MATHEMATICS

E. A. BREDIKHINA

SOME QUESTIONS ON THE APPROXIMATION OF ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRUM

(Presented by Academician V. I. Smirnov on 4 XII 1959)

1. We denote by \(S\) the class of almost-periodic functions \(f(x)\) (here and below uniformly almost-periodic functions are meant) and introduce the corresponding Fourier series:
   \[ f(x)\sim \sum_{k=-\infty}^{\infty} A_k e^{i\Lambda_k x} \]
   \[ (\Lambda_0=0;\quad \Lambda_{k+1}<\Lambda_k\ \text{for } k>0;\quad \lim_{k\to\infty}\Lambda_k=0;\quad \Lambda_{-k}=-\Lambda_k;\quad A_0=0;\quad (1) \]
   \[ |A_k|+|A_{-k}|>0\quad \text{for } k\ne 0). \]

We denote by \(L=L(f)\) the sequence \(\{\Lambda_k\}\) \((k=1,2,\ldots)\).

We shall say that an almost-periodic function \(f(x)\) belongs to the class \(S_n\) \((n=1,2,\ldots)\), if there exist functions \(f_0(x), f_1(x),\ldots,f_n(x)\) possessing the following properties: \(f_0(x)=f(x)\), \(f'_{m+1}(x)=f_m(x)\) \((m=0,1,\ldots,n-1)\), \(f_m(x)\in S\) \((m=0,1,\ldots,n)\). The inclusions
\[ S\subset S_1\subset S_2\subset\cdots \]
are obvious.

Put
\[ R_\varepsilon(f)=\operatorname{Sup}_{x}\left|f(x)-\sum_{|\Lambda_k|>\varepsilon} A_k e^{i\Lambda_k x}\right|, \]
\[ e_\varepsilon(f)=\operatorname{Inf}_{c_k}\left\{\operatorname{Sup}_{x}\left|f(x)-\sum_{|\Lambda_k|>\varepsilon} c_k e^{i\Lambda_k x}\right|\right\},\qquad E_\varepsilon(f)=\operatorname{Inf}_{F(x)\in Q_\varepsilon}\left\{\operatorname{Sup}_{x}|f(x)-F(x)|\right\}, \]
where \(Q_\varepsilon\) is the class of almost-periodic functions whose Fourier exponents \(\{\lambda_k\}\) satisfy the condition \(|\lambda_k|>\varepsilon\).

Let
\[ \Omega_f(N)= \begin{cases} \displaystyle \operatorname{Sup}_{|T|\ge N}\left\{\operatorname{Sup}_{x}\left|\frac1T\int_0^T f(x+t)\,dt\right|\right\}, & N>0,\\[1.2em] \displaystyle \operatorname{Sup}|f(x)|, & N=0; \end{cases} \]
\(\Omega_f(N)\) is a continuous, bounded, nonincreasing function. For every function \(f(x)\in S\),
\[ \lim_{N\to\infty}\Omega_f(N)=0. \]

1. Theorem 1. If \(f(x)\in S\), then
   \[ e_\varepsilon(f)\le C_0\Omega_f\left(\frac1\varepsilon\right), \tag{2} \]
   where \(C_0\) is an absolute constant.

Proof. Put
\[ \varphi_\varepsilon(t)= \begin{cases} 1, & 0\le |t|\le \varepsilon,\\[0.4em] \displaystyle 1-6\left(1-\frac{|t|}{\varepsilon}\right)^2-6\left(1-\frac{|t|}{\varepsilon}\right)^3, & \varepsilon<|t|<\dfrac{3\varepsilon}{2},\\[0.8em] \displaystyle 2\left(2-\frac{|t|}{\varepsilon}\right)^3, & \dfrac{3\varepsilon}{2}<|t|\le 2\varepsilon,\\[0.8em] 0, & |t|>2\varepsilon; \end{cases} \]

then

\[ \Psi_\varepsilon(u)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\varphi_\varepsilon(t)e^{-iut}\,dt = \frac{12}{\pi}\, \frac{ 8\sin^3\frac{\varepsilon u}{4}\sin\frac{5\varepsilon u}{4} +\sin\varepsilon u\left(2\sin\frac{\varepsilon u}{2}-\varepsilon u\right) }{ \varepsilon^3u^4 }, \]

\[ \rho_\varepsilon(f,x)=\int_{-\infty}^{\infty} f(x+u)\Psi_\varepsilon(u)\,du \sim \sum_{|\Lambda_k|\leq\varepsilon} A_k e^{i\Lambda_k x} + \sum_{\varepsilon<|\Lambda_k|\leq 2\varepsilon} \varphi_\varepsilon(\Lambda_k)A_k e^{i\Lambda_k x}. \]

It is easy to see that

\[ e_\varepsilon(f)\leq \operatorname{Sup}_{x}|\rho_\varepsilon(f,x)|. \tag{3} \]

Integrating by parts, we obtain

\[ \rho_\varepsilon(f,x)=-\int_{-\infty}^{\infty} F(u,x)\Psi'_\varepsilon(u)\,du, \qquad \text{where } F(u,x)=\int_0^u f(x+t)\,dt. \]

For any real \(c\) the inequality holds

\[ \operatorname{Sup}_{x}|F(cu,x)|\leq (2+|c|)|u|\,\Omega_f(|u|), \]

therefore

\[ \rho_\varepsilon(f,x)\leq C_0\Omega_f\left(\frac{1}{\varepsilon}\right), \tag{4} \]

where

\[ C_0=\frac{12}{\pi}\int_{-\infty}^{\infty}(2+|v|) \left| \left[ \frac{ 8\sin^3\frac{v}{4}\sin\frac{5v}{4} +\sin v\left(2\sin\frac{v}{2}-v\right) }{ v^4 } \right]' \right|\,dv. \]

From (3) and (4) follows the estimate (2) to be proved.

Corollary. If there exist constants \(A\) and \(\alpha\) \((0<\alpha\leq 1)\) such that

\[ \left|\int_0^u f(x+t)\,dt\right|<A|u|^{1-\alpha}, \tag{5} \]

then

\[ e_\varepsilon(f)\leq \operatorname{const}\cdot \varepsilon^\alpha. \tag{6} \]

Proof. In view of (5), \(\Omega_f\left(\frac{1}{\varepsilon}\right)\leq A\varepsilon^\alpha\). Note that the estimate (6) follows from the results of the paper \((^1)\) (see also \((^2)\)).

Theorem 2. If \(f(x)\in S_n\), then

\[ e_\varepsilon(f)\leq C_n\varepsilon^n\Omega_{f_n}\left(\frac{1}{\varepsilon}\right), \tag{7} \]

where \(C_n\) is a constant depending only on \(n\).

The proof of the theorem is carried out by the induction method and is based on the following lemma.

Lemma. If \(f(x)\in S_1\), then

\[ e_\varepsilon(f)\leq C\varepsilon \operatorname{Sup}|f_1(x)|, \]

where

\[ C=\frac{12}{\pi}\int_{-\infty}^{\infty} \left| \left[ \frac{ 8\sin^3\frac{v}{4}\sin\frac{5v}{4} +\sin v\left(2\sin\frac{v}{2}-v\right) }{ v^4 } \right]' \right|\,dv. \]

In the estimate (7), \(C_n=C_0C^n\).

In view of the obvious inequality \(e_\varepsilon(f)\geqslant E_\varepsilon(f)\), in estimates (2), (6), (7) one may replace \(e_\varepsilon(f)\) by \(E_\varepsilon(f)\).

1. The following theorem gives an estimate of the deviation of the partial sums of the Fourier series from an almost-periodic function of class \(S\).

Theorem 3. Let \(0<\eta<\varepsilon,\ f(x)\in S\). Then

\[ R_\varepsilon(f)\leqslant 2E_\varepsilon(f)\left\{1+\frac{2}{\pi}+N_L(\eta)-N_L(\varepsilon)+\frac{1}{\pi}\ln\frac{\varepsilon+\eta}{\varepsilon-\eta}\right\}, \tag{8} \]

where

\[ N_L(\varepsilon)=\sum_{\Lambda_k\geqslant\varepsilon}1. \]

Proof. Let

\[ \varphi_{\eta,\varepsilon}(t)= \begin{cases} 1, & |t|<\eta,\\[4pt] \dfrac{1}{\varepsilon-\eta}(\varepsilon-|t|), & \eta\leqslant |t|\leqslant \varepsilon,\\[6pt] 0, & |t|>\varepsilon; \end{cases} \]

then

\[ \Psi_{\eta,\varepsilon}(u) =\frac{1}{2\pi}\int_{-\infty}^{\infty}\varphi_{\eta,\varepsilon}(t)e^{-iut}\,dt = \frac{2\sin\frac{\varepsilon-\eta}{2}u\,\sin\frac{\varepsilon+\eta}{2}u}{\pi(\varepsilon-\eta)u^2}, \]

\[ f_{\eta,\varepsilon}(x) = \int_{-\infty}^{\infty} f(x+u)\Psi_{\eta,\varepsilon}(u)\,du \sim \sum_{|\Lambda_k|<\eta} A_k e^{i\Lambda_k x} + \sum_{\eta<|\Lambda_k|\leqslant\varepsilon} \varphi_{\eta,\varepsilon}(\Lambda_k)A_k e^{i\Lambda_k x}. \]

It is easy to see that

\[ f(x)-f_{\eta,\varepsilon}(x) = \sum_{\eta<|\Lambda_k|\leqslant\varepsilon} A_k\left[1-\varphi_{\eta,\varepsilon}(\Lambda_k)\right]e^{i\Lambda_k x} + \sum_{|\Lambda_k|>\varepsilon} A_k e^{i\Lambda_k x}, \]

therefore

\[ R_\varepsilon(f)\leqslant \operatorname{Sup}_{x}|f_{\eta,\varepsilon}(x)| + 2\max_{\eta<|\Lambda_k|\leqslant\varepsilon}|A_k|[N(\eta)-N(\varepsilon)+1]. \tag{9} \]

Let \(\varepsilon_1=0\); there exists a function \(F^*(x)\in Q_\varepsilon\) such that

\[ |f(x)-F^*(x)|\leqslant E_\varepsilon(f)+\varepsilon_1. \]

Then

\[ \left| \int_{-\infty}^{\infty} f(x+u)\Psi_{\eta,\varepsilon}(u)\,du - \int_{-\infty}^{\infty} F^*(x+y)\Psi_{\eta,\varepsilon}(u)\,du \right| \leqslant \]

\[ \leqslant [E_\varepsilon(f)+\varepsilon_1]\int_{-\infty}^{\infty}|\Psi_{\eta,\varepsilon}(u)|\,du \leqslant 2[E_\varepsilon(f)+\varepsilon_1]\left(\frac{2}{\pi}+\frac{1}{\pi}\ln\frac{\varepsilon+\eta}{\varepsilon-\eta}\right), \]

and, by the arbitrariness of \(\varepsilon_1\),

\[ |f_{\eta,\varepsilon}(x)| \leqslant 2E_\varepsilon(f)\left(\frac{2}{\pi}+\frac{1}{\pi}\ln\frac{\varepsilon+\eta}{\varepsilon-\eta}\right). \tag{10} \]

For \(|\Lambda_k|\leqslant\varepsilon\),

\[ A_k=\lim_{T\to\infty}\frac{1}{T}\int_0^T [f(x)-F^*(x)]e^{-i\Lambda_k x}\,dx, \]

hence \(|A_k|\leqslant E_\varepsilon(f)+\varepsilon_1\), and, since \(\varepsilon_1\) is arbitrary,

\[ \eta\leqslant \max_{|\Lambda_k|<\varepsilon}|A_k|\leqslant E_\varepsilon(f). \tag{11} \]

From (9), (10), and (11) follows (8).

In applications of Theorem 3, the choice of the parameter $\eta$ is determined to a known extent (see (4)) by the character of the sequence $L(f)$.

1. Let us consider some applications of the estimates obtained above.

Theorem 4. Let $\theta(x)$ be nonincreasing for $x \geqslant 0$, $\lim\limits_{x\to\infty}\theta(x)=0$, and

\[ \left|\frac{1}{u}\int_0^u f(x+t)\,dt\right|<\theta(|u|). \tag{12} \]

Then the series (1) converges uniformly, if

\[ \lim_{n\to\infty}\theta\!\left(\frac{1}{\Lambda_n}\right)\ln\frac{\Lambda_n+\Lambda_{n+1}}{\Lambda_n-\Lambda_{n+1}}=0. \]

Proof. In consequence of (12), $\Omega_f(N)\leqslant\theta(N)$; applying (2) and (8) with $\varepsilon=\Lambda_n$, $\eta=\Lambda_{n+1}$, we obtain

\[ R_{\Lambda_n}(f)\leqslant 2C_0\theta\!\left(\frac{1}{\Lambda_n}\right) \left(2+\frac{2}{\pi}+\frac{1}{\pi}\ln\frac{\Lambda_n+\Lambda_{n+1}}{\Lambda_n-\Lambda_{n+1}}\right), \]

which proves the theorem.

Theorem 5. Let there exist a constant $A$ such that

\[ \left|\int_0^u f(x+t)\,dt\right|<A. \tag{13} \]

Then the series (1) converges uniformly, if

\[ \Lambda_n\ln\frac{\Lambda_n+\Lambda_{n+1}}{\Lambda_n-\Lambda_{n+1}}=O(1). \tag{14} \]

Proof. By virtue of (13), $f(x)\in S_1$; from (7) and (8) it follows that

\[ R_{\Lambda_n}(f)\leqslant 2C_1\Lambda_n\Omega_{f_1}\!\left(\frac{1}{\Lambda_n}\right) \left(2+\frac{2}{\pi}+\frac{1}{\pi}\ln\frac{\Lambda_n+\Lambda_{n+1}}{\Lambda_n-\Lambda_{n+1}}\right), \]

and, in consequence of (14),

\[ \lim_{n\to\infty} R_{\Lambda_n}(f)=0. \]

From Theorem 4, with $\theta(x)=\dfrac{1}{x^\alpha}$ $(0<\alpha\leqslant 1)$, there follows the convergence criterion of B. M. Levitan $(^{1,2})$. Theorem 5 is a refinement of this criterion for $\alpha=1$.

Theorem 6. If there exist a natural number $m$ and $\theta>1$ such that

\[ \frac{\Lambda_n}{\Lambda_{n+m}}\geqslant \theta \quad (n=1,2,\ldots), \tag{15} \]

then the series (1) converges uniformly.

Proof. From (15) it follows (see $(^3)$) that $N_L(\varepsilon)-N_L(2\varepsilon)=O(1)$. Putting in (8) $\varepsilon=\Lambda_n$, $\eta=\tfrac12\Lambda_n$, we obtain $R_{\Lambda_n}(f)=O[E_{\Lambda_n}(f)]$.

Corollary. Under the condition of Theorem 6, the series (1) converges absolutely $(^{4,5})$; moreover, the order equalities hold

\[ E_\varepsilon(f)\sim R_\varepsilon(f)\sim \alpha_\varepsilon(f), \]

where

\[ \alpha_\varepsilon(f)=\sum_{|\Lambda_k|\leqslant\varepsilon}|A_k|. \]

Kuibyshev
Aviation Institute

Received
1 XII 1959

CITED LITERATURE

$^1$ B. M. Levitan, Zap. Nauch.-issl. inst. matem. i mekh., Kharkovsk. matem. tov., 14, ser. 4, 105 (1937).
$^2$ B. M. Levitan, Almost-periodic functions, Moscow, 1953.
$^3$ S. B. Stechkin, Izv. AN SSSR, ser. matem., 20, No. 3 (1956).
$^4$ E. A. Bredikhina, DAN, 123, No. 2 (1958).
$^5$ E. A. Bredikhina, DAN, 111, No. 6 (1956).