mehrperiodische

Mehrperiodische, or mehrperiodizität, describes functions that possess more than one period. Formally, a function f: R -> R is mehrperiodisch if there exist real numbers T1, ..., Tk > 0 with k ≥ 2 such that f(x + Ti) = f(x) for all real x and all i = 1, ..., k. If all ratios Ti/Tj are rational, the periods are commensurate and f is periodic with a fundamental period equal to the greatest common divisor of the set {T1, ..., Tk} in the additive sense. If at least two periods are incommensurate (Ti/Tj irrational), the situation depends on regularity: for continuous (or more generally reasonably regular) functions, having two incommensurate periods forces f to be constant. Without regularity assumptions, non-constant examples can be constructed.

Multi-periodicity generalizes to higher dimensions: a function on R^n may be periodic with respect to a lattice Γ, meaning f(x + γ) = f(x) for all γ in Γ.

Relation to related notions: quasi-periodic and almost periodic functions describe signals that exhibit regularity without a single fundamental period; quasi-periodic functions typically arise from sums of incommensurate periodic components, often leading to dense frequency sets.

Applications: multi-periodic models appear in signal processing, harmonic analysis, and dynamical systems, where the interplay of several periodic components is studied.