A quasiperiodic function in the finite-frequency sense is a function of the form

where is a continuous function with period in each variable, and the real frequencies are linearly independent over the rational numbers. Thus is obtained by evaluating a function on the -dimensional torus, the Cartesian product of circles, along a linear flow (Lochak and Meunier 1988, p. 32). This convention includes continuous periodic functions; the term is also used more narrowly for nonperiodic examples with two or more independent frequencies. For example, has no nonzero period. Every such is an almost periodic function, but an almost periodic function need not have only finitely many basic frequencies (Lochak and Meunier 1988, p. 32).

The discrete version samples a periodic function along an irrational rotation. For example, with , a continuous 1-periodic , and irrational is a quasiperiodic sequence. The potential in the almost Mathieu operator is the special case . Quasiperiodic motion is the corresponding dynamical notion, in which the state evolves along an irrational linear flow on a torus and continuous observations of that state are quasiperiodic functions.

In complex analysis, quasiperiodicity also refers to a change by a prescribed additive term or multiplicative factor under translation, rather than exact repetition. This usage is distinct from the finite-frequency definition above. For example, the Weierstrass zeta function satisfies

where is a half-period and . The Weierstrass sigma function satisfies

The Jacobi theta functions and Riemann theta function obey related translation laws (NIST DLMF).

See also

Almost Periodic Function, Jacobi Theta Functions, Periodic Function, Quasiperiodic Motion, Riemann Theta Function, Weierstrass Sigma Function, Weierstrass Zeta Function

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References

Lochak, P. and Meunier, C. Multiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems. New York: Springer-Verlag, 1988. https://doi.org/10.1007/978-1-4612-1044-3.National Institute of Standards and Technology. "Definitions and Periodic Properties." §§20.2 and 23.2 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/20.2 and https://dlmf.nist.gov/23.2.National Institute of Standards and Technology. "Symmetry and Quasi-Periodicity." §21.3 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/21.3.

Cite this as:

Weisstein, Eric W. "Quasiperiodic Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuasiperiodicFunction.html

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