E. A. BREDIKHINA

ON THE CONVERGENCE OF FOURIER SERIES OF ALMOST PERIODIC FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 27 VI 1964)

1. This note sets forth a method that provides a number of new criteria for the uniform and absolute convergence of Fourier series of almost periodic (a.p.) functions. It is based on representing the given function as the sum of a finite number or a countable set of a.p. functions with bounded spectra, and is applicable mainly to the class \(Q\) of uniform a.p. functions whose Fourier exponents have a finite number of limit points on each segment of the real axis.

Let the Fourier series of the uniform a.p. function \(f(x)\) be written in symmetric form:

\[ f(x)\sim \sum_{k=-\infty}^{\infty} A_{\lambda_k}e^{i\lambda_k x} \qquad (\lambda_0=0,\quad \lambda_k=-\lambda_{k};\quad \lambda_k>0 \text{ for } k>0). \tag{1} \]

Put \(\|f\|=\operatorname{Sup}_{x}|f(x)|\); by \(L(f)\) denote the set of absolute values of the Fourier exponents of the function \(f(x)\). We shall call any \(\sigma>0\) an isolated point from the right with respect to the set \(L(f)\), if there exists \(\varepsilon>0\) such that the interval \((\sigma,\sigma+\varepsilon)\) contains no points of the set \(L(f)\). For \(f(x)\in Q\) introduce into consideration the sequence \(\bar L=\{\Lambda_j\}\) \((j=1,2,\ldots;\ 0<\Lambda_j<\Lambda_{j+1})\) of all finite nonzero limit points of the set \(L(f)\). Let the numbers \(\varepsilon_j\) \((j=0,1,\ldots)\) be chosen so that the intervals \(I_j=(\Lambda_j-\varepsilon_j,\Lambda_j+\varepsilon_j)\) \((j=0,\pm1,\pm2,\ldots,\ \Lambda_{-j}=-\Lambda_j,\ \Lambda_0=0,\ \varepsilon_{-j}=\varepsilon_j)\) have no common points. By \(\eta_j\) denote the distance between the intervals \(I_j\) and \(I_{j+1}\) \((j\ge 0)\) (if \(\Lambda_0=0\) is not a limit point of the set \(L(f)\), then in defining \(\eta_0\) we take \(\varepsilon_0=0\), and subsequently we shall put \(j\ne 0\)).

Put \(I=\bigcup_j I_j\) and let \(M=L(f)\cap CI\), where \(CI\) is the complement of the set \(I\).

The set \(L_j(f)=L(f)\cap I_j\) may, without loss of generality, be regarded as symmetric with respect to \(\Lambda_j\). The points \(\lambda\) of the set \(L_j(f)\) satisfying the condition \(\lambda-\Lambda_j>0\) are renumbered in decreasing order; as a result we obtain the sequence

\[ l^{(j)}=\{\lambda_k^{(j)}-\Lambda_j\} \quad (k=1,2,\ldots;\ \Lambda_j<\lambda_{k+1}^{(j)},\ \lim_{k\to\infty}\lambda_k^{(j)}=\Lambda_j). \]

We shall assign the function \(f(x)\in Q\) to the class \(Q_0\) if its spectrum is bounded; we shall assign the function \(f(x)\in Q\) to the class \(Q_1\) if the Fourier exponents of the function \(f(x)\) have a finite number of finite limit points and a limit point at infinity (in both cases the sequence \(\bar L\) is finite). For \(f(x)\in Q_1\), for sufficiently large \(\sigma=\sigma(f)\), the set \(L_\sigma(f)=L(f)\cap[\sigma,\infty)\) has a unique limit point at infinity; renumbering its terms in increasing order, we obtain the sequence

\[ l^{(\sigma)}=\{\lambda_k^{(\sigma)}\} \quad (k=1,2,\ldots;\ \sigma\le \lambda_k^{(\sigma)}<\lambda_{k+1}^{(\sigma)};\ \lim_{k\to\infty}\lambda_k^{(\sigma)}=\infty). \]

Let \(l=\{a_k\}\) \((k=1,2,\ldots)\) be a decreasing (increasing) sequence of positive numbers. We shall call this sequence \(\theta\)-lacunary if there exists \(\theta>1\) such that \(a_k/a_{k+1}\geq \theta\) \((a_{k+1}/a_k\geq \theta)\); we shall call it \(\theta,r\)-lacunary if it admits a representation

\[ l=\bigcup_{i=1}^{s} l_i,\quad \text{where } s\leq r,\; l_i \text{ is a } \theta\text{-lacunary sequence} \]

(see \((^3,^5,^6)\)).

Let \(0<\Lambda_1<\Lambda_2<\cdots<\Lambda_m\) be the nonzero (finite) limit points of the set \(L(f)\) of the function \(f(x)\in Q_0\) \((f(x)\in Q_1)\). We assign \(f(x)\in Q_0\) to the class \(\mathcal L_0(\theta,r)\) if the sequences \(l^{(j)}\) \((j=0,\pm1,\pm2,\ldots,\pm m)\) are \(\theta,r\)-lacunary; we assign \(f(x)\in Q_1\) to the class \(\mathcal L_1(\theta,r)\) if the sequences \(l^{(\sigma)}l^{(j)}\) \((j=0,\pm1,\pm2,\ldots,\pm m)\) are \(\theta,r\)-lacunary.

Let \(f(x)\in Q\) and let the sequence \(\overline L\) be infinite; if the set \(M\) is infinite, renumber its elements in increasing order; as a result we obtain the sequence \(l^{(M)}=\{\lambda_k(M)\}\) \((k=1,2,\ldots)\). We assign \(f(x)\) to the class \(\mathcal L(\theta,r)\) if: 1) the sequences \(\overline L,\, l^{(j)}\) \((j=0,\pm1,\pm2,\ldots)\) are \(\theta,r\)-lacunary and, in the case when the set \(M\) is infinite, the sequence \(l^{(M)}\) is also \(\theta,r\)-lacunary; 2) there exist \(\varepsilon>0\) and \(\eta<0\) such that \(\varepsilon_j\geq\varepsilon,\; \eta_j\geq\eta\) \((j=0,1,\ldots)\).

2. Lemma 1. If \(f(x)\in Q_0\), then

\[ f(x)=P(x)+\sum_{j=-m}^{m} f_j(x), \tag{2} \]

where

\[ P(x)=\sum_{\lambda_k\in I} A_{\lambda_k}e^{i\lambda_k x},\qquad f_j(x)\in Q_0,\qquad f_j(x)\sim \sum_{\lambda_k\in I_j} A_{\lambda_k}e^{i\lambda_k x}; \]

moreover, the estimate

\[ \|f_j\|\leq \left[ \frac{4}{\pi}+\frac{2}{\pi}\ln\left(1+\frac{2\varepsilon}{\eta}\right)+2N \right]\|f\|, \tag{3} \]

holds, where \(\varepsilon=\max_j\{\varepsilon_j\}\), \(\eta=\min_j\{\eta_j\}\), and \(N\) is the number of points belonging to the set \(M\).

We shall write the Fourier series of the functions \(f_j(x)\) \((j=0,\pm1,\pm2,\ldots,\pm m)\) and \(f(x)\in Q_0\) in the following form:

\[ f_j(x)\sim \sum_{k=-\infty}^{\infty} A_{\lambda_k^{(j)}} e^{i\lambda_k^{(j)}x} \]

\[ \left(\lambda_j<\lambda_{k+1}^{(j)}<\lambda_k^{(j)} \text{ for } k\geq 0;\quad \lim_{k\to\infty}\lambda_k^{(j)}=\Lambda_j;\quad \lambda_{-k}^{(j)}=2\Lambda_j-\Lambda_k^{(j)}; \right. \]

\[ \left. |A_{\lambda_k^{(j)}}|+|A_{\lambda_{-k}^{(j)}}|>0 \text{ for } k>0 \right), \tag{4} \]

\[ f(x)\sim P(x)+\sum_{k=-\infty}^{\infty}\sum_{j=-m}^{m} A_{\lambda_k^{(j)}}e^{i\lambda_k^{(j)}x}. \tag{1'} \]

Let \(\sigma\) be a point isolated on the right with respect to the set \(L(f)\), \(\mu>\sigma\); enclose all points of the set \(L_{\sigma,\mu}=L(f)\cap(\sigma,\mu]\) in disjoint intervals \(\delta_i=[\alpha_i,\beta_i]\) \((\sigma<\alpha_i\leq\beta_i\leq\mu,\; i=1,2,\ldots,n;\; n\geq1)\), renumbered in increasing order of their left endpoints. Choose numbers \(\nu_i>0\) so that the intervals \(\Delta_i=(\alpha_i-\nu_i,\beta_i+\nu_i]\) \((i=1,2,\ldots,n)\) do not intersect and belong to the semi-interval \((\sigma,\mu]\) (when \(\beta_n=\mu\), the last of the intervals is replaced by the semi-interval \((\alpha_n-\nu_n,\beta_n]\)). By \(\Delta_f=\Delta_f(\sigma,\mu)\) denote any system of intervals \(\Delta_i\supset\delta_i\) \((i=1,2,\ldots,n)\) covering the set \(L_{\sigma,\mu}\).

Lemma 2. Let \(f(x)\) be a uniformly a.p. function, and let \(\sigma\) be a point isolated on the right with respect to the set \(L(f)\); then there exists a uniformly a.p. function

\[ f_\sigma(x)\sim \sum_{|\lambda_k|\leq\sigma} A_{\lambda_k}e^{i\lambda_k x} \]

and the estimate

\[ \|f-f_\sigma\|\leq \Phi(\sigma,\mu,\Delta_f)A_\sigma(f), \tag{5} \]

holds.

where \(A_\sigma(f)\) is the best approximation of the function \(f(x)\) by entire functions of degree \(\leqslant \sigma\),

\[ \Phi(\sigma,\mu,\Delta_f)=1+\frac{4}{\pi}+\frac{2}{\pi}\ln\frac{\mu+\sigma}{|\mu-\sigma|} +\frac{1}{\pi(\mu-\sigma)} \]

\[ {}\cdot \int_{-\infty}^{\infty} \left| \sum_{\alpha_i\ne\beta_i} \left[ \frac{\beta_i-\alpha_i}{\nu_i}\, \frac{\cos(\alpha_i-\nu_i)u-\cos\alpha_i u}{u^2} - \frac{\cos\alpha_i u-\cos\beta_i u}{u^2} \right] \right. \]

\[ \left. \qquad\qquad + \sum_{\beta_i\leqslant\mu} \frac{\mu-\beta_i}{\nu_i}\, \frac{\cos(\alpha_i-\nu_i)u-\cos\alpha_i u-\cos\beta_i u+\cos(\beta_i+\nu_i)u}{u^2} \right|\,du . \]

The proof of Lemmas 1 and 2 is based on the integral representation (1) of a.p. functions \(f_j(x)\) and \(f_\sigma(x)\) with kernels that are linear combinations of the kernels of N. I. Akhiezer—B. M. Levitan \((^{2,7})\).

Lemma 3. If for a uniformly a.p. function \(f(x)\) there exist an increasing sequence \(\{\sigma_k\}\) \((k=1,2,\ldots;\ \lim_k\sigma_k=\infty)\) of points isolated on the right with respect to the set \(L(f)\), a sequence \(\{\mu_k\}\) \((k=1,2,\ldots;\ \mu_k>\sigma_k)\), and a sequence of systems of intervals
\[ \Delta_f^{(k)}\quad (k=1,2,\ldots;\ \Delta_f^{(k)}=\Delta_f(\sigma_k,\mu_k)) \]
such that
\[ \sum_{k=1}^{\infty}\Phi(\sigma_k,\mu_k,\Delta_f^{(k)})A_{\sigma_k}(f)<\infty, \]
then, uniformly on the entire real axis,
\[ f(x)=\sum_{k=1}^{\infty} f_k(x), \]
where
\[ f_k(x)=f_{\sigma_k}(x)-f_{\sigma_{k-1}}(x)\quad (k=2,3,\ldots),\qquad f_1(x)=f_{\sigma_1}(x); \]
moreover,
\[ \sum_{k=1}^{\infty}\|f_k\|<\infty . \]

1. In what follows it is assumed that the limit points of the set of Fourier exponents of the function \(f(x)\) do not belong to the spectrum; however, this restriction can be removed without substantially complicating the proofs. Lemma 1 makes it possible to extend criteria for uniform and absolute convergence of Fourier series of a.p. functions whose Fourier exponents have a single limit point \(\Lambda^*\ne\infty\) to functions of the class \(Q_0\). We shall confine ourselves in this direction to the formulation of two theorems.

Theorem 1. If for \(f(x)\in Q_0\) there exist constants \(a>0\) and \(C>0\) such that

\[ \left| \frac{1}{u}\int_x^{x+u} f(t)e^{-i\lambda_j t}\,dt \right| < \frac{C}{|u^\alpha|} \qquad (0<\alpha\leqslant 1;\ j=0,\pm1,\ldots,\pm m), \tag{6} \]

\[ N_j\!\left(\frac{\varepsilon}{1+a\varepsilon}\right)-N_j(\varepsilon)=O(1) \qquad (j=0,\pm1,\ldots,\pm m), \tag{7} \]

where
\[ N_j(\varepsilon)= \sum_{\varepsilon\leqslant|\lambda_k-\Lambda_j|\leqslant\varepsilon_j} 1, \]
then the series \((1')\) converges uniformly.

Proof. Conditions (6) and (7) imply, by \(((6),\ \text{Theorem }8)\), the uniform convergence of the series (4). By virtue of (2), the series \((1')\) converges uniformly.

Theorem 2. If \(f(x)\in \Pi_0(\theta,r)\), then

\[ \sum_{k=-\infty}^{\infty} |A_{\lambda_k}|\leqslant \widetilde C\|f\|, \tag{8} \]

where
\[ \widetilde C=(2m+1)C(\theta,r) \left[ \frac{4}{\pi} +\frac{2}{\pi}\ln\left(1+\frac{2\varepsilon}{\eta}\right) +2N \right]+N \]
and the constant \(C(\theta,r)\) depends only on \(\theta\) and \(r\).

Proof. By virtue of Theorem 7 of paper \({}^{6}\),

\[ \sum_{k=-\infty}^{\infty} \left| A_{\lambda_k^{(j)}}(i) \right| \leq C(\theta,r)\|f_j\|; \]

applying (2) and (3), we obtain (8). Let \(Q_\infty\) be the class of uniformly a.p. functions whose Fourier exponents have a unique limit point at infinity. For \(f(x)\in Q_1\), Lemma 2 implies the equality
\(f(x)=f_\sigma(x)+\tilde f_\sigma(x)\), where \(f_\sigma(x)\in Q_0\), \(\tilde f_\sigma(x)\in Q_\infty\); it allows one to extend to the class \(Q_1\) the criteria for convergence of Fourier series known for the classes \(Q_0\) and \(Q_1\) \({}^{1,4-6}\). We give an example of such a generalization.

Theorem 3. If \(f(x)\in \mathcal L_1(\theta,r)\), then the series (1) converges absolutely.

The proof follows from Theorem 2 of the present note and Theorem 9 of paper \({}^{5}\).

On the basis of Lemma 3 one can obtain a number of criteria for absolute convergence of Fourier series in the case when the set \(L(f)\) is not everywhere dense. In particular, for \(f(x)\in \mathcal L(\theta,r)\), in estimate (8), applied to the functions \(f_k(x)\in \mathcal L_0(\theta,r)\) \((k=1,2,\ldots)\), the constants \(\tilde C\) are uniformly bounded, and it is possible to choose sequences \(\{\sigma_k\}\), \(\{\Delta_f^{(k)}\}\) \((k=1,2,\ldots;\ \sigma_k\geq 2^k)\) such that
\(\Phi(\sigma_k,2\sigma_k,\Delta_f^{(k)})=O(1)\). Therefore the following holds.

Theorem 4. The Fourier series of a function \(f(x)\in \mathcal L(\theta,r)\) converges absolutely if \(f(x)\in \operatorname{Lip} a\) \((a>0)\).

Kuibyshev Aviation
Institute

Received
22 V 1964

REFERENCES

\({}^{1}\) B. M. Levitan, Almost Periodic Functions, Moscow, 1953.
\({}^{2}\) N. I. Akhiezer, Lectures on Approximation Theory, Moscow–Leningrad, 1947.
\({}^{3}\) S. B. Stechkin, Izv. Acad. Sci. USSR, Ser. Math., 20, No. 3 (1956).
\({}^{4}\) N. I. Kuptsov, Mat. Sb., 40 (82), 2 (1956).
\({}^{5}\) E. A. Bredikhina, Mat. Sb., 50 (92), 3 (1960).
\({}^{6}\) E. A. Bredikhina, Mat. Sb., 56 (98), 1 (1962).
\({}^{7}\) E. A. Bredikhina, Dokl. Akad. Nauk, 145, No. 1 (1962).