Abstract Generated abstract
This note studies approximation of Stepanov almost periodic functions by uniformly almost periodic functions and entire functions of bounded degree, using convolution kernels with controlled Fourier transforms. It establishes estimates in the Stepanov metric for truncated spectral approximants, relates the error to the Stepanov modulus of continuity and best approximation quantities, and derives bounds for deviations of Fourier partial sums under assumptions on the distribution of Fourier exponents. The results include pointwise convergence at Lebesgue points for suitable approximants, a criterion yielding almost everywhere convergence for lacunary Fourier series that generalizes Kolmogorov’s theorem, and a uniqueness theorem for Stepanov almost periodic functions. Further consequences identify conditions under which Stepanov almost periodic functions with bounded or suitably structured spectra are uniformly almost periodic.
Full Text
E. A. Bredikhina
ON THE APPROXIMATION OF STEPANOV ALMOST PERIODIC FUNCTIONS
(Presented by Academician A. N. Kolmogorov on 11 II 1965)
1. In this note estimates are given for deviations, in the Stepanov metric, of uniformly almost periodic (u.a.p.) functions with bounded spectra from Stepanov almost periodic (S-a.p.) functions. Some applications of these estimates are indicated to problems concerning the structure and approximation of S-a.p. functions and to the study of convergence of Fourier series; in particular, with their aid A. N. Kolmogorov’s theorem \((^{1,2})\) on lacunary Fourier series is generalized to S-a.p. functions.
Let the Fourier series of an S-a.p. function \(f(x)\) be written in the following form:
\[ f(x)\sim \sum_k A_k e^{i\lambda_k x} \quad \left(A_k=M\{f(x)e^{-i\lambda_k x}\}\right). \tag{1} \]
Put
\[ \|f-g\|_S=\sup_{-\infty<x<\infty}\int_x^{x+1}|f(t)-g(t)|\,dt; \]
\[ \omega_S(\delta,f)= \sup_{|h|\leq\delta} \left\{ \sup_{-\infty<x<\infty}\int_x^{x+1}|f(t+h)-f(t)|\,dt \right\}; \]
by virtue of the \(S\)-uniform continuity \((^3)\) of \(f(x)\),
\[ \lim_{\delta\to 0}\omega_S(\delta,f)=0. \tag{2} \]
It is easy to see that \(\omega_S(\delta,f)\) is a nondecreasing subadditive function of \(\delta\); therefore, for any \(\lambda>0\),
\[ \omega_S(\lambda\delta,f)\leq (1+\lambda)\omega_S(\delta,f). \tag{3} \]
Introduce the function
\[ \varphi_{\lambda,\mu}(t)= \begin{cases} 1, & |t|\leq \lambda,\\[4pt] \dfrac12-\dfrac{2}{\mu-\lambda} \left(|t|-\dfrac{\lambda+\mu}{2}\right) -\dfrac{2}{(\mu-\lambda)^2} \left(|t|-\dfrac{\lambda+\mu}{2}\right)^2, & \lambda<|t|\leq\dfrac{\lambda+\mu}{2},\\[8pt] \dfrac12-\dfrac{2}{\mu-\lambda} \left(|t|-\dfrac{\lambda+\mu}{2}\right) +\dfrac{2}{(\mu-\lambda)^2} \left(|t|-\dfrac{\lambda+\mu}{2}\right)^2, & \dfrac{\lambda+\mu}{2}<|t|\leq\mu,\\[8pt] 0, & |t|>\mu, \end{cases} \]
and its Fourier transform
\[ \Psi_{\lambda,\mu}(u) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \varphi_{\lambda,\mu}(t)e^{-iut}\,dt = \frac{16}{\pi(\mu-\lambda)^2} \frac{\sin^2 \dfrac{\mu-\lambda}{4}u \, \sin \dfrac{\mu+\lambda}{2}u}{u^3}. \]
The estimates hold
\[ \int_{-\infty}^{\infty}|\Psi_{\lambda,\mu}(u)|\,du \leq A\ln\frac{\mu+\lambda}{\mu-\lambda}+B, \qquad \int_0^{\infty}u|\Psi_{\lambda,\mu}(u)|\,du \leq \frac{4A}{\mu-\lambda}; \tag{4} \]
where \(A=\dfrac{2}{\pi}\), \(B=\dfrac{3}{\pi}+\dfrac{2}{\pi}\ln 2\); their proof is analogous to the proof of Lemma 1.10.1 from \((^3)\).
As a consequence of the Fourier inversion formula,
\[ \varphi_{\lambda,\mu}(\Lambda)=\int_{-\infty}^{\infty}\Psi_{\lambda,\mu}(u)e^{-i\Lambda u}\,du \]
and, in particular,
\[ \int_{-\infty}^{\infty}\Psi_{\lambda,\mu}(u)\,du=\varphi_{\lambda,\mu}(0)=1. \tag{5} \]
2. Let us formulate the principal theorem of the note.
Theorem 1. If \(f(x)\) is an \(S\)-a.p. function having the Fourier series (1), then
\[ f_{\lambda,\mu}(x)=\int_{-\infty}^{\infty} f(x+u)\Psi_{\lambda,\mu}(u)\,du \]
is a uniformly a.p. function with Fourier series
\[ f_{\lambda,\mu}(x)\sim \sum_k A_k\varphi_{\lambda,\mu}(\lambda_k)e^{i\lambda_k x} \tag{6} \]
and the estimate
\[ \|f_{\lambda,\mu}-f\|_S \le \omega_S\!\left(\frac{1}{\mu-\lambda},f\right) \left[ A\ln\frac{\mu+\lambda}{\mu-\lambda}+C \right], \tag{7} \]
where \(C=B+8A\).
The proof of the first assertion of the theorem is constructed according to the scheme of the proof of Lemma 1.10.2 from \((^3)\); (7) follows from (3), (4), (5), and known \((^3)\) properties of \(S\)-a.p. functions.
We note that, by the definition of \(\varphi_{\lambda,\mu}(t)\), the spectrum of the function \(f_{\lambda,\mu}(x)\) belongs to the interval \((-\mu,\mu)\), and, by (2), \(\|f_{\lambda,\mu}-f\|_S\to 0\) as \(\mu\to\infty\), if \(\lambda/\mu\le \theta<1\).
3. Let
\[ E_\mu^{(S)}(f)=\inf_{g\in B_\mu}\{\|f-g\|_S\}, \]
where \(B_\mu\) is the class of entire functions of degree \(\le \mu\), bounded on the real axis.
Theorem 2. For an \(S\)-a.p. function \(f(z)\),
\[ E_\mu^{(S)}(f)\le C\omega_S\!\left(\frac{1}{\mu},f\right). \tag{8} \]
Theorem 3. For an \(S\)-a.p. function \(f(x)\),
\[ \|f_{\lambda,\mu}-f\|_S \le E_\lambda^{(S)}(f) \left[ A\ln\frac{\mu+\lambda}{\mu-\lambda}+B+1 \right]. \tag{9} \]
Theorem 4. If the Fourier exponents of the \(S\)-a.p. function \(f(x)\) with Fourier series (1) have a unique limit point at infinity, then
\[ \|f-S_\lambda\|_S \le E_\lambda^{(S)}(f) \left[ A\ln\frac{\mu+\lambda}{\mu-\lambda} +N_f(\mu)-N_f(\lambda)+B+1 \right], \tag{10} \]
where
\[ S_\lambda(x)=\sum_{|\lambda_k|\le \lambda} A_k e^{i\lambda_k x}, \qquad N_f(\lambda)=\sum_{|\lambda_k|\le \lambda} 1. \]
Concerning the proof of these theorems we note the following. Estimate (8) follows from (7), since \(f_{0,\mu}(x)\in B_\mu\). The proof of estimate (9) is based to a considerable extent on (4) and on the application of the equality \(g_{\lambda,\mu}(x)=g(x)\), which holds \((^{4,6})\) for any function \(g(x)\in B_\lambda\). On the basis of Lemma 2 of paper \((^6)\), \(|A_k|\le E_\lambda^{(S)}(f)\) if \(|\lambda_k|>\lambda\); therefore, when fulfilled
conditions of Theorem 4
\[ \sup_x | f_{\lambda,\mu}(x)-S_\lambda(x)|\leq E_\lambda^{(S)}(f)[N_f(\mu)-N_f(\lambda)], \tag{11} \]
and (10) follows from (9) and (11).
Theorem 4 is an analogue of Theorem 1 of the paper [7], which gives an estimate of the deviation of a partial sum of the Fourier series from a uniformly almost periodic function in the space \(C(-\infty,\infty)\).
In view of the topological equivalence of the \(S\)-distances
\[ \|f-g\|_S^{(\alpha)} = \sup_{-\infty<x<\infty} \left\{ \frac{1}{\alpha} \int_x^{x+\alpha} |f(t)-g(t)|\,dt \right\} \quad (\|f-g\|_S=\|f-g\|_S^{(1)}) \]
the estimate (10), after the corresponding specification, becomes applicable for establishing convergence criteria for Fourier series of \(2\pi\)-periodic functions in the space \(\mathscr L(0,2\pi)\) [2].
4. Theorem 5. Let \(x\) be a Lebesgue point of the \(S\)-a.p. function \(f(x)\); then
\[ f_{\mu/2,\mu}(x)-f(x)=o(1) \qquad (\mu\to\infty). \tag{12} \]
We outline the proof of this theorem. By the properties of the kernel \(\Psi_{\lambda,\mu}(u)\) and the \(S\)-boundedness (3) of the function \(f(x)\),
\[ f_{\mu/2,\mu}(x)-f(x) = \int_0^1 \varphi_x(u)\Psi_{\mu/2,\mu}(u)\,du + O\left(\frac{1}{\mu^2}\right), \tag{13} \]
where \(\varphi_x(u)=f(x+u)+f(x-u)-2f(x)\).
At every Lebesgue point \(\Phi_x(u)=\int_0^u |\varphi_x(t)|\,dt=o(u)\), therefore
\[ \int_0^{1/\mu} |\varphi_x(u)|\,|\Psi_{\mu/2,\mu}(u)|\,du \leq \frac{3}{4}\mu\Phi_x\left(\frac{1}{\mu}\right) = o(1). \tag{14} \]
Integrating by parts the right-hand side of the inequality
\[ \int_{1/\mu}^1 |\varphi_x(u)|\,|\Psi_{\mu/2,\mu}(u)|\,du \leq \frac{64}{\pi\mu^2} \int_{1/\mu}^1 |\varphi_x(u)|\,\frac{du}{u^3}, \]
we obtain the estimate
\[ \int_{1/\mu}^1 |\varphi_x(u)|\,|\Psi_{\mu/2,\mu}(u)|\,du=o(1). \tag{15} \]
From (13), (14), and (15) follows (12).
Theorem 6. If the Fourier exponents of the \(S\)-a.p. function \(f(x)\) with Fourier series (1) have a unique limit point at infinity, \(x\) is a Lebesgue point of the function \(f(x)\), and
\[ E_\mu^{(S)}(f)[N_f(2\mu)-N_f(\mu)]=o(1) \qquad (\mu\to\infty), \tag{16} \]
then
\[ S_\mu(x)-f(x)=o(1) \qquad (\mu\to\infty). \tag{17} \]
Proof. By (11) and (16), at any point \(x\),
\[ |f_{\mu/2,\mu}(x)-S_{\mu/2}(x)| \leq E_{\mu/2}^{(S)}(f)[N_f(\mu)-N_f(\mu/2)] = o(1). \tag{18} \]
From (12) and (18) we obtain (17).
An increasing sequence of positive numbers \(L=\{l_k\}\) \((k=1,2,\ldots)\) will be called \(\theta,s\)-lacunary if there exist a natural number \(s\) and \(\theta>1\) such that \(l_{k+s}/l_k\geq \theta\) \((k=1,2,\ldots)\).
Denote by \(L(f)\) the set of absolute values of the Fourier exponents of the function \(f(x)\). We shall call the series (1) \(\theta, s\)-lacunary if it contains no constant term, the set \(L(f)\) has a unique limit point at infinity and, when renumbered in increasing order of its elements, forms a \(\theta, s\)-lacunary sequence.
The condition
\[ N_f(2\mu)-N_f(\mu)=O(1) \tag{19} \]
is necessary and sufficient \((^{5,7})\) for the series (1) to be \(\theta, s\)-lacunary.
By virtue of (19), (8), (2), and (18), the following generalization of A. N. Kolmogorov’s theorem holds:
Theorem 7. If the Fourier series of an \(S\)-a.p. function \(f(x)\) is \(\theta, s\)-lacunary, then it converges to \(f(x)\) almost everywhere.
- From Theorem 5 and the uniqueness theorem for uniformly a.p. functions \((^3)\) there follows the uniqueness theorem for \(S\)-a.p. functions (for \(S_2\)-a.p. functions it is true by Parseval’s equality \((^3)\)).
Theorem 8. If all Fourier coefficients of an \(S\)-a.p. function \(f(x)\) are equal to zero, then \(f(x)=0\) almost everywhere.
We note that this assertion can be obtained as an immediate consequence of Theorem 1.
Below we give two theorems which make it possible to establish that certain \(S\)-a.p. functions belong to the class of uniformly a.p. functions (here equivalent functions are regarded as identical).
Theorem 9. An \(S\)-a.p. function with bounded spectrum is a uniformly a.p. function.
This theorem follows from Theorems 1 and 8.
Theorem 10. A bounded \(S\)-a.p. function \(f(x)\) is a uniformly a.p. function if one of the following conditions is satisfied: 1) the Fourier exponents of \(f(x)\) are linearly independent; 2) the Fourier coefficients of \(f(x)\) are nonnegative.
Proof. It is enough, in view of Theorem 8, to prove the absolute convergence of the Fourier series (1) of the function \(f(x)\); under each of the two conditions of the theorem this follows from the estimate
\[ \sum_{|\lambda_k|\leq \mu/2}|A_k| \leq \sup_x |f_{\mu/2,\mu}(x)| \leq (A\ln 3+B)\sup_x |f(x)|, \]
which follows from (4), (6), and the known \((^3)\) criteria for absolute convergence of Fourier series of uniformly a.p. functions.
Theorem 10 can be strengthened if the set \(L(f)\) is not everywhere dense. Indeed, if the interval \((\sigma,\sigma+\varepsilon)\) \((\sigma>0,\ \varepsilon>0)\) contains no points of the set \(L(f)\), then, by Theorems 1 and 9, it is enough that either of the two conditions of Theorem 10 be satisfied for the \(S\)-a.p. function
\[ \tilde f_\sigma(x)\sim \sum_{|\lambda_k|>\sigma} A_k e^{i\lambda_k x}. \]
In conclusion we note that, as a consequence of Theorem 1 of the paper \((^8)\) and Theorem 8, a bounded \(S\)-a.p. function with a \(\theta, s\)-lacunary Fourier series turns out to be a uniformly a.p. function.
Kuibyshev Aviation Institute
Received
4 I 1965
CITED LITERATURE
\(^1\) A. N. Kolmogorov, Fund. Math., 5, 96 (1924).
\(^2\) N. K. Bari, Trigonometric Series, Moscow, 1961.
\(^3\) B. M. Levitan, Almost-Periodic Functions, Moscow, 1953.
\(^4\) N. I. Akhiezer, Lectures on Approximation Theory, Moscow–Leningrad, 1947.
\(^5\) S. B. Stechkin, Izv. Akad. Nauk SSSR, 20, No. 3, 385 (1956).
\(^6\) E. A. Bredikhina, Dokl. Akad. Nauk SSSR, 171, No. 1, 17 (1957).
\(^7\) E. A. Bredikhina, Mat. Sb., 50, 92, 3, 359 (1960).
\(^8\) E. A. Bredikhina, Uspekhi Mat. Nauk, 19, 133 (1964).