supportterm

Supportterm is a term used in mathematical modeling and signal processing to describe a component of a model that emphasizes the support of a function or signal. It refers to a piece of a representation whose influence is confined to a localized region of the domain.

Definition and construction. A supportterm is a component whose support—the set where it is nonzero or above a threshold—is localized. In decompositions, a function f may be represented as f = sum_i a_i t_i, where each t_i is a potential supportterm designed to have confined support. Supportterms can be built by selecting basis elements with localized support, truncating a function to a region, multiplying a base function by an indicator of a region, or using partition-of-unity techniques.

Properties. Supportterms are typically localized, can be sparse, and their supports may be disjoint or overlap. The combination of multiple supportterms aims to reproduce the target function or signal under the chosen modeling framework. Their localized nature often makes them suitable for capturing regional phenomena or features.

Applications. They appear in image and audio processing to model localized features, in sparse representations to identify active regions, and in numerical methods such as finite element or mesh-based models where basis functions have compact support. They are also used in regional modeling and piecewise approximations.

Variants and note. The exact definition and usage of the term supportterm vary between authors and disciplines; it is not a universally standardized term, but it serves as a descriptive label for components with localized support.

See also. Support of a function, sparse representation, basis function, finite element method.