# Almost Periodic Function

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 .

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 ?
The topics include a unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals, the well-posedness of solutions with finite energy for nonlocal equations of porous medium type, the decay of almost periodic solutions of anisotropic degenerate parabolic-hyperbolic equations, dispersion estimates for spherical Schr|ding equations with critical angular momentum, and Helge Holden's seven guidelines for scientific computing and development of open-source community software.
The almost periodic function introduced seminally by Bohr in 1925 plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, and ecosphere.
Gorbacuk, Pilipovic, and Taguchi's results have been applied in almost periodic ultradistributions of Roumieu type and Beurling type [13,14] and signal analysis [15].
Almost periodic and asymptotically almost periodic solutions of differential equations in Banach spaces have been considered by many authors so far (for the basic information on the subject, we refer the reader to the monographs [1-10]).
Almost periodic functions, which are an important generalization of periodic functions, were introduced into the field of mathematics by Bohr [1, 2].
Agarwal, "Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales," Communications in Nonlinear Science and Numerical Simulation, vol.
Up to present, several authors in [3,5,28] have researched the exponential extinction, permanence and existence of positive periodic and almost periodic solutions for the following delayed Nicholson's blowflies system with nonlinear density-dependent mortality terms and patch structure:
Almost periodic, asymptotically almost periodic, and pseudoalmost periodic solutions for differential, and difference equations arise naturally in biology, economics and physics [9, 10, 11, 12, 13].
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].
Dos Santos, "Asymptotically almost periodic and almost periodic solutions for a class of partial integrodifferential equations," Electronic Journal of Differential Equations, vol.

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