Almost Periodic Function

(redirected from Almost periodic)

Almost Periodic Function


a function whose value is approximately repeated when its argument is increased by properly selected constants (the almost periods). More precisely, a continuous function f(x) defined for all real values of 5 x is called almost periodic if for every > 0 there exists an l = l (∊) such that in every interval of length l on the x-axis at least one number τ = τ(∊) can be found for which the inequality ǀf (x + τ) – f(x)ǀ < ∊ is satisfied for all x. The numbers τ are called the almost periods of the function f(x). Periodic functions are special cases of almost periodic functions; simple examples of almost periodic functions that are not periodic can be obtained by adding periodic functions with incommensurable periods—for example, cos x + cos Almost Periodic Function.

The following are some important properties of almost periodic functions:

(1) An almost periodic function is bounded and uniformly continuous on the entire x-axis.

(2) The sum and product of a finite number of almost periodic functions is an almost periodic function.

(3) The limit of a uniformly convergent sequence of almost periodic functions is an almost periodic function.

(4) Every almost periodic function has a mean value (over the entire x-axis):

(5) To every almost periodic function we can associate a Fourier series:

where λ1, λ2, …, λn, … can be any sequence of distinct real numbers and

An = M {f (x)enx}

(6) The Parseval equality: for every almost periodic function

(7) Uniqueness theorem: if f(x) is a continuous almost periodic function and if for all real λ

M {f (x)eiλx} = 0

then f(x) ≡ 0. In other words, a Fourier series uniquely determines an almost periodic function.

(8) Approximation theorem: for every ∊ > 0, there exists a finite trigonometric polynomial

(where μκ is a real number) such that the inequality ǀf(x) – P (x)ǀ < ∊ is satisfied for all values of x; conversely, every function f(x) with this property is an almost periodic function.

The first construction of almost periodic functions was given by the Danish mathematician H. Bohr in 1923. Even earlier, in 1893, the Latvian mathematician P. Bohl studied a special case of almost periodic functions—quasi-periodic functions. A new construction of the theory of almost periodic functions was provided by N. N. Bogoliubov in 1930. The theory of almost periodic functions was generalized to include discontinuous functions first by V. V. Stepanov in 1925 and subsequently by H. Weyl and A. S. Besicovitch. A generalization of a different kind was given by the Soviet mathematician B. M. Levitan in 1938.


Bohr, H. Pochti periodicheskie funktsii. Moscow-Leningrad, 1934. (Translated from German.)
Levitan, B. M. Pochti-periodicheskie funktsii. Moscow, 1953.
References in periodicals archive ?
Their topics include friction laws in modeling dynamical systems, the smooth approximation of discontinuous stick-slip solutions, impacts in the chaotic motion of a particle on a non-flat billiard, almost periodic solutions for jumping discontinuous systems, and controlling stochastically excited systems with an approximate discontinuity.
Therefore, the stability problem for SICNNs has been one of the most active areas of research and there exist some results on the existence and stability of periodic and almost periodic solutions for the SICNNs with delays [2-11].
Objective: ResMet is dedicated to applications of resampling methods in statistical inference for the processes with periodic and almost periodic structures.
The theory of almost periodic functions has been developed in connection with problems of differential equations, stability theory, dynamical systems and many others.
The almost periodic conflicts in the south and sometimes wars between the north and the south were waged for ideological reasons, the President said during his meeting with chairman of the Arab Organization for Human Rights Raji Sourani.
Abstract: This paper presents an axiomatic approach for the construction of spaces of almost periodic functions (Poincare, Bohr, Besicovitch [2] and [3]; all these spaces are classical in the theory of almost periodic functions).
The authors give sharp conditions for exponential stability, which are suitable in the case that the coefficient function a(t) is periodic, almost periodic or asymptotically almost periodic, as often encountered in applications.
Infinite sequence A is almost periodic if and only if there is a function f such that for every natural n every word w of length n either doesn't occur in A or occurs in every fragment of A with length more than f(n).
They discuss the homogenization of almost periodic nonlinear parabolic operators, the boundary stabilization of a compactly coupled system of nonlinear wave equations, a discrete form of the backward heat problem on a plane, nonlinear kinetic equations for rigid spheres with weak Poisson coupling and diffusion, characterizing and generating local C-cosine and C-sine functions, Jacobian feedback loops analysis, the Laplace transform of functions with bounded averages, and other topics.
j](n)) are uniformly bounded by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the left hand side is an almost periodic function.
In this paper we use the theory of semigroup of bounded linear operators in a complex Banach space to establish the existence and uniqueness of a pseudo almost periodic mild solution of a retarded functional differential equation.