Mittefinitne

Mittefinitne is a term used in some German-language mathematical writings to describe a property of functions or sequences related to the location and extent of their nonzero values. The word blends Mitte (middle) and finite, reflecting the idea that the nonzero region is contained within a finite central interval of the index set.

Definition and notation

Let f be a function from the integers Z to a field or abelian group. f is said to be mittefinitne if there exist integers a and b with a ≤ b such that f(n) = 0 for all n < a and for all n > b. Equivalently, the support of f, the set {n ∈ Z : f(n) ≠ 0}, is a finite subset of Z contained in the interval [a, b]. Some authors emphasize placing the nonzero region in a central position (for example around 0), while others require only finiteness of the support.

Properties

The class of mittefinitne functions on Z coincides with the standard notion of finite-support sequences. It forms a vector subspace under pointwise addition and scalar multiplication. If f and g are mittefinitne, then any linear combination αf + βg is auch mittefinitne. The convolution of two mittefinitne sequences yields another sequence with finite support, whose support is contained in the sum of the original supports.

Examples

A sequence that is 1 at n = 0 and 0 elsewhere is mittefinitne. More generally, any sequence that is nonzero only on a finite interval, such as nonzero on [-N, M] and zero elsewhere, is mittefinitne.

Relation to other concepts

Mittefinitne is closely related to finite support and compact support concepts used in discrete mathematics, signal processing, and algebra. In practice, it often overlaps with the standard language of finite-support sequences and can be viewed through the lens of polynomials or transforms that encode finite information.

Note

The term mittefinitne is not universally standardized and may be described simply as finite-support or compact-support in many texts.