In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the ...
and later generalized by
Vyacheslav Stepanov
Vyacheslav Vassilievich Stepanov (Вячеслав Васильевич Степанов; 4 September 1889, Smolensk – 22 July 1950, Moscow) was a mathematician, specializing in analysis. He was from the Soviet Union.
Stepanov was the son o ...
,
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is a ...
and
Abram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch (or Besikovitch) (russian: link=no, Абра́м Само́йлович Безико́вич; 23 January 1891 – 2 November 1970) was a Russian mathematician, who worked mainly in England. He was born in Berdyansk ...
, amongst others. There is also a notion of almost periodic functions on
locally compact abelian group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the ...
s, first studied by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
.
Almost periodicity is a property of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s that appear to retrace their paths through
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
, but not exactly. An example would be a
planetary system
A planetary system is a set of gravitationally bound non- stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems may also con ...
, with
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a yo ...
s in
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
s moving with
periods that are not
commensurable (i.e., with a period vector that is not
proportional to a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
of
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
). A
theorem of Kronecker from
diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by ...
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
A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...
to the positions they once were in.
Motivation
There are several inequivalent definitions of almost periodic functions. The first was given by
Harald Bohr
Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the ...
. His interest was initially in finite
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyt ...
. In fact by truncating the series for the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
''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
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
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 In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The ...
with independent frequencies,
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied i ...
was applied to discuss the closure of this set of basic functions, in various
norms.
The theory was developed using other norms by
Besicovitch,
Stepanov,
Weyl,
von Neumann Von Neumann may refer to:
* John von Neumann (1903–1957), a Hungarian American mathematician
* Von Neumann family
* Von Neumann (surname), a German surname
* Von Neumann (crater), a lunar impact crater
See also
* Von Neumann algebra
* Von Neu ...
,
Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
,
Bochner and others in the 1920s and 1930s.
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
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when t ...
:
(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. 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
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
''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 sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
of translations of ''f'' has a subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
that converges uniformly
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
for ''t'' in (−∞, +∞).
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''
''p'' 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
:
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'').
Weyl almost periodic functions
The space ''W''
''p'' of Weyl almost periodic functions (for ''p'' ≥ 1) was introduced by Weyl (1927). It contains the space ''S''
''p'' 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 ''B''
''p'' of Besicovitch almost periodic functions was introduced by Besicovitch (1926).
[A.S. Besicovitch, "On generalized almost periodic functions" Proc. London Math. Soc. (2), 25 (1926) pp. 495–512]
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 ''B''
2 have an expansion (not necessarily convergent) as
:
with Σ''a'' finite and ''λ''
''n'' real. Conversely every such series is the expansion of some Besicovitch periodic function (which is not unique).
The space ''B''
''p'' of Besicovitch almost periodic functions (for ''p'' ≥ 1) contains the space ''W''
''p'' of Weyl almost periodic functions. If one quotients out a subspace of "null" functions, it can be identified with the space of ''L''
''p'' 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
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
,
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), t ...
and
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 (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
s) a general theory became possible. The general idea of almost-periodicity in relation to a
locally compact abelian group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the ...
''G'' becomes that of a function ''F'' in ''L''
∞(''G''), such that its translates by ''G'' form a
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
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 ''L''
''p'' 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 processing
Speech processing is the study of speech signals and the processing methods of 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 ...
,
audio signal processing
Audio signal processing is a subfield of signal processing that is concerned with the electronic manipulation of audio signals. Audio signals are electronic representations of sound waves— longitudinal waves which travel through air, consist ...
, and
music synthesis, a quasiperiodic signal, sometimes called a quasiharmonic signal, is a
waveform
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electroni ...
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
An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s are
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic' ...
(that is all overtones are at frequencies that are an integer multiple of a
fundamental frequency
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. ...
of the tone).
When a signal
is fully periodic with period
, then the signal exactly satisfies
:
or
:
The
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
representation would be
: