of degree (\lambda) (((^4),) p. 376).
Let
[
F_0(z)=\sum_{k=0}^{\infty} c_k\frac{z^k}{k!};
]
put
[
F_1(z)=\sum_{k=0}^{\infty}\frac{c_k+\overline{c_k}}{2}\frac{z^k}{k!};
]
obviously, (F_1(z)\in B_\lambda) and (\operatorname{Re}F_0(x)=F_1(x)). From (9) it follows
[
(-1)^l{\operatorname{Re} f(x_l)-F_1(x_l)}\geq L.
\tag{13}
]
Consequently, by (13) (((^4),) p. 376, Theorem IV) (E_\lambda(f)\geq L).
Theorem 3. If (f(x)\in \mathcal L), (\theta>3), and (\arg A_{-k}=-\arg A_k), then
[
\alpha_\lambda(f)\leq \frac{1}{\cos \dfrac{\pi}{\theta-1}}\,E_\lambda(f).
]
Proof ((7), Theorem 5). Let
[
x_p^{(k)}=\frac{(4+p)\pi-\varphi_k}{\lambda_k},
]
where (k) is a natural number, (p=0,1,2,\ldots), and (\varphi_k=\arg A_k). Considering (p) and (k) fixed, denote by (x_{p+2r_1}^{(k+1)}) the point of the form (x_{p+2r}^{(k+1)}), where (r) is an integer, nearest to (x_p^{(k)}). Obviously,
[
\left|x_p^{(k)}-x_{p+2r_1}^{(k+1)}\right|\leq \pi/\lambda_{k+1}.
]
Denote by (x_{p+2r_2}^{(k+2)}) the point of the form (x_{p+2r}^{(k+2)}) nearest to the point (x_{p+2r_1}^{(k+1)}), and so on.
Thus we obtain a sequence ({x_{p+2r_l}^{(k+l)}}) ((l=0,1,2,\ldots;\ r_0=0)) such that
[
\left|x_{p+2r_l}^{(k+l)}-x_{p+2r_{l+1}}^{(k+l+1)}\right|\leq \pi/\lambda_{k+l+1}.
]
There exists
[
\lim_{l\to\infty} x_{p+2r_l}^{(k+l)}=\widetilde{x}_p^{(k)}
]
and the inequality holds
[
\left|x_{p+2r_l}^{(k+l)}-\widetilde{x}p^{(k)}\right|\leq
\frac{\pi}{\lambda.}}\frac{1}{\theta-1
\tag{14}
]
Putting (l=0) in inequality (14), we obtain the following properties of the numbers (\widetilde{x}_p^{(k)}) ((p=0,1,2,\ldots)):
[
\widetilde{x}0^{(k)}>0,\qquad
\widetilde{x}_p^{(k)}<\widetilde{x},\qquad}^{(k)
\lim_{p\to\infty}\frac{p}{\widetilde{x}_p^{(k)}}=\frac{\lambda_k}{\pi}.
\tag{15}
]
Let (\lambda_{k-1}\leq \lambda<\lambda_k); then
[
\operatorname{Re}\left{f(\widetilde{x}p^{(k)})-
\sum\right}} A_\nu e^{i\lambda_\nu \widetilde{x}_p^{(k)}
=(-1)^p L_p,
\tag{16}
]
where, by virtue of (14),
[
L_p=\sum_{l=0}^{\infty}{|A_{k+l}|+|A_{-(k+l)}|}
\cos \lambda_{k+l}\bigl(\widetilde{x}p^{(k)}-x\bigr)}^{(k+l)
\geq
\cos\frac{\pi}{\theta-1}\,\alpha_\lambda(f).
]
By Theorem 2, from (15) and (16) it follows that
[
E_\lambda(f)\geq \cos\frac{\pi}{\theta-1}\,\alpha_\lambda(f).
]
Corollary. If (f(x)\in L) and (q_k\to\infty), then the asymptotic equalities hold
[
E_\lambda(f)\simeq R_\lambda(f)\simeq \alpha_\lambda(f).
]
The corollary of Theorem 3 is a generalization of Theorem 4 of paper (8).
I express my deep gratitude to Prof. I. P. Natanson for supervising the work.
Kuibyshev
Aviation Institute
Received
15 V 1957
References
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