Almost periodic function
Encyclopedia
In mathematics
, an almost periodic function is, loosely speaking, a function of a real number 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 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 period
s 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 a second of arc to the positions they once were in.
An almost periodic function is a complex-valued function of a real variable that has the properties expected of a function on a phase space describing the time evolution of such a system. There have in fact been a number of definitions given, beginning with that of 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
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
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 norm
s.
The theory was developed using other norms by Besicovitch
, Stepanov
, Weyl
, von Neumann
, Turing
, Bochner
and others in the 1920s and 1930s.
(on continuous functions f on R). He proved that this definition was equivalent to the existence of a relatively dense set of ε almost-periods, for all ε > 0: that is, translation
s T(ε) = T of the variable t making
An alternative definition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to state:
The Bohr almost periodic functions are essentially the same as continuous functions on the Bohr compactification
of the reals.
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 r).
It is the closure of the trigonometric polynomials under the seminorm
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.
It is the closure of the trigonometric polynomials under the seminorm
Warning: there are nonzero functions ƒ with ||ƒ||B,p = 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 B2 have an expansion (not necessarily convergent) as
with Σ an2 finite and λn real. Conversely every such series is the expansion of some Besicovitch periodic function (which is not unique).
The space Bp of Besicovitch almost periodic functions (for p ≥ 1) contains the space Wp of Weyl almost periodic functions. If one quotients out a subspace of "null" functions, it can be identified with the space of Lp functions on the Bohr compactification of the reals.
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∞(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.
, 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
in the sense of the Wikipedia article of that name, 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 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
is fully periodic with period 
, then the signal exactly satisfies
or
The Fourier series
representation would be
or
where
is the fundamental frequency and the Fourier coefficients are
The fundamental frequency
, and Fourier coefficient
s
, 
, 
, or 
, are constant, not functions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.
When
is quasiperiodic then
or
where
Now the Fourier series representation would be
or
or
where
is the possibly time-varying fundamental frequency and the Fourier coefficients are
and the instantaneous frequency for each partial
is
Whereas in this quasiperiodic case, the fundamental frequency
, the harmonic frequencies 
, and the Fourier coefficients 
, 
, 
, or 
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
to be considered to be quasiperiodic.
The partial frequencies
are very nearly harmonic but not necessarily exactly so. The time-derivative of 
, that is 
, has the effect of detuning the partials from their exact integer harmonic value 
. A rapidly changing 
means that the instantaneous frequency for that partial is severely detuned from the integer harmonic value which would mean that 
is not quasiperiodic.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an almost periodic function is, loosely speaking, a function of a real number that is periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr
Harald Bohr
Harald August Bohr was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr...
and later generalized by Vyacheslav Stepanov, Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
and Abram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch was a Russian mathematician, who worked mainly in England. He was born in Berdyansk on the Sea of Azov to a Karaite family.-Life and career:...
, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
.
Almost periodicity is a property of dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s that appear to retrace their paths through phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, but not exactly. An example would be a planetary system
Planetary system
A planetary system consists of the various non-stellar objects orbiting a star such as planets, dwarf planets , asteroids, meteoroids, comets, and cosmic dust...
, with planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s in orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s moving with period
Orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...
s that are not commensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...
(i.e., with a period vector that is not proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
to a vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of integers). A theorem of Kronecker from diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
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 a second of arc to the positions they once were in.
Definition and properties
There are several inequivalent definitions of almost periodic functions.An almost periodic function is a complex-valued function of a real variable that has the properties expected of a function on a phase space describing the time evolution of such a system. There have in fact been a number of definitions given, beginning with that of Harald Bohr
Harald Bohr
Harald August Bohr was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels 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
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
Taking a finite sum of such terms avoids difficulties of analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
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
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...
with independent frequencies, mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
was applied to discuss the closure of this set of basic functions, in various norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
s.
The theory was developed using other norms by Besicovitch
Abram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch was a Russian mathematician, who worked mainly in England. He was born in Berdyansk on the Sea of Azov to a Karaite family.-Life and career:...
, Stepanov
Stepanov
Stepanov is a common Russian and Serbian surname and may refer to the following people:*Aleksei Stepanov , Russian footballer*Aleksandr Stepanov , several people...
, Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
, Bochner
Salomon Bochner
Salomon Bochner was an American mathematician of Austrian-Hungarian origin, known for wide-ranging work in mathematical analysis, probability theory and differential geometry.- Life :...
and others in the 1920s and 1930s.
Uniform or Bohr or Bochner almost periodic functions
defined the uniformly almost-periodic functions as the closure of the trigonometric polynomials with respect to the uniform norm(on continuous functions f on R). He proved that this definition was equivalent to the existence of a relatively dense set of ε almost-periods, for all ε > 0: that is, translation
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...
s T(ε) = T of the variable t making
An alternative definition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to state:
A function f is almost periodic if every sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
{ƒ(t + Tn)} of translations of f has a subsequenceSubsequenceIn mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
that converges uniformly for t in (−∞, +∞).
The Bohr almost periodic functions are essentially the same as continuous functions on the Bohr compactification
Bohr compactification
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H...
of the reals.
Stepanov almost periodic functions
The space Sp of Stepanov almost periodic functions (for p ≥ 1) was introduced by V.V. . It contains the space of Bohr almost periodic functions. It is the closure of the trigonometric polynomials under the normfor 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 r).
Weyl almost periodic functions
The space Wp of Weyl almost periodic functions (for p ≥ 1) was introduced by . It contains the space Sp of Stepanov almost periodic functions.It is the closure of the trigonometric polynomials under the seminorm
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 .It is the closure of the trigonometric polynomials under the seminorm
Warning: there are nonzero functions ƒ with ||ƒ||B,p = 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 B2 have an expansion (not necessarily convergent) as
with Σ an2 finite and λn real. Conversely every such series is the expansion of some Besicovitch periodic function (which is not unique).
The space Bp of Besicovitch almost periodic functions (for p ≥ 1) contains the space Wp of Weyl almost periodic functions. If one quotients out a subspace of "null" functions, it can be identified with the space of Lp functions on the Bohr compactification of the reals.
Almost periodic functions on a locally compact abelian group
With these theoretical developments and the advent of abstract methods (the Peter–Weyl theorem, Pontryagin dualityPontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
and Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
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∞(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
Bohr compactification
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H...
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.
Quasiperiodic signals in audio and music synthesis
In speech processingSpeech processing
Speech processing is the study of speech signals and the processing methods of these signals.The signals are usually processed in a digital representation, so speech processing can be regarded as a special case of digital signal processing, applied to speech signal.It is also closely tied to...
, audio signal processing
Audio signal processing
Audio signal processing, sometimes referred to as audio processing, is the intentional alteration of auditory signals, or sound. As audio signals may be electronically represented in either digital or analog format, signal processing may occur in either domain...
, and music synthesis
Synthesizer
A synthesizer is an electronic instrument capable of producing sounds by generating electrical signals of different frequencies. These electrical signals are played through a loudspeaker or set of headphones...
, a quasiperiodic signal, sometimes called a quasiharmonic signal, is a waveform
Waveform
Waveform means the shape and form of a signal such as a wave moving in a physical medium or an abstract representation.In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph...
that is virtually periodic
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
microscopically, but not necessarily periodic macroscopically. This does not give a quasiperiodic function
Quasiperiodic function
In mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition.A simple case is if the function obeys the equation:...
in the sense of the Wikipedia article of that name, 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 overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
s are harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
(that is all overtones are at frequencies that are an integer multiple of a fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
of the tone).
When a signal
is fully periodic with period , then the signal exactly satisfies
or
The Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
representation would be
or
where
is the fundamental frequency and the Fourier coefficients are- where
can be any time: 
.
The fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
, and Fourier coefficientCoefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s
, , , or , are constant, not functions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.When
is quasiperiodic then
or
where
Now the Fourier series representation would be
or
or
where
is the possibly time-varying fundamental frequency and the Fourier coefficients areand the instantaneous frequency for each partial
Partial
Partial may refer to:*partial derivative, in mathematics** ∂, the partial derivative symbol, often read as "partial"*partial function, in mathematics*partial algorithm, in computer science*part score, in contract bridge...
is
Whereas in this quasiperiodic case, the fundamental frequency
, the harmonic frequencies , and the Fourier coefficients , , , or are not necessarily constant, and are functions of time albeit slowly varying functions of time. Stated differently these functions of time are bandlimitedBandlimited
Bandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...
to much less than the fundamental frequency for
to be considered to be quasiperiodic.The partial frequencies
are very nearly harmonic but not necessarily exactly so. The time-derivative of , that is , has the effect of detuning the partials from their exact integer harmonic value . A rapidly changing means that the instantaneous frequency for that partial is severely detuned from the integer harmonic value which would mean that is not quasiperiodic.See also
- Quasiperiodic functionQuasiperiodic functionIn mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition.A simple case is if the function obeys the equation:...
- Aperiodic function
- Quasiperiodic tilingQuasiperiodic tilingA quasiperiodic tiling is a tiling of the plane that exhibits local periodicity under some transformations; we can slide or rotate it such that a finite number of tiles overlap perfectly, yet the entire tiling will not.See...
- Fourier seriesFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
- Additive synthesisAdditive synthesisAdditive synthesis is a technique of sound synthesis that creates musical timbre by explicitly adding sinusoidal overtones together.The timbre of an instrument is composed of multiple harmonic or inharmonic partials , of different frequencies and amplitudes, that change over time...
- Harmonic series (music)Harmonic series (music)Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
- Computer musicComputer musicComputer music is a term that was originally used within academia to describe a field of study relating to the applications of computing technology in music composition; particularly that stemming from the Western art music tradition...