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 ?
Then A is uniformly almost periodic family if it is uniformly bounded, equicontinuous and for every [epsilon] > 0 there exists a number l > 0 such that every interval of length l contains a number [tau] such that
Then the family [LAMBDA] is uniformly almost periodic if and only if V [member of] AP(R, R).
Wang, "Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales," Abstract and Applied Analysis, vol.
Wang, Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstr.
Agarwal, Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations, Adv.
(1) The set of sequences {t.sup.j.sub.k]}, [t.sup.j.sub.k] = [t.sub.k+j] - [t.sub.k], where k [member of] Z, j [member of] Z, and {[t.sub.k]} [member of] [??], is uniformly almost periodic.
(H6) The set of sequences {[t.sup.j.sub.k]}, [t.sup.j.sub.k] = [t.sub.k+j] - [t.sub.k], where k [member of] Z, j [member of] Z, and {[t.sub.k]} [member of] [??], is uniformly almost periodic and there exists [theta] > 0 such that [inf.sub.k[member of]Z][t.sup.l.sub.k] = [theta] > 0.
Further development, mathematically more challenging, would be a detailed description of the interspike-interval sequence for IF systems with an almost periodic input, for which a formal framework was prepared in [18] and [24]: Namely, we proved that in linear systems [??] = -[sigma]x + f(t) the almost periodic input function f (t) (in the sense of Stepanov, to include also not continuous input functions) induces the firing map with uniformly almost periodic displacement.
The set of sequences {[[tau].sup.j.sub.k]}, [[tau].sup.j.sub.k] = [[tau].sup.j.sub.k+j] - [[tau].sub.k], k [member of], k [member of] Z, j [member of] Z, {[[tau].sub.k]} [member of] I, is said to be uniformly almost periodic if for arbitrary [epsilon] > 0 there exists a relatively dense set of e-almost periods common for any sequences.
(1) The set of sequences {[[tau].sup.j.sub.k]}, [[tau].sup.j.sub.k] = [[tau].sup.j.sub.k+j] - [[tau].sub.k], k [member of] Z, j [member of] Z, {[[tau].sub.k]} [member of] I, is uniformly almost periodic.
f(n, x) is said to be almost periodic in n uniformly for x [member of] D, or uniformly almost periodic for short, if for any [epsilon] > 0 and any compact set S in D, there exists a positive integer l([epsilon], S) such that any interval of length l([epsilon], S) contains a integer [tau] for which
The set of sequences [mathematical expression not reproducible] is said to be uniformly almost periodic if, for arbitrary [epsilon] > 0, there exists a relatively dense set in R of [epsilon]-almost periodic common for all of the sequences; here [mathematical expression not reproducible].