BumpFunktion

BumpFunktion, commonly called a bump function or smooth cutoff function, is a smooth function with compact support used to localize analysis on Euclidean spaces and manifolds. Formally, a bump function is a C∞ function φ: R^n → R that vanishes outside a compact set. It is often chosen to be nonnegative and to equal 1 on a neighborhood of a point or set, providing a localized “window.”

Canonical examples and construction: In one dimension, there exist smooth functions that are identically 1 on an interval and vanish outside a larger interval. A typical approach uses standard smooth transition constructions based on exponentials to create φ with φ(x) = 1 for |x| ≤ 1/2 and φ(x) = 0 for |x| ≥ 1, with a smooth transition in between. In higher dimensions, a common method is to define φ(x) = ψ(|x|) where ψ: [0, ∞) → [0, 1] is smooth, equals 1 on [0, r], and vanishes for |x| beyond some radius, ensuring φ has compact support.

Properties: Bump functions are infinitely differentiable, nonnegative, and have compact support. They can be scaled and translated to fit around any point or region. They are stable under multiplication and addition, making them useful building blocks.

Applications: They are central in constructing partitions of unity, smoothing or localizing functions, and defining approximate identities in analysis. On manifolds, bump functions exist around every point, enabling partitions of unity subordinate to open covers and facilitating the transfer of local data to global constructions.