## Almost-Periodic Function / Quasiperiodic Function

**Almost-periodic functions**, are an important class of aperiodic functions. They can be represented by a sum of two or more periodic functions. In other words, they can be formed by summing two or more harmonic parts. Two of the parts must be frequencies that aren’t rational multiples of one another (Depner & Rasmussen, 2017).

As an example, the following almost periodic function has two distinct harmonic parts:

f(t) = 6 sin(4*t*) + 14 cos(6√4*t*).

## Quasi-Periodic Function

Quasi-periodic functions are a **special case** of almost periodic functions. They are a *not* periodic; They are a combination of periodic functions of different frequencies that never match exactly.

Perhaps **the simplest way to create one is just to add two periodic functions**: one with a rational period and one with an irrational period (Ong, 2020). Fourier transforms of quasi-periodic functions are discrete sets of delta functions; they can always be expressed as a series of sines and cosines with non matching lengths—or with an amount of arithmetically independent basis vectors that exceed the number of independent variables (Cahn, 2001).

There are **several ways to define quasiperiodic functions** mathematically. One fairly straightforward way (Jorba & Simo, 1984):

“A function

fis a quasiperiodic function with basic frequencies ω_{1}, …, ω_{r}if f(t) = F(θ_{1},…, θ_{r}) where F is 2π periodic in all its arguments and θ_{j}= ω_{j}t for j = 1, …,r”

## References

Cahn, J. (2001). Quasicrystals. Journal of Research of the National Institute of Standards and Technology. 106, 975–982.

Depner, J. & Rasmussen, T. (2017). Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media, Geophysical Monograph 224. American Geophysical Union.

Jorba, A. & Simo, C. (1984). On Quasiperiodic Perturbations of Elliptic Equilibrium Points. Retrieved November 13, 2020 from: https://upcommons.upc.edu/bitstream/handle/2117/901/9501jorba.pdf

Ong, D. (2020). Quasiperiodic music. Retrieved November 13, 2020 from: https://export.arxiv.org/ftp/arxiv/papers/2009/2009.04667.pdf

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