In mathematics, an almost periodic function is, loosely speaking, a function of a real variable that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.
Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within, say, a second of arc to the positions they once were in.
Contents
Motivation
There are several inequivalent definitions of almost periodic functions. The first was given by Harald Bohr. His interest was initially in finite Dirichlet series. In fact by truncating the series for the Riemann zeta function ζ(s) to make it finite, one gets finite sums of terms of the type
e
s
log
n
{\displaystyle e^{s\log n}\,}
with s written as σ + it – the sum of its real part σ and imaginary part it. Fixing σ, so restricting attention to a single vertical line in the complex plane, we can see this also as
n
σ
e
(
log
n
)
i
t
.
{\displaystyle n^{\sigma }e^{(\log n)it}.\,}
Taking a finite sum of such terms avoids difficulties of analytic continuation to the region σ < 1. Here the "frequencies" log n will not all be commensurable (they are as linearly independent over the rational numbers as the integers n are multiplicatively independent – which comes down to their prime factorizations).
With this initial motivation to consider types of trigonometric polynomial with independent frequencies, mathematical analysis was applied to discuss the closure of this set of basic functions, in various norms.
Definitions
Uniform or Bohr or Bochner almost periodic functions
Bohr (1925) defined the uniformly almost-periodic functions as the closure of the trigonometric polynomials with respect to the uniform norm
‖
f
‖
∞
=
sup
x
|
f
(
x
)
|
{\displaystyle \|f\|_{\infty }=\sup _{x}|f(x)|}
(on bounded functions f on R). In other words, a function f is uniformly almost periodic if for every ε > 0 there is a finite linear combination of sine and cosine waves that is of distance less than ε from f with respect to the uniform norm. The sine and cosine frequencies can be arbitrary real numbers. Bohr proved that this definition was equivalent to the existence of a relatively dense set of ε almost-periods, for all ε > 0: that is, translations T(ε) = T of the variable t making
|
f
(
t
+
T
)
−
f
(
t
Stepanov almost periodic functions
The space Sp of Stepanov almost periodic functions (for p ≥ 1) was introduced by V.V. Stepanov (1925). It contains the space of Bohr almost periodic functions. It is the closure of the trigonometric polynomials under the norm
‖
f
‖
S
,
r
,
p
=
sup
x
(
1
r
∫
x
x
+
r
|
f
(
s
)
|
p
d
s
)
1
/
p
{\displaystyle \|f\|_{S,r,p}=\sup _{x}\left({1 \over r}\int _{x}^{x+r}|f(s)|^{p}\,ds\right)^{1/p}}
Weyl almost periodic functions
The space Wp of Weyl almost periodic functions (for p ≥ 1) was introduced by Weyl (1927). It contains the space Sp of Stepanov almost periodic functions.
It is the closure of the trigonometric polynomials under the seminorm
‖
f
‖
W
,
p
=
lim
r
→
∞
‖
f
‖
S
,
r
,
p
{\displaystyle \|f\|_{W,p}=\lim _{r\to \infty }\|f\|_{S,r,p}}
Warning: there are nonzero functions ƒ with ||ƒ||W,p = 0, such as any bounded function of compact support, so to get a Banach space one has to quotient out by these functions.
Besicovitch almost periodic functions
The space Bp of Besicovitch almost periodic functions was introduced by Besicovitch (1926).
It is the closure of the trigonometric polynomials under the seminorm
‖
f
‖
B
,
p
=
lim sup
x
→
∞
(
1
2
x
∫
−
x
x
|
f
(
s
)
|
p
d
s
)
1
/
p
Almost periodic functions on a locally compact group
With these theoretical developments and the advent of abstract methods (the Peter–Weyl theorem, Pontryagin duality and Banach algebras) a general theory became possible. The general idea of almost-periodicity in relation to a locally compact abelian group G becomes that of a function F in L∞(G), such that its translates by G form a relatively compact set.
Equivalently, the space of almost periodic functions is the norm closure of the finite linear combinations of characters of G. If G is compact the almost periodic functions are the same as the continuous functions.
The Bohr compactification of G is the compact abelian group of all possibly discontinuous characters of the dual group of G, and is a compact group containing G as a dense subgroup. The space of uniform almost periodic functions on G can be identified with the space of all continuous functions on the Bohr compactification of G. More generally the Bohr compactification can be defined for any topological group G, and the spaces of continuous or Lp functions on the Bohr compactification can be considered as almost periodic functions on G.
For locally compact connected groups G the map from G to its Bohr compactification is injective if and only if G is a central extension of a compact group, or equivalently the product of a compact group and a finite-dimensional vector space.
A function on a locally compact group is called weakly almost periodic if its orbit is weakly relatively compact in
L
∞
{\displaystyle L^{\infty }}
.
Given a topological dynamical system
(
X
,
Quasiperiodic signals in audio and music synthesis
In speech processing, audio signal processing, and music synthesis, a quasiperiodic signal, sometimes called a quasiharmonic signal, is a waveform that is virtually periodic microscopically, but not necessarily periodic macroscopically. This does not give a quasiperiodic function, but something more akin to an almost periodic function, being a nearly periodic function where any one period is virtually identical to its adjacent periods but not necessarily similar to periods much farther away in time. This is the case for musical tones (after the initial attack transient) where all partials or overtones are harmonic (that is all overtones are at frequencies that are an integer multiple of a fundamental frequency of the tone).
When a signal
x
(
t
)
{\displaystyle x(t)\ }
is fully periodic with period
P
{\displaystyle P\ }
, then the signal exactly satisfies
x
(
t
)
=
x
(
t
+
P
)
∀
t