Almost Periodic Function

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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.

REFERENCES

Bohr, H. Pochti periodicheskie funktsii. Moscow-Leningrad, 1934. (Translated from German.)
Levitan, B. M. Pochti-periodicheskie funktsii. Moscow, 1953.
References in periodicals archive ?
They begin with such preliminaries as evolution equations and semigroups of linear operators and almost periodic functions.
Keywords: Pseudo almost periodic functions, almost periodic functions, exponentially stable evolution operator.
t]([theta]) = u(t + [theta]), [theta] [member of] R, f : R x X x PAP(X) [right arrow] X, PAP(X) is the set of all pseudo almost periodic functions from R to X and the family {A(t) : t [member of] R} of operators in X generates an exponentially stable evolution system {U(t, s), t [greater than or equal to] s}.
1999, Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain, Ann.
Besicovitch, Almost periodic functions, Dover, New York, 1954.
In this paper we develop a theory of almost periodic functions with values in the non-locally convex spaces called p-Frechet spaces, 0 < p < 1, including the [l.
Key words and phrases: p-Frechet space, almost periodic functions, functions with values in p-Frechet spaces, 0 < p < 1.
In the proof of existence, uniqueness and stability for almost periodic solutions of differential (partial differential) equations (or systems) basic properties of almost periodic functions are used, for instance the boundedness, the relatively compactness of the range, the property of normality and the property of closure with respect to uniform convergence.
Other applications of almost periodic functions are, for example, in Statistics, Number Theory, the Theory of Spectrum of Bounded Functions, the Theory of Semigroups of Bounded Linear Operators, Dynamical Systems and Ergodic Theory.
The theory of almost periodic functions was strongly extended to abstract spaces, see for example the monographs [6], [16], [17] for Banach space valued functions and [5], [16], [18] for complete locally convex (Frechet) space valued functions.
The purpose of this paper is to extend in Section 3 the main properties of almost periodic functions with values in Banach spaces, to the class of almost periodic functions with values in other important abstract spaces in Functional Analysis, namely the p-Frechet spaces, 0 < p < 1, which are non-locally convex spaces.
In this section, starting from a Bohr-kind definition for the almost periodicity, we develop a theory of almost periodic functions with values in a p-Frechet space, 0 < p < 1, similar to that for functions with values in Banach space.