Periodic function

A function is periodic if there is a nonzero shift that leaves its values unchanged at every input. Its periods form a repeating structure, with a least positive period when one exists.

Period (mathematics): A nonzero shift that leaves a function’s values unchanged wherever both inputs are defined. This shift is the defining quantity in the periodicity condition.
Fourier series: A representation of a periodic function as a sum of sinusoidal harmonics under suitable conditions. Its harmonics encode a function’s repetition through integer multiples of a base frequency.
Sine: A trigonometric function that gives the vertical coordinate on the unit circle. Its repetition over 2π makes it a basic model of smooth periodic variation.
Quasiperiodic function: A function with structured oscillations that never repeat exactly at a single period. It can look recurrent without possessing a true period.
Fundamental period: The least positive period of a periodic function, when such a least positive period exists. It captures the smallest repeating interval rather than any multiple of it.
Fourier transform: An integral transform that represents a function through its continuous spectrum of frequencies. Unlike Fourier series, it is commonly used for nonperiodic functions on unbounded domains.
Square wave: A periodic waveform that alternates between two levels, typically with abrupt transitions. It shows how periodicity can persist despite discontinuities.
Almost periodic function: A function whose translates approximate it arbitrarily closely with relatively dense shifts. It generalizes recurrence while allowing the absence of an exact period.
Function domain: The set of inputs for which a function is defined. A shift must preserve the function’s domain as well as its values.
Harmonic analysis: The study of functions and signals through their frequencies and oscillatory components. It provides the broader framework for decomposing periodic functions into modes.

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