View source for Almost periodic function

{{Short description|Function that "converges" to periodicity}}
{{Distinguish|Quasiperiodic function}}
{{inline|date=March 2025}}
In [[mathematics]], an '''almost periodic function''' is, loosely speaking, a [[function (mathematics)|function]] of a [[real number|real]] variable that is [[periodic function|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 group]]s, first studied by [[John von Neumann]].

'''Almost periodicity''' is a property of [[dynamical system]]s that appear to retrace their paths through [[phase space]], but not exactly. An example would be a [[planetary system]], with [[planet]]s in [[orbit]]s moving with [[Orbital period|period]]s that are not [[commensurability (mathematics)|commensurable]] (i.e., with a period vector that is not [[Proportionality (mathematics)|proportional]] to a vector of [[integer]]s). A [[Kronecker's theorem on diophantine approximation|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.

==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

:<math>e^{s\log n}\,</math>

with ''s'' written as ''σ''&nbsp;+&nbsp;''it'' &ndash; 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

:<math>n^\sigma e^{(\log n)it}.\,</math>

Taking a ''finite'' sum of such terms avoids difficulties of [[analytic continuation]] to the region σ&nbsp;<&nbsp;1. Here the "frequencies" log ''n'' will not all be commensurable (they are as linearly independent over the [[rational number]]s as the integers ''n'' are multiplicatively independent &ndash; 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 [[norm (mathematics)|norm]]s.

The theory was developed using other norms by [[Abram Samoilovitch Besicovitch|Besicovitch]], [[Vyacheslav Stepanov|Stepanov]], [[Hermann Weyl|Weyl]], [[John von Neumann|von Neumann]], [[Alan Turing|Turing]], [[Salomon Bochner|Bochner]] and others in the 1920s and 1930s.

==Definitions==

===Uniform or Bohr or Bochner  almost periodic functions===

Bohr (1925)<ref>H. Bohr,   "Zur Theorie der fastperiodischen Funktionen I"  Acta Math., 45  (1925)  pp.&nbsp;29–127</ref> defined the '''uniformly almost-periodic functions''' as the closure of the trigonometric polynomials with respect to the [[uniform norm]] 
:<math>\|f\|_\infty = \sup_x|f(x)|</math>
(on [[bounded function]]s ''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 [[mathematical proof|proved]] that this definition was equivalent to the existence of a [[relatively dense set]] of '''''ε''&nbsp;almost-periods''', for all ''ε''&nbsp;>&nbsp;0: that is, [[Translation (geometry)|translations]] ''T''(''ε'')&nbsp;=&nbsp;''T'' of the variable ''t''  making

:<math>\left|f(t+T)-f(t)\right|<\varepsilon.</math>

An alternative definition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to state:
<blockquote>A function ''f'' is almost periodic if every [[sequence]] {''&fnof;''(''t''&nbsp;+&nbsp;''T''<sub>''n''</sub>)} of translations of ''f'' has a [[subsequence]] that [[uniform convergence|converges uniformly]] for ''t'' in (&minus;&infin;,&nbsp;+&infin;).</blockquote>

The Bohr almost periodic functions are essentially the same as continuous functions on the [[Bohr compactification]] of the reals.

===Stepanov almost periodic functions===
The space ''S''<sup>''p''</sup> of Stepanov almost periodic functions (for ''p''&nbsp;≥&nbsp;1) was introduced by V.V. Stepanov (1925).<ref>W.  Stepanoff(=V.V. Stepanov),   "Sur quelques généralisations des fonctions presque périodiques"  C. R. Acad. Sci. Paris, 181  (1925)  pp.&nbsp;90–92; W.  Stepanoff(=V.V. Stepanov),   "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen"  Math. Ann., 45  (1925)  pp.&nbsp;473–498</ref> It contains the space of Bohr almost periodic functions. It is the closure of the trigonometric polynomials under the norm

:<math>\|f\|_{S,r,p}=\sup_x \left({1\over r}\int_x^{x+r} |f(s)|^p \, ds\right)^{1/p}</math>

for any fixed positive value of ''r''; for different values of ''r'' these norms give the same topology and so the same space of almost periodic functions (though the norm on this space depends on the choice of&nbsp;''r'').

===Weyl almost periodic functions===
The space ''W''<sup>''p''</sup> of Weyl almost periodic functions (for ''p''&nbsp;≥&nbsp;1) was introduced by Weyl (1927).<ref>H. Weyl,   "Integralgleichungen und fastperiodische Funktionen"  Math. Ann., 97  (1927)  pp.&nbsp;338–356</ref> It contains the space ''S''<sup>''p''</sup> of Stepanov almost periodic functions. 
It is the closure of the trigonometric polynomials under the seminorm

:<math>\|f\|_{W,p}=\lim_{r\to\infty}\|f\|_{S,r,p}</math>

Warning: there are nonzero functions ''&fnof;'' with ||''&fnof;''||<sub>''W'',''p''</sub>&nbsp;=&nbsp;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 ''B''<sup>''p''</sup> of Besicovitch almost periodic functions was introduced by Besicovitch (1926).<ref>A.S. Besicovitch,   "On generalized almost periodic functions"  Proc. London Math. Soc. (2), 25  (1926)  pp.&nbsp;495–512</ref>
It is the closure of the trigonometric polynomials under the seminorm

:<math>\|f\|_{B,p}=\limsup_{x \to\infty}\left({1\over 2x} \int_{-x}^x |f(s)|^p \, ds \right)^{1/p}</math>

Warning: there are nonzero functions ''&fnof;'' with ||''&fnof;''||<sub>B,''p''</sub>&nbsp;=&nbsp;0, such as any bounded function of compact support, so to get a Banach space one has to quotient out by these functions.

The Besicovitch almost periodic functions in ''B''<sup>2</sup> have an expansion (not necessarily convergent) as

:<math>\sum a_ne^{i\lambda_n t}</math>

with Σ''a''{{supsub|2|''n''}} finite and ''λ''<sub>''n''</sub> real. Conversely every such series is the expansion of some Besicovitch periodic function (which is not unique).

The space ''B''<sup>''p''</sup> of Besicovitch almost periodic functions (for ''p''&nbsp;≥&nbsp;1) contains the space ''W''<sup>''p''</sup> of Weyl almost periodic functions. If one quotients out a subspace of "null" functions, it can be identified with the space of ''L''<sup>''p''</sup> functions on the Bohr compactification of the reals.

===Almost periodic functions on a locally compact group===
With these theoretical developments and the advent of abstract methods (the [[Peter&ndash;Weyl theorem]], [[Pontryagin duality]] and [[Banach algebra]]s) 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''<sup>∞</sup>(''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&nbsp;''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&nbsp;''G''. More generally the Bohr compactification can be defined for any topological group&nbsp;''G'', and the spaces of continuous or ''L''<sup>''p''</sup> functions on the Bohr compactification can be considered as almost periodic functions on&nbsp;''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 <math>L^\infty</math>.

Given a topological dynamical system <math>(X,G)</math> consisting of a compact topological space ''X'' with an action of the locally compact group ''G'', a continuous function on ''X'' is (weakly) almost periodic if its orbit is (weakly) precompact in the Banach space <math>C(X)</math>.

== Quasiperiodic signals in audio and music synthesis ==

In [[speech processing]], [[audio signal processing]], and [[synthesizer|music synthesis]], a '''quasiperiodic''' signal, sometimes called a '''quasiharmonic''' signal, is a [[waveform]] that is virtually [[Frequency|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 [[Harmonic series (music)#Partial|partial]]s or [[overtone]]s are [[harmonic]] (that is all overtones are at frequencies that are an integer multiple of a [[fundamental frequency]] of the tone).

When a signal <math> x(t) \ </math> is '''fully periodic''' with period <math> P \ </math>, then the signal exactly satisfies

: <math> x(t) = x(t + P) \qquad \forall t \in \mathbb{R} </math>

or

: <math> \Big| x(t) - x(t + P) \Big| = 0 \qquad \forall t \in \mathbb{R}. \ </math>

The [[Fourier series]] representation would be

: <math>x(t) = a_0 + \sum_{n=1}^\infty \big[a_n\cos(2 \pi n f_0 t) - b_n\sin(2 \pi n f_0 t)\big]</math>

or

: <math>x(t) = a_0 + \sum_{n=1}^\infty r_n\cos(2 \pi n f_0 t + \varphi_n)</math>

where <math> f_0 = \frac{1}{P} </math> is the fundamental frequency and the Fourier coefficients are

:<math>a_0 = \frac{1}{P} \int_{t_0}^{t_0+P} x(t) \, dt  \ </math>

:<math>a_n = r_n \cos \left( \varphi_n \right) = \frac{2}{P} \int_{t_0}^{t_0+P} x(t) \cos(2 \pi n f_0 t) \, dt  \qquad n \ge 1 </math>

:<math>b_n = r_n \sin \left( \varphi_n \right)  = - \frac{2}{P} \int_{t_0}^{t_0+P} x(t) \sin(2 \pi n f_0 t) \, dt \ </math>

:where  <math> t_0 \ </math> can be any time:  <math> -\infty < t_0 < +\infty \ </math>.

The [[fundamental frequency]] <math> f_0 \ </math>, and Fourier [[coefficient]]s <math> a_n \ </math>,  <math> b_n \ </math>, <math> r_n \ </math>, or <math> \varphi_n \ </math>,  are constants, i.e. they are not functions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.

When <math> x(t) \ </math> is '''quasiperiodic''' then

: <math> x(t) \approx x \big( t + P(t) \big) \ </math>

or

: <math> \Big| x(t) - x \big( t + P(t) \big) \Big| < \varepsilon \ </math>

where

: <math> 0 < \epsilon \ll \big \Vert x \big \Vert = \sqrt{\overline{x^2}} = \sqrt{ \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} x^2(t)\, dt }. \ </math>

Now the Fourier series representation would be

: <math>x(t) = a_0(t) \ + \ \sum_{n=1}^\infty \left[a_n(t)\cos \left(2 \pi n \int_{0}^{t} f_0(\tau)\, d\tau \right) - b_n(t)\sin \left( 2 \pi n \int_0^t  f_0(\tau)\, d\tau \right) \right]</math>

or

: <math>x(t) = a_0(t) \ + \ \sum_{n=1}^\infty r_n(t)\cos \left( 2 \pi n \int_0^t f_0(\tau)\, d\tau + \varphi_n(t) \right) </math>

or

: <math>x(t) = a_0(t) + \sum_{n=1}^\infty r_n(t)\cos \left( 2 \pi \int_0^t f_n(\tau)\, d\tau + \varphi_n(0) \right)</math>

where <math> f_0(t) = \frac{1}{P(t)} </math> is the possibly ''time-varying'' fundamental frequency and the  ''time-varying'' Fourier coefficients are

:<math>a_0(t) = \frac{1}{P(t)} \int_{t-P(t)/2}^{t+P(t)/2} x(\tau) \, d\tau  \  </math>

:<math>a_n(t) = r_n(t) \cos\big(\varphi_n(t)\big) = \frac{2}{P(t)} \int_{t-P(t)/2}^{t+ P(t)/2} x(\tau) \cos\big( 2 \pi n f_0(t) \tau \big) \, d\tau  \qquad n \ge 1  </math>

:<math>b_n(t) = r_n(t) \sin\big(\varphi_n(t)\big) = -\frac{2}{P(t)} \int_{t-P(t)/2}^{t+P(t)/2} x(\tau) \sin\big( 2 \pi n f_0(t) \tau \big) \, d\tau  \  </math>

and the [[instantaneous phase#Instantaneous frequency|instantaneous frequency]] for each [[Harmonic series (music)#Partial|partial]] is

:<math> f_n(t) = n f_0(t) + \frac{1}{2 \pi} \varphi_n^\prime(t). \, </math>

Whereas in this quasiperiodic case, the fundamental frequency <math> f_0(t) \ </math>, the harmonic frequencies <math> f_n(t) \ </math>, and the Fourier coefficients <math> a_n(t) \ </math>,  <math> b_n(t) \ </math>,  <math> r_n(t) \ </math>, or <math> \varphi_n(t) \ </math> are '''not''' necessarily constant, and '''are''' functions of time albeit ''slowly varying'' functions of time.  Stated differently these functions of time are [[bandlimited]] to much less than the fundamental frequency for <math> x(t) \ </math> to be considered to be quasiperiodic.

The partial frequencies <math> f_n(t) \ </math> are very nearly harmonic but not necessarily exactly so.  The time-derivative of <math> \varphi_n(t) \ </math>, that is <math> \varphi_n^\prime(t) \ </math>, has the effect of detuning the partials from their exact integer harmonic value <math> n f_0(t) \ </math>.  A rapidly changing <math> \varphi_n(t) \ </math> means that the instantaneous frequency for that partial is severely detuned from the integer harmonic value which would mean that <math> x(t) \ </math> is not quasiperiodic.

==See also==
*[[Additive synthesis]]
*[[Aperiodic function]]
*[[Computer music]]
*[[Fourier series]]
*[[Harmonic series (music)]]
*[[Quasiperiodic function]]
*[[Quasiperiodic tiling]]

==References==
{{reflist}}

==Bibliography==
*{{Citation
 | last1 =Amerio
 | first1 =Luigi
 | author-link =Luigi Amerio
 | last2 =Prouse
 | first2 =Giovanni
 | author2-link =Giovanni Prouse
 | title =Almost-periodic functions and functional equations
 | place =New York–Cincinnati–Toronto–London–Melbourne
 | publisher =[[Van Nostrand Reinhold]]
 | series =[[The University Series in Higher Mathematics]]
 | year =1971
 | pages =viii+184
 | isbn =0-442-20295-4  
 | mr =275061
 | zbl =0215.15701
}}.
*A.S. Besicovitch,   "Almost periodic functions", Cambridge Univ. Press  (1932)
* {{citation | first=S.|last= Bochner | title=Beitrage zur Theorie der fastperiodischen Funktionen | journal=Math. Annalen | year=1926 | volume=96 | pages=119&ndash;147 | doi=10.1007/BF01209156 |s2cid= 118124462 }}
*S. Bochner and J. von Neumann, "Almost Periodic Function in a Group II", Trans. Amer. Math. Soc., 37 no. 1 (1935)  pp.&nbsp;21–50
* H. Bohr,   "Almost-periodic functions", Chelsea, reprint  (1947) 
*{{eom|title=Almost-periodic function|first=E.A.|last= Bredikhina}}
*{{eom|title=Besicovitch almost-periodic functions|first=E.A.|last= Bredikhina}}
*{{eom|title=Bohr almost-periodic functions|first=E.A.|last= Bredikhina}}
*{{eom|title=Stepanov almost-periodic functions|first=E.A.|last= Bredikhina}}
*{{eom|title=Weyl almost-periodic functions|first=E.A.|last= Bredikhina}}
*J. von Neumann, "Almost Periodic Functions in a Group I", Trans. Amer. Math. Soc., 36 no. 3 (1934)  pp.&nbsp;445–492

==External links==
*{{planetmath reference|urlname=AlmostPeriodicFunctionEquivalentDefinition|title=Almost periodic function (equivalent definition)}}

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[[Category:Complex analysis]]
[[Category:Digital signal processing]]
[[Category:Audio engineering]]
[[Category:Real analysis]]
[[Category:Topological groups]]
[[Category:Fourier analysis]]
[[Category:Types of functions]]