Almost-Periodic Function / Quasiperiodic Function
Almost periodic functions are a subtype of aperiodic functions.
As an example, the following almost periodic function has two distinct harmonic parts:
f(t) = 6 sin(4t) + 14 cos(6√4t).
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 f is a quasiperiodic function with basic frequencies ω1, …, ωr if f(t) = F(θ1,…, θr) where F is 2π periodic in all its arguments and θj = ωjt 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
Stephanie Glen. "Almost Periodic Function, Quasiperiodic" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/almost-periodic-function-quasiperiodic/
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