Almost periodic function

In mathematics, almost periodic functions are functions of a real number that are periodic up to a small error, first studied by Harald Bohr. There are generalizations to almost periodic functions on locally compact abelian groups.

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.

Definition and properties

There are several different 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. 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

$e^\{(sigma+it)log\; n\},$

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

$n^sigma\; e^\{(log\; n)it\}.,$

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.

The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner 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

$||f||\_infty\; =\; sup\_x|f(x)|$

(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, 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 {f(t_n+T)} 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.

Stepanov almost periodic functions

The space S^p 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 norm

$||f||\_\{S,r,p\}=sup\_x\; left(\{1over\; r\}int\_x^\{x+r\}\; |f(x)|^p\; dxright)^\{1/p\}$

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 . It contains the space S^p of Stepanov almost periodic functions. It is the closure of the trigonometric polynomials under the seminorm

$||f||\_\{W,p\}=lim\_\{rmapstoinfty\}||f||\_\{S,r,p\}$

Warning: there are nonzero functions f with ||f||=0, such as any bounded function of compact support, so to get a Banach space one has to quotient out by these functions.

Besikovitch almost periodic functions

The space B^p of Besicovitch almost period functions was introduced by . It is the closure of the trigonometric polynomials under the seminorm

$||f||\_\{B,p\}=limsup\_\{xmapstoinfty\}left(\{1over\; 2x\}int\_\{-x\}^\{x\}\; |f(x)|^p\; dxright)^\{1/p\}$

Warning: there are nonzero functions f with ||f||_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 Besikovitch almost periodic functions in B²have an expansion (not necessarily convergent) as

$sum\; a\_ne^\{ilambda\_n\}$

with Σ a_n² finite and λ_n real. Conversely every such series is the expansion of some Besivovitch periodic function (which is not unique).

The space B^p of Besikovitch 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, 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 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, 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. It should be noted that 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 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 $x(t)$ is fully periodic with period $T$, then the signal exactly satisfies

$x(t)\; =\; x(t\; +\; T)$

or

$left|\; x(t)\; -\; x(t\; +\; T)\; right|\; =\; 0$ for all $t$.

The Fourier series representation would be

$x(t)\; =\; frac\{1\}\{2\}a\_0\; +\; sum\_\{n=1\}^inftyleft[a\_ncos(2\; pi\; n\; f\_0\; t)\; -\; b\_nsin(2\; pi\; n\; f\_0\; t)right]$

or

$x(t)\; =\; frac\{1\}\{2\}a\_0\; +\; sum\_\{n=1\}^inftyleft[r\_ncos(2\; pi\; n\; f\_0\; t\; +\; varphi\_n)right]$

where $f\_0\; =\; frac\{1\}\{T\}$ is the fundamental frequency and the Fourier coefficients are

$a\_n\; =\; r\_n\; cos\; left(varphi\_n\; right)\; =\; frac\{2\}\{T\}\; int\_\{t\_0\}^\{t\_0+T\}\; x(t)\; cos(2\; pi\; n\; f\_0\; t)\; dt$

$b\_n\; =\; r\_n\; sin\; left(varphi\_n\; right)\; =\; -\; frac\{2\}\{T\}\; int\_\{t\_0\}^\{t\_0+T\}\; x(t)\; sin(2\; pi\; n\; f\_0\; t)\; dt$

where $t\_0$ can be any time: .

The fundamental frequency $f\_0$, and Fourier coefficients $a\_n$, $b\_n$, $r\_n$, or $varphi\_n$, are constant, not functions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.

When $x(t)$ is quasiperiodic then

$x(t)\; approx\; x\; left(t\; +\; T(t)\; right)$

or

where

Now the Fourier series representation would be

$x(t)\; =\; frac\{1\}\{2\}a\_0(t)\; +\; sum\_\{n=1\}^inftyleft[a\_n(t)cos\; left(2\; pi\; n\; int\_\{0\}^\{t\}\; f\_0(tau),\; dtau\; right)\; -\; b\_n(t)sin\; left(2\; pi\; n\; int\_\{0\}^\{t\}\; f\_0(tau),\; dtau\; right)\; right]$

or

$x(t)\; =\; frac\{1\}\{2\}a\_0(t)\; +\; sum\_\{n=1\}^inftyleft[r\_n(t)cos\; left(2\; pi\; n\; int\_\{0\}^\{t\}\; f\_0(tau),\; dtau\; +\; varphi\_n(t)\; right)\; right]$

or

$x(t)\; =\; frac\{1\}\{2\}a\_0(t)\; +\; sum\_\{n=1\}^inftyleft[r\_n(t)cos\; left(2\; pi\; int\_\{0\}^\{t\}\; f\_n(tau),\; dtau\; +\; varphi\_n(0)\; right)\; right]$

where $f\_0(t)\; =\; frac\{1\}\{T(t)\}$ is the possibly time-varying fundamental frequency and the Fourier coefficients are

$a\_n(t)\; =\; r\_n(t)\; cos\; left(varphi\_n(t)\; right)$

$b\_n(t)\; =\; r\_n(t)\; sin\; left(varphi\_n(t)\; right)$

and the instantaneous frequency for each partial is

$f\_n(t)\; =\; n\; f\_0(t)\; +\; frac\{1\}\{2\; pi\}\; varphi\_n^prime(t)$.

Whereas in this quasiperiodic case, the fundamental frequency $f\_0(t)$, the harmonic frequencies $f\_n(t)$, and the Fourier coefficients $a\_n(t)$, $b\_n(t)$, $r\_n(t)$, or $varphi\_n(t)$ 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 $x(t)$ to be considered to be quasiperiodic.

The partial frequencies $f\_n(t)$ are very nearly harmonic but not necessarily exactly so. The time-derivative of $varphi\_n(t)$, that is $varphi\_n^prime(t)$, has the effect of detuning the partials from their exact integer harmonic value $n\; f\_0(t)$. A rapidly changing $varphi\_n(t)$ means that the instantaneous frequency for that partial is severely detuned from the integer harmonic value which would mean that $x(t)$ is not quasiperiodic.

See also

• Quasiperiodic function
• Aperiodic function
• Quasiperiodic tiling
• Fourier series
• Additive synthesis
• Harmonic series (music)
• Computer music

External links

References

• A.S. Besicovitch, "On generalized almost periodic functions" Proc. London Math. Soc. (2) , 25 (1926) pp. 495-512
• A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)
• H. Bohr, "Zur Theorie der fastperiodischen Funktionen I" Acta Math., 45 (1925) pp. 29–127
• H. Bohr, "Almost-periodic functions" , Chelsea, reprint (1947)
• W. Stepanoff(=V.V. Stepanov), "Sur quelques généralisations des fonctions presque périodiques" C.R. Acad. Sci. Paris , 181 (1925) pp. 90–92
• W. Stepanoff(=V.V. Stepanov), "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen" Math. Ann. , 45 (1925) pp. 473–498
• H. Weyl, "Integralgleichungen und fastperiodische Funktionen" Math. Ann. , 97 (1927) pp. 338–356

Notes