Final properties of integrals of quasiperiodic functions related to the problem of small divisors. (English. Russian original) Zbl 0782.42008

Moshchevitin, N. G.

Mosc. Univ. Math. Bull. 43, No. 5, 80-83 (1988); translation from Vestn. Mosk. Univ., Ser. I 1988, No. 5, 94-96 (1988).

From the text: H. Poincaré posed the problem of the behavior of the integral \(I(t)= \int^t_0 f(\omega_1t,\ldots,\omega_rt)\,dt\), where the function \(f(x_1,\ldots,x_r)\) is defined on a torus \(T^r\), \(\int_{T^r} f(x_1,\ldots,x_r)\,dx_1\ldots dx_r=0\), and \(\omega_1,\ldots,\omega_r \in \mathbb{R}\) are incommensurable. Of special interest is the problem of zeros of the function \(I(t)\). Suppose that \(f(0,\ldots,0) \ne 0\). It has been established that if the function \(f(x_1, \ldots,x_r)\) is a trigonometric polynomial, then \(I(t)\) has an infinite number of zeros as \(t \to \infty\). In addition, if the linear form \(m_1 \omega_1+\ldots+m_r \omega_r\) allows a lower estimate on a certain set of vectors \((\omega_1,\ldots,\omega_r)\), then for vectors from this set, given a sufficient smoothness of the function \(f(x_1,\ldots,x_r)\), a similar statement about the zeros of \(I(t)\) is true.
V. V. Kozlov [Methods of qualitative analysis in the dynamics of a rigid body. Moskva: Izdat. Mosk. Univ. (1980; Zbl 0557.70009)] obtained a theorem on the zeros for \(r=2\) for any incommensurable \(\omega_1\), \(\omega_2\) and \(f(x_1,x_2) \in C^2(T^2)\). E. A. Sidorov [Usp. Mat. Nauk 34, No. 6(210), 184–187 (1979; Zbl 0433.54033)] extended this result to functions absolutely continuous on \(T^2\). The classical Poincaré examples show that for continuous functions the theorem on zeros does not generally hold. Kozlov’s hypothesis on the zeros of the integral \(I(t)\) for the case of a smooth function \(f\) with zero mean and for an incommensurable set of frequencies has not yet been proved.
In the present paper we obtain a theorem on the zeros for a class of vectors \((\omega_1,\ldots,\omega_r)\), that does not admit a lower estimate of the form \(m_1\omega_1+\ldots+m_r \omega_r\), where the function \(f(x_1,\ldots,x_r)\) is of a certain smoothness.

MSC:

42A75	Classical almost periodic functions, mean periodic functions
42B99	Harmonic analysis in several variables
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H05	Hamilton’s equations

Keywords:

integrals of quasiperiodic functions; small divisors; trigonometric polynomial; smoothness; zeros

Citations:

Zbl 0557.70009; Zbl 0433.54033