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 a second of arc to the positions they once were in.
There are several inequivalent definitions of almost periodic functions.
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.
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.
The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner and others in the 1920s and 1930s.
Bohr (1925)[1] defined the uniformly almost-periodic functions as the closure of the trigonometric polynomials with respect to the uniform norm (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 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 {ƒ(t + Tn)} of translations of f has a subsequence that converges uniformly for t in (−∞, +∞).The Bohr almost periodic functions are essentially the same as continuous functions on the Bohr compactification of the reals.
The space Sp of Stepanov almost periodic functions (for p ≥ 1) was introduced by V.V.
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).
[3] It contains the space Sp of Stepanov almost periodic functions.
The Besicovitch almost periodic functions in B2 have an expansion (not necessarily convergent) as with Σa2n finite and λn real.
Conversely every such series is the expansion of some Besicovitch periodic function (which is not unique).
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.
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.
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
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 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).
, then the signal exactly satisfies or The Fourier series representation would be or where
Stated differently these functions of time are bandlimited to much less than the fundamental frequency for
, has the effect of detuning the partials from their exact integer harmonic value