intervallets

Intervallets is a term used in some mathematical and computational contexts to denote small, elementary one-dimensional intervals that serve as building blocks for partitions of a real line or similar spaces. Formally, an intervallet can be described as a closed interval [a, b] with a ≤ b, often with the additional constraint that its length b − a is small, for example b − a ≤ ε for a chosen tolerance ε > 0.

Properties of intervallets typically include convexity and compactness, since they are closed intervals on the real line. When used to cover a larger set, the intervallets may be arranged to be disjoint or to form a finer overlap as dictated by the application. In the common case of a disjoint cover, the total length of the intervallets equals the measure of the covered set (in one dimension). Intervallets can be refined by splitting an intervallet into two smaller intervallets, a process akin to adaptive refinement in numerical methods.

Construction and variants: intervallets arise naturally from partitions of an interval, as in the subdivision of [a, b] into subintervals. They can also appear in adaptive mesh generation or piecewise-constant/linear approximations. A related idea is the use of small intervals in interval arithmetic, where computations are performed with ranges rather than exact numbers. Higher-dimensional analogues are sometimes informally referred to as hyperintervallets or boxes.

Applications: intervallets are used in numerical integration, function approximation, data segmentation, and one-dimensional mesh generation, where a problem domain is represented as a collection of small, manageable intervals.

See also: Interval, Interval arithmetic, Partition of unity, Mesh generation, Piecewise approximation.