Reports of the Academy of Sciences of the USSR
1958. Volume 123, No. 2

MATHEMATICS

E. A. BREDIKHINA

FOURIER SERIES AS A DEVICE FOR APPROXIMATING ALMOST-PERIODIC FUNCTIONS

(Presented by Academician V. I. Smirnov on 30 V 1958)

1. One of the questions in the theory of harmonic approximation is that of establishing the dependence between the deviations of the partial sums of the Fourier series of a continuous (2\pi)-periodic function and the best approximations of this function by trigonometric polynomials.

In the present note we briefly set forth some results of considering an analogous question for almost-periodic functions. Namely, a dependence is established between the deviations of the partial sums of the Fourier series of an almost-periodic function, whose Fourier exponents have no finite limit points, and the best approximations of this function by entire functions of finite degree.

Let the Fourier series of the almost-periodic function (f(x)) be written in the following form:

[
f(x)\sim \sum_{k=-\infty}^{\infty} A_k e^{i\Lambda_k x}
\tag{1}
]

[
\left(\Lambda_0=0;\quad \Lambda_k<\Lambda_{k+1}\ \text{for } k=0,1,2,\ldots;\right.
]

[
\left.
\lim_{k\to\infty}\Lambda_k=\infty,\ \Lambda_k=-\Lambda_{-k};\quad |A_k|+|A_{-k}|\ne 0\ \text{for } k\ne 0
\right).
]

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

[
R_\lambda(f)=\sup_x \left| f(x)-\sum_{|\Lambda_k|\le \lambda} A_k e^{i\Lambda_k x}\right|;
]

[
\alpha_\lambda(f)=\sum_{|\Lambda_k|>\lambda}|A_k|,\qquad
E_\lambda(f)=\inf_{F(z)\in B_\lambda}\left{\sup_x |f(x)-F(x)|\right},
]

where (B_\lambda) is the class of entire functions of degree (\le \lambda), bounded on the real axis.

Let (l={\lambda_k}) ((k=1,2,\ldots;\ 0<\lambda_k<\lambda_{k+1})) be an increasing sequence of positive numbers. Put

[
N_l(\lambda)=\sum_{\lambda_k\le \lambda}1.
]

We shall say that the sequence (l) belongs to the class (A) if there is an (a>0) such that (\lambda_{k+1}-\lambda_k>a) ((k=1,2,\ldots)). Denote by (A_\sigma) the class of all sequences (l), each of which can be partitioned into a finite number of sequences belonging to the class (A). Thus, (l\in A_\sigma) if the representation

[
l=\bigcup_{j=1}^{r} l^{(j)},\quad \text{where } l^{(j)}\in A\ (j=1,2,\ldots,r).
]

is possible.

Let (\mathcal{L}) be the class of all lacunary sequences (l). A sequence (l) belongs to the class (\mathcal{L}) if there exists (\theta>1) such that

(\lambda_{k+1}/\lambda_k \geqslant \theta). By (\mathcal{L}\sigma) we denote the class of all sequences (l), each of which can be divided into a finite number of lacunary ones. Thus, (\mathcal{L}\sigma) is the class of sequences (l) admitting the representation

[
l=\bigcup_{j=1}^{r} l^{(j)}, \quad \text{where } l^{(j)}\in \mathcal{L}\ (j=1,2,\ldots,r).
]

The inclusions (A_\sigma \supset A), (\mathcal{L}_\sigma \supset \mathcal{L}) are obvious.

1. We formulate the main theorem of the note.

Theorem 1. Let (0<\lambda<\mu); then, for any almost-periodic function (f(x)) whose Fourier exponents have no finite limit points, the inequality

[
R_\lambda(f)\leqslant \Phi(\lambda,\mu)E_\lambda(f),
\tag{2}
]

holds, where

[
\Phi(\lambda,\mu)=1+\frac{4}{\pi}+2[N_L(\mu)-N_L(\lambda)]
+\frac{2}{\pi}\ln\frac{\mu+\lambda}{\mu-\lambda}.
]

The method of proof of Theorem 1 is a generalization and development of the method set out in the author’s paper ((^7)).

In applications of Theorem 1 the parameter (\mu) is chosen so as to minimize, as far as possible, the factor (\Phi(\lambda,\mu)).

From Theorem 1, formulated under very general conditions, a number of theorems follow. Let us consider some of them.

1. Let (f(x)) be a continuous (2\pi)-periodic function, and let (E_n^(f)) be the best approximation of this function by trigonometric polynomials of order (n). Then (see ((^1)), pp. 374–375) (E_n^(f)=E_n(f)). Putting in inequality (2) (\lambda=n), (\mu=n+1-\varepsilon), where (\varepsilon) is sufficiently small, we obtain Lebesgue’s theorem ((^2), pp. 193–194).

The following theorem is a generalization of Lebesgue’s theorem to the almost-periodic case.

Theorem 2. Let the sequence (L(f)) have the following property: there exists a function (\varphi(\lambda)), nonnegative for (\lambda>\lambda_0), such that

[
N_L\left(\lambda+\frac{\lambda}{e^{\varphi(\lambda)}}\right)-N_L(\lambda)
=O[1+\varphi(\lambda)].
]

Then (R_\lambda(f)\leqslant \Phi(\lambda)E_\lambda(f)), where (\Phi(\lambda)=O[1+\varphi(\lambda)]).

For the proof of Theorem 2 it is enough to put in inequality (2)

[
\mu=\lambda+\frac{\lambda}{e^{\varphi(\lambda)}}.
]

We note two theorems contained in Theorem 2.

Theorem 3. If (L(f)\in A_\sigma), then (R_\lambda(f)\leqslant \Phi(\lambda)E_\lambda(f)), where (\Phi(\lambda)=O(\ln\lambda)).

Proof. For (L(f)\in A_\sigma) there exists (a>0) such that

[
N_L(\lambda+a)-N_L(\lambda)=O(1).
]

Putting in Theorem 2 (\varphi(\lambda)=\ln \dfrac{\lambda}{a}), we obtain the assertion to be proved.

Theorem 4. If (L(f)\in \mathcal{L}\sigma), then (R\lambda(f)=O[E_\lambda(f)]).

Proof. It is known ((^4)) that for (L(f)\in \mathcal{L}_\sigma) we have (N_L(2\lambda)-N_L(\lambda)=O(1)); therefore the condition of Theorem 2 is fulfilled for (\varphi(\lambda)=0).

This theorem is a generalization, to almost-periodic functions, of Theorem 4 of the paper ((^4)).

1. Let us consider theorems providing new criteria for the uniform and absolute convergence of Fourier series of almost-periodic functions. Here the following theorem is the starting point.

Theorem 5. The Fourier series (1) of an almost-periodic function (f(x)) converges uniformly if there exists a numerical sequence ({\mu_n}) ((n=1,2,\ldots)) satisfying the conditions:

1) (\mu_n>\Lambda_n) for (n>n_0);

2) (\displaystyle \lim_{n\to\infty} E_{\Lambda_n}(f)\,[N_L(\mu_n)-N_L(\Lambda_n)]=0;)

3) (\displaystyle \lim_{n\to\infty} E_{\Lambda_n}(f)\ln\frac{\mu_n+\Lambda_n}{\mu_n-\Lambda_n}=0.)

To prove Theorem 5 it is enough to put (\mu=\mu_n,\ \lambda=\Lambda_n) in inequality (2) and take into account that (\displaystyle \lim_{\lambda\to\infty} E_\lambda(f)=0) (\bigl((^1),) pp. 371–372\bigr).

Theorem 6. The Fourier series (1) of the almost-periodic function (f(x)) converges uniformly if

[
\lim_{n\to\infty} E_{\Lambda_n}(f)\ln\frac{\Lambda_{n+1}+\Lambda_n}{\Lambda_{n+1}-\Lambda_n}=0.
]

Theorem 6 follows from Theorem 5 for (\mu_n=\Lambda_{n+1}).

Let us note a consequence of Theorem 6, which holds by virtue of the inequality

[
E_\lambda(f)\leq 2\pi\omega_f\left(\frac{1}{\lambda}\right),\qquad
\text{where }\ \omega_f(\delta)=\sup_{|x-y|\leq\delta}|f(x)-f(y)|
]

(\bigl((^1),) pp. 371–373\bigr).

Corollary. The Fourier series (1) of the almost-periodic function (f(x)) converges uniformly if

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

Theorem 6, which is a generalization of Bohr’s criterion (\bigl((^3),) pp. 81–83\bigr), makes it possible to use more fully the properties of the modulus of continuity of an almost-periodic function when investigating its Fourier series for uniform convergence.

It is easy to see that the condition of Theorem 6 ensuring uniform convergence, when the sequence (L(f)) of frequencies contains gaps whose lengths decrease arbitrarily rapidly, can be fulfilled only at the expense of a correspondingly rapid decrease of the best approximation of the almost-periodic function.

From the theorems given below it follows that this undesirable circumstance is caused not by the essence of the matter, but by the choice of the sequence ({\mu_n}) in Theorem 6.

Theorem 7. The Fourier series (1) of the almost-periodic function (f(x)) converges uniformly if there exists a function (\varphi(\lambda)\geq 0) for (\lambda>\lambda_0), satisfying the conditions:

1) (\displaystyle N_L\left(\Lambda_n+\frac{\Lambda_n}{e^{\varphi(\Lambda_n)}}\right)-N_L(\Lambda_n)=O[1+\varphi(\Lambda_n)];)

2) (\displaystyle \lim_{n\to\infty}\varphi(\Lambda_n)E_{\Lambda_n}(f)=0.)

Theorem 7 follows from Theorem 5 for

[
\mu_n=\Lambda_n+\frac{\Lambda_n}{e^{\varphi(\Lambda_n)}}.
]

Theorem 8. Suppose that the sequence (L(f)) has the following property: there are numbers (a>0,\ m\geq 0) such that

[
N_L\left(\Lambda_n+\frac{a}{\Lambda_n^m}\right)-N_L(\Lambda_n)=O(\ln\Lambda_n).
]

Then the Fourier series (1) of the almost-periodic function (f(x)) converges uniformly if (\displaystyle \lim_{\delta\to 0}\omega_f(\delta)\ln\delta=0.)

Theorem 8 follows from Theorem 7 for (\displaystyle \varphi(\lambda)=\ln\frac{\lambda^{m+1}}{a}).

Corollary. If (L(f)\in A_{\sigma}) and (\lim_{\delta\to 0}\omega_f(\delta)\ln\delta=0), then the Fourier series (1) of the almost-periodic function (f(x)) converges uniformly.

Theorem 8 is a generalization to the almost-periodic case of the Dini–Lipschitz criterion ((^{5,2})).

Theorem 9. If (L(f)\in Л_{\sigma}), then the Fourier series (1) of the almost-periodic function (f(x)) converges absolutely; moreover, if (A_0=0), then

[
\sum_{k=-\infty}^{\infty}|A_k|\leq C_L\sup_x |f(x)|,
]

where (C_L) is a constant depending only on the sequence (L(f)).

The proof of Theorem 9 is analogous to the proof of Theorem 1 of the author’s note ((^{6})). In the proof of Theorem 9, Theorem 1 of the paper ((^{4})) and the uniform convergence of the series (1), following from Theorem 7 for (\varphi(\lambda)=0), are used in an essential way.

1. In conclusion we formulate a theorem generalizing the results of the author’s paper ((^{7})) concerning best approximations of almost-periodic functions representable by lacunary series.

Theorem 10. If (L(f)\in Л_{\sigma}), then the order equalities

[
E_{\lambda}(f)\sim R_{\lambda}(f)\sim \alpha_{\lambda}(f).
]

hold.

The proof of Theorem 10 is based on the application of Theorems 4 and 9 of the present note.

Kuibyshev Aviation
Institute

Received
28 V 1958

CITED LITERATURE

(^{1}) S. N. Bernstein, Collected Works, 2, 1954.
(^{2}) I. P. Natanson, Constructive Theory of Functions, Moscow–Leningrad, 1949.
(^{3}) B. M. Levitan, Almost-Periodic Functions, Moscow, 1953.
(^{4}) S. B. Stechkin, Izv. AN SSSR, Ser. Matem., 20, No. 3 (1956).
(^{5}) A. Zygmund, Trigonometric Series, Moscow–Leningrad, 1939.
(^{6}) E. A. Bredikhina, DAN, 111, No. 6 (1956).
(^{7}) E. A. Bredikhina, DAN, 117, No. 1 (1957).