UDC 517.512

MATHEMATICS

E. A. BREDIKHINA

ON THE DIVERGENCE OF FOURIER SERIES OF ALMOST PERIODIC FUNCTIONS

(Presented by Academician A. N. Kolmogorov, 5 II 1966)

1. Denote by \(B^*\) the class of uniformly almost periodic (a.p.) functions \(f(x)\) whose Fourier exponents have a single limit point \(\lambda^*\). If \(\lambda^* \ne \infty\), then without loss of generality one may assume that \(\lambda^* = 0\), \(M\{f(x)\}=0\), since otherwise it would suffice to consider the function
\[ F(x)=f(x)e^{-i\lambda^*x}-M\{f(x)e^{-i\lambda^*x}\}. \]
Let \(\{\lambda_k\}\) \((k=1,2,\ldots;\ \lambda_k>0)\) be the monotone sequence formed by the absolute values of the Fourier exponents of the function \(f(x)\in B^*\); then the Fourier series of this function is naturally written in the form
\[ f(x)\sim \sum_{k=-\infty}^{\infty} A_k e^{i\lambda_k x} \quad (\lambda_0=0,\ \lambda_{-k}=-\lambda_k,\ |A_k|+|A_{-k}|>0\ \text{for }k>0). \tag{1} \]

We shall assign \(f(x)\in B^*\) to the class \(B_0^*\) if \(\lambda_k \downarrow 0\) and \(M\{f(x)\}=0\); we shall assign \(f(x)\in B^*\) to the class \(B_\infty^*\) if \(\lambda_k \uparrow \infty\). We shall say that \(f(x)\in B_0^*\) belongs to the class \(B_0^{*[p]}\) \((p=1,2,\ldots)\) if there exist functions \(f_0(x), f_1(x),\ldots, f_p(x)\) possessing the following properties: \(f_0(x)=f(x)\), \(f'_{m+1}(x)=f_m(x)\) \((m=0,1,2,\ldots,p-1)\), \(f_m(x)\in B_0^*\) \((m=0,1,2,\ldots,p)\). Put \(B_0^*=B_0^{*[0]}\); the inclusions
\[ B_0^{*[0]}\supset B_0^{*[1]}\supset B_0^{*[2]}\supset\cdots \]
are obvious. It is easy to see that a function \(f(x)\in B_0^{*[m]}\) belongs to the class \(B_0^{*[m+1]}\) if and only if the indefinite integral of the function
\[ f_m(x)\sim \sum_{k=-\infty}^{\infty}\frac{A_k}{(i\lambda_k)^m}e^{i\lambda_k x} \]
is bounded.

We shall assign \(f(x)\in B_\infty^*\) to the class \(B_\infty^{*(p)}\) if on the whole number axis there exists for \(f(x)\) a uniformly continuous derivative of order \(p\). Put \(B_\infty^*=B_\infty^{*(0)}\); the inclusions
\[ B_\infty^{*(0)}\supset B_\infty^{*(1)}\supset B_\infty^{*(2)}\supset\cdots \]
are obvious.

2. The following criteria for convergence of Fourier series of uniform a.p. functions of the class \(B^*\) hold.

Theorem A. Let \(f(x)\in B_\infty^*\), \(f(x)\in \operatorname{Lip}\alpha\) \((0<\alpha<1)\). If
\[ \lambda_n^{-\alpha}\ln\frac{\lambda_{n+1}+\lambda_n}{\lambda_{n+1}-\lambda_n}=o(1), \tag{2} \]
then the series (1) converges uniformly to \(f(x)\) on the whole real axis.

Theorem B. Let \(f(x)\in B_0^*\) and let there exist a constant \(C\) such that
\[ \left|\int_0^u f(x+w)\,dw\right|<C|u|^{1-\alpha} \quad (0<\alpha\le 1). \tag{3} \]

If

\[ \lambda_n^\alpha \ln \frac{\lambda_n+\lambda_{n+1}}{\lambda_n-\lambda_{n+1}}=o(1), \tag{2'} \]

then the series (1) converges uniformly to \(f(x)\) for all real \(x\).

The first of these theorems is due to S. Bochner \((^{1,2})\), the second to B. M. Levitan \((^{2,3})\).

In this note it is asserted (Theorems 1 and 2) that Theorem A and Theorem B for \(\alpha<1\) are final in the sense defined below; new final criteria (Theorems 3 and 4) are given for the uniform convergence of Fourier series of functions of the classes \(B_\infty^{*(p)}\), \(B_0^{*[p]}\).

1. Theorem 1. There exists a function \(f(x)\in B_\infty^*\), \(f(x)\in \operatorname{Lip}\alpha\) \((0<\alpha<1)\) such that

\[ \lambda_n^{-\alpha}\ln\frac{\lambda_{n+1}+\lambda_n}{\lambda_{n+1}-\lambda_n}=O(1) \tag{4} \]

and the Fourier series (1) of this function diverges at the point \(x=0\).

Theorem 2. There exists a function \(f(x)\in B_0^*\), satisfying condition (3) for \(0<\alpha<1\), for which

\[ \lambda_n^\alpha\ln\frac{\lambda_n+\lambda_{n+1}}{\lambda_n-\lambda_{n+1}}=O(1) \tag{4'} \]

and the Fourier series (1) diverges at every point of the real axis.

The proofs of Theorems 1 and 2 are based on the following constructions. Let \(\{\Lambda_n\}\) \((n=1,2,\ldots)\) be a monotone sequence of positive numbers; let the numbers \(\varepsilon_n>0\) \((n=1,2,\ldots)\) be chosen so that the intervals \((\Lambda_n-\varepsilon_n,\Lambda_n+\varepsilon_n)\) \((n=1,2,\ldots)\) do not intersect, and let \(\{N_n\}\) \((n=1,2,\ldots)\) be an increasing sequence of natural numbers. We introduce into consideration trigonometric polynomials analogous to Fejér polynomials \((^4)\):

\[ Q(x,n)=P(x,n)-\widetilde P(x,n), \]

where

\[ P(x,n)=\sum_{k=1}^{N_n}\frac1k\cos\left(\Lambda_n-k\frac{\varepsilon_n}{N_n}\right)x,\qquad \widetilde P(x,n)=\sum_{k=1}^{N_n}\frac1k\cos\left(\Lambda_n+k\frac{\varepsilon_n}{N_n}\right)x. \]

Lemma 1. For all values of \(x\) and \(n\),

\[ |Q(x,n)|<C, \tag{5} \]

where \(C\) is an absolute constant.

Lemma 2. If \(\Lambda_n\uparrow\infty\), then for all \(n\)

\[ P(0,n)>\ln N_n. \tag{6} \]

If \(\Lambda_n\downarrow0\), then for each \(x\) there exists \(M>0\) such that for \(n>M\)

\[ P(x,n)>\tfrac12\ln N_n. \tag{7} \]

Put

\[ f(x)=\sum_{n=1}^{\infty}\frac{Q(x,n)}{2^{n\alpha}}\qquad (0<\alpha<1). \tag{8} \]

In view of (5), \(f(x)\in B^*\), and the series (8) is the Fourier series of the function \(f(x)\); it is easy to see that this series can be written in the form (1) by first introducing in (8) a single enumeration of the frequencies (where the frequencies of the polynomials \(Q(x,n)\) are arranged in decreasing order when \(\Lambda_n\downarrow0\), and in increasing order when \(\Lambda_n\uparrow\infty\)).

Lemma 3. For \(\Lambda_n=2^n\), \(\varepsilon_n=2^{n-2}\), and any natural \(N_n\),

\[ f(x)\in \operatorname{Lip}\alpha. \]

Lemma 4. For \(\Lambda_n=2^{-n}\), \(\varepsilon_n=2^{-n-2}\), and any natural \(N_n\), estimate (3) holds, where \(C=C(\alpha)\) is a constant depending only on \(\alpha\).

Proof of Theorem 1. The conditions of the theorem are satisfied by the function \(f(x)\), defined by the series (8), for \(\Lambda_n=2^n\), \(\varepsilon_n=2^{n-2}\), \(N_n=[2^{n\alpha}]\). Indeed, by Lemma 3, \(f(x)\in \operatorname{Lip}\alpha\); for the sequence of its Fourier exponents

\[ \left\{\Lambda_n-k\frac{\varepsilon_n}{N_n}\right\}\quad (k=N_n,N_n-1,\ldots,1) \]

\[ \left\{\Lambda_n+k\frac{\varepsilon_n}{N_n}\right\}\quad (k=1,2,\ldots,N_n) \]

\[ (n=1,2,\ldots) \]

condition (4) is fulfilled; by (6),

\[ \lim_{n\to\infty} P(0,n)/2^{n\alpha}\ne 0, \]

therefore Cauchy’s convergence criterion is not fulfilled, and the Fourier series (1) of the function \(f(x)\) diverges at the point \(x=0\).

Proof of Theorem 2. The conditions of the theorem are satisfied by the function \(f(x)\), defined by the series (8), for \(\Lambda_n=2^{-n}\), \(\varepsilon_n=2^{-n-2}\), \(N_n=[2^{n\alpha}]\). Indeed, by Lemma 4, \(f(x)\) satisfies condition (3); for the sequence of its Fourier exponents

\[ \left\{\Lambda_n+k\frac{\varepsilon_n}{N_n}\right\}\quad (k=N_n,N_n-1,\ldots,1) \]

\[ \left\{\Lambda_n-k\frac{\varepsilon_n}{N_n}\right\}\quad (k=1,2,\ldots,N_n) \]

\[ (n=1,2,\ldots) \]

condition \((4')\) is fulfilled, and, by (7), the Fourier series (1) of the function \(f(x)\) diverges at every point of the real axis.

Thus, conditions (2) and \((2')\) in Theorems A and B (for \(\alpha<1\)) for the classes \(B_\infty^*\) and \(B_0^*\) in general cannot be weakened without imposing additional conditions on the structural properties of the function \(f(x)\in B^*\).

For \(\alpha=1\), Theorem B admits a refinement, for, by (3), \(f(x)\in B_0^{*[1]}\) and, by Theorems 2 and 5 of paper \({}^{6}\), uniform convergence of the series (1) will be ensured if in condition \((2')\) one replaces \(o(1)\) by \(O(1)\).

1. There are generalizations \({}^{5,6}\) of Theorems A and B that make it possible to use more fully both the behavior of the Fourier exponents of the function \(f(x)\in B^*\) and its structural properties. We give here two new convergence criteria, which are in fact contained in the results of papers \({}^{5,6}\) and which contain Theorems A and B.

A direct consequence of Theorem 6 of paper \({}^{5}\) is

Theorem 3. Let \(f(x)\in B_\infty^{*(p)}\) \((p=0,1,2,\ldots)\), \(f^{(p)}(x)\in \operatorname{Lip}\alpha\) \((0<\alpha<1)\). If

\[ \lambda_n^{-(p+\alpha)}\ln\frac{\lambda_{n+1}+\lambda_n}{\lambda_{n+1}-\lambda_n}=o(1), \tag{9} \]

then the series (1) converges uniformly to \(f(x)\) on the entire real axis.

From the corollary of Theorem 2 and Theorem 5 of paper \({}^{6}\) it follows that

Theorem 4. Let \(f(x)\in B_0^{*[p]}\) \((p=0,1,2,\ldots)\), and suppose there exists a constant \(C\) such that

\[ \left|\int_0^u f_p(x+u)\,du\right|<C|u|^{1-\alpha}\quad (0<\alpha\le 1). \]

If

\[ \lambda_n^{p+\alpha}\ln\frac{\lambda_n+\lambda_{n+1}}{\lambda_n-\lambda_{n+1}}=o(1), \tag{9'} \]

then the series (1) converges uniformly to \(f(x)\) for all real \(x\).

Theorems 3 and 4 are final in the same sense as Theorems A and B: in (9) and \((9')\) (for \(\alpha<1\)), generally speaking, one cannot replace \(o(1)\) by \(O(1)\). The proof of this assertion is carried out in the same way as ...

scheme, as in the proofs of Theorems 1 and 2, but requires more complicated preliminary constructions.

Let us note in conclusion that, in contrast to the periodic case, no improvement of the structural properties of the function \(f(x)\in B^*\) by itself ensures uniform convergence of the Fourier series (1). Thus, for example, the function \(f(x)\) defined by the series (8) with \(\Lambda_n=n\), \(\varepsilon_n=1/3\), and \(N_n=[2^{n^\alpha}]\), is infinitely differentiable on the entire real axis and has a Fourier series (1) diverging at the point \(x=0\).

The author expresses gratitude to P. L. Ulyanov for posing the problem and for his attention to the work.

Kuibyshev Aviation
Institute

Received
5 II 1966

References

1. S. Bochner, London Math. Soc., (2), 26, 433 (1926).
2. B. M. Levitan, Almost periodic functions, Moscow, 1953.
3. B. M. Levitan, Zap. Nauch.-issl. inst. matem. i mekh., Kharkov State Univ., 14, 105 (1937).
4. N. K. Bari, Trigonometric series, Moscow, 1961.
5. E. A. Bredikhina, Matem. sborn., 50 (92), 3, 369 (1960).
6. E. A. Bredikhina, Matem. sborn., 50 (98), 1, 59 (1962).