Almost Periodic Functions: Their Limit Sets and Various Applications

Abstract

In the present paper, we introduce and study the limit sets of the almost periodic functions f: ${{\mathbb {R}}}\rightarrow {{\mathbb {C}}}$. It is interesting, that $r=\inf |f(x)|$ and $R=\sup |f(x)|$ may be expressed in exact form. In particular, the formula for r coincides with the well known partition problem formula. We show that the ring $r\le |z|\le R$ is the limit set of the almost periodic function f(x) (under some natural conditions on f). We obtain interesting applications to the partition problem in number theory and to trigonometric series theory. We extend classical results for periodic functions (estimation of Fourier coefficients and Bernstein’s theorem about absolute convergence of trigonometric series) to almost periodic functions. Using our results on almost periodic functions, we propose a new approach to interesting problems of the stability of motion. An application of our function theoretic results to the spectral problems in operator theory is studied as well. The results and corresponding computations are illustrated by several figures.

Appendices

Appendix: Examples and Figures

In the appendix, we consider various almost periodic functions and the corresponding parametric plots.

Example A.1

We begin with the case $n=2$.

In this case, the function f(x) has the form

$$\begin{aligned} f(x)=c_1e^{ix\lambda _1}+c_2e^{ix\lambda _2},\qquad c_k=|c_k|e^{i\alpha _k},\qquad \lambda _1\ne \lambda _2. \end{aligned}$$

(A.1)

Hence f(x) admits the representation $f(x)=A(x)+iB(x)$, where

$$\begin{aligned} A(x)= & {} |c_1|\cos (x\lambda _1+\alpha _1)+|c_2|\cos (x\lambda _2+\alpha _2), \end{aligned}$$

(A.2)

$$\begin{aligned} B(x)= & {} |c_1|\sin (x\lambda _1+\alpha _1)+|c_2|\sin (x\lambda _2+\alpha _2). \end{aligned}$$

(A.3)

Thus (see also (2.22)), we have

$$\begin{aligned} |f(x)|^{2}=A^{2}(x)+B^{2}(x)=|c_1|^{2}+|c_2|^{2}+2|c_{1}c_{2}|\cos [(\lambda _1-\lambda _2)x+\alpha _1- \alpha _2].\nonumber \\ \end{aligned}$$

(A.4)

Relations (A.2)–(A.4) imply the next proposition.

Proposition A.2

• (1) If $n=2$ and $x_k=(2k\pi -\alpha _1+\alpha _2)/(\lambda _1-\lambda _2)$, where k is integer, then $f(x_k)=e^{i(x_{k}\lambda _1+\alpha _1)}(|c_{1}|+|c_{2}|)$ and we obtain

  $$\begin{aligned} \sup |f(x)|=|c_{1}|+|c_{2}|. \end{aligned}$$

  (A.5)

• (2) If $n=2$ and ${\tilde{x}}_k=((2k+1)\pi -\alpha _1+\alpha _2)/(\lambda _1-\lambda _2)$, where k is integer, then $f({\tilde{x}}_k)=e^{i({\tilde{x}}_{k}\lambda _1+\alpha _1)}(|c_{1}|-|c_{2}|)$ and we obtain

  $$\begin{aligned} \inf |f(x)|=\big |\, |c_{1}|-|c_{2}|\, \big |. \end{aligned}$$

  (A.6)

Example A.3

In the special (periodic) case of $n=2$:

$$\begin{aligned} f(x)=e^{ix} +ae^{2ix}, \qquad a>0. \end{aligned}$$

(A.7)

Proposition A.2 yields

$$\begin{aligned} R=1+a,\qquad r=|1-a|. \end{aligned}$$

The function $f(x)=\xi +i \eta$ can be represented in the parametric form

$$\begin{aligned} \xi =\cos {x}+a\cos {2x},\qquad \eta =\sin {x}+a\sin {2x}. \end{aligned}$$

The corresponding parametric plot (for $a=2$, that is, for $r=1$ and $R=3$) is depicted in Fig. 1 (in this appendix). Several figures for the case (A.7) are given in [29, p. 104].

Example A.4

In the special case of $n=2$, where the numbers $\lambda _1$ and $\lambda _2$ are linearly independent over the field of rational numbers:

$$\begin{aligned} f(x)=e^{ix}+\frac{1}{2}e^{i\sqrt{2}x}, \qquad -\infty<x<\infty , \end{aligned}$$

(A.8)

Proposition A.2 yields

$$\begin{aligned} r=\frac{1}{2},\qquad R=\frac{3}{2}. \end{aligned}$$

The function $f(x)=\xi +i\eta$ is again represented in the parametric form

$$\begin{aligned} \xi =\cos (x)+\frac{1}{2}\cos \big (\sqrt{2}\, x\big ),\qquad \eta =\sin (x)+\frac{1}{2}\sin \big (\sqrt{2}\,x\big ). \end{aligned}$$

For the illustration of this parametric plot see Fig. 2.

Fig. 1

Example A.3

Fig. 2

Example A.4

Fig. 3

Example A.5

Fig. 4

Example A.6

Fig. 5

Case I (colour figure online)

Fig. 6

Case II

Fig. 7

Case III

Example A.5

In the case

$$\begin{aligned} f(x)=e^{ix}+e^{i\sqrt{2}x}, \qquad -\infty<x<\infty , \end{aligned}$$

(A.9)

we have

$$\begin{aligned} r=0,\qquad R=2. \end{aligned}$$

(A.10)

The function f(x) is represented in the parametric form

$$\begin{aligned} \xi =\cos (x)+\cos \big (\sqrt{2}\,x\big ),\qquad \eta =\sin (x)+\sin \big (\sqrt{2}\, x\big ). \end{aligned}$$

This parametric plot is depicted in Fig. 3.

Example A.6

Consider the case $n=3$, where f(x) is given by

$$\begin{aligned} f(x)=e^{ix}+e^{2ix}+\frac{1}{10}e^{i\sqrt{3}x}. \end{aligned}$$

This is an intermediate case because the function f(x) is not periodic, but the system $\lambda _1=1,\,\lambda _2=2,\,\lambda _3=\sqrt{3}$ is not linearly independent. Here, we have : 

$$\begin{aligned} R=2.1,\qquad r= 0.1. \end{aligned}$$

The function f(x) has the parametric form

$$\begin{aligned}&\xi =\cos ({x})+\cos ({2}x)+\frac{1}{10}\cos \big (\sqrt{3}x\big ), \\&\eta =\sin ({x})+\sin ({2}x)+\frac{1}{10}\sin \big (\sqrt{3}\,x\big ). \end{aligned}$$

This parametric plot is depicted in Fig. 4.

Stability of Motion

Let us shortly discuss a question of stability which may be modelled without resorting to the Hamiltonian treatment.

We consider the motion along an orbit which belongs to some fixed annulus. This motion is stable if the orbit of the perturbed motion belongs to this annulus too. The possible form of the perturbed orbits is illustrated in the example below.

Our definition of stability differs from the classical ones because we handle the orbits instead of the equations which generated these orbits.

Example B.1

Choose the original orbit $f(x)=e^{ix}+3e^{ix\lambda }$. Then, we may fix the corresponding annulus (1.3) such that $R_f=4$, $r_f=2$. Now, we consider the perturbed orbits of the form $g(x)=e^{ix}+3e^{ix\mu }$, where $\mu \ne \lambda$, $\mu \in {{\mathbb {R}}}$. Clearly, the perturbed orbits belong to the same annulus.

If the number $\mu$ is irrational, then the corresponding perturbed orbit passes through all the points of the corresponding annulus; we will call such an orbit essential. If the number $\mu$ is rational, then the corresponding perturbed orbit does not pass through all the points of the corresponding annulus; we will call such an orbit inessential. The essential orbits form an uncountable set (continuum set). The inessential orbits form a countable set. From the physical point of view the inessential orbits may be neglected. The division of the class of orbits into two subclasses (of essential and inessential orbits) is prompted by the KAM theory.

In our example, the motion along the orbit $f(x)=e^{ix}+3e^{ix\lambda }$ is stable with respect to the perturbed orbits of the form $g(x)=e^{ix}+3e^{ix\mu }$. It is not required that the perturbations should be small. Indeed, we have $\sup |f(x)-g(x)|=6$. The results of Sect. 2 enable us to study much more general unperturbed almost periodic orbits than $f(x)=e^{ix}+3e^{ix\lambda }$ and much more general perturbed cases than $g(x)=e^{ix}+3e^{ix\mu }$.

The plots are depicted for three cases. Figure 5 (Case I) depicts the unperturbed orbit $f(x)=e^{ix}+3e^{i\sqrt{2}\,x}$ and perturbed orbit $g(x)=e^{ix}+3e^{ 2 i x}$ (in red). Figure 6 (Case II) depicts the perturbed orbit $g(x)=e^{ix}+3e^{ 2 i x}$. Finally, Fig. 7 (Case III) depicts the perturbed orbit $g(x)=e^{ix}+3e^{i (7/5)x}$.