Trägerfunktion

Trägerfunktion is a term used in German mathematics and engineering to denote a function that carries or confines another quantity, commonly describing the region where a function is nonzero or “supported.” The idea is that the Träger specifies where the function has influence or can be observed.

In analysis, for a function f: X → R or C on a topological space X, the Träger (support) of f is supp(f) = closure of { x ∈ X | f(x) ≠ 0 }. The Trägerfunktion often refers to the indicator of the support, χ_{supp(f)}(x) = 1 if x ∈ supp(f) and 0 otherwise. The Träger of a function is the smallest closed set outside which the function vanishes. If f is continuous with compact support, supp(f) is compact.

In distribution theory, the concept extends to distributions: the Träger of a distribution T is the smallest closed set outside which T vanishes in the sense of test functions. Thus the carrier describes where T can interact nontrivially with test functions.

In signal processing and communications, the term Trägerfunktion can denote the carrier signal used in modulation: a high-frequency waveform c(t) (for example cos(2π f_c t) or e^{i 2π f_c t}) that multiplies a baseband signal. The modulated signal is typically s(t) = b(t) c(t).

Examples help illustrate the idea: a Gaussian f(x) = e^{-x^2} has supp(f) = ℝ (nonzero everywhere), whereas f(x) = max(0, 1 − |x|) has supp(f) = [−1, 1].

See also: support, indicator function, carrier signal, modulation.