Abstract Generated abstract
This note studies harmonic approximation for almost-periodic functions whose Fourier exponents have no finite limit points, relating the deviation of Fourier partial sums to best approximation by entire functions of finite degree. It proves a general inequality bounding the remainder of a Fourier sum by the best approximation error and a factor determined by the local counting function of the frequency sequence, then derives analogues of Lebesgue’s theorem for separated, finitely separated, and lacunary frequency sets. Further consequences give criteria for uniform and absolute convergence of Fourier series in terms of best approximation, frequency gaps, and the modulus of continuity, including almost-periodic versions of Bohr and Dini-Lipschitz type criteria. For frequency sequences that are finite unions of lacunary sequences, the note also obtains order equivalence between best approximation, Fourier remainder, and the tail sum of Fourier coefficients.
Full Text
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.
- 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.
- 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 \((^4)\).
- 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.
- 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).