Tperiodic

Tperiodic is a term used in mathematics and related disciplines to describe objects whose values repeat after a fixed interval, denoted T. The more common spelling is T-periodic, but the unhyphenated form appears in some texts and software contexts. In continuous time, a real-valued function f is Tperiodic if f(t + T) = f(t) for all real t, with T > 0. In discrete time, a sequence x[n] is N-periodic if x[n + N] = x[n] for all integers n, with N > 0.

The fundamental period is the smallest positive interval T for which the periodicity relation holds. If a function has two periods T1 and T2 whose ratio T2/T1 is rational, the function has a fundamental period equal to the least common multiple of T1 and T2. If the ratio is irrational, the function may be periodic only with arbitrarily large periods or not at all, depending on the context. A constant function is trivially periodic with any period.

Common examples of Tperiodic functions include f(t) = sin(2π t / T) and f(t) = cos(2π t / T), which have fundamental period T. More generally, any finite sum of harmonically related sinusoids with period T is Tperiodic. The set of Tperiodic functions forms a vector space, and many such functions admit Fourier series representations with fundamental frequency ω0 = 2π / T.

Applications of Tperiodic concepts appear in signal processing, physics, and engineering, where repeating phenomena are modeled as periodic signals or forces. They are central to solving differential equations with periodic forcing and to the analysis of systems subject to periodic boundary conditions. See also periodic function, fundamental period, Fourier series, and harmonic.