periodsiset

Periodsiset is a mathematical construct used to describe the set of positive periods of a function defined on the integers. Given a function f: Z → X, the positive periods are the set P = { t ∈ N : f(n+t) = f(n) for all n ∈ Z }. When f is purely periodic with a least (fundamental) period p > 0, this set of periods takes the form P = { p, 2p, 3p, … } = pN, i.e., all positive multiples of p. The term periodsiset is used to refer to this collection of periods.

Properties: If f is not periodic, P is empty. If f is periodic with least period p, then P is infinite and is exactly the set of positive multiples of p. The number p is unique and called the fundamental period; it determines the structure of P. In particular, every element of P is a multiple of p, and if q ∈ P then p divides q.

Examples: A constant function on Z has least period p = 1, giving P = {1, 2, 3, …}. A sequence repeating every 4 samples has P = {4, 8, 12, …}. A non-periodic function has P = ∅.

Relations and uses: The concept connects to the broader study of periodic sequences, harmonic analysis, and the theory of numerical semigroups when viewed as the semigroup pN. In practice, identifying the periodsiset can simplify the analysis of repeated patterns and aid in reconstruction from finite data segments.

See also: periodic function; fundamental period; discrete-time signal; numerical semigroup.