Integrationskugeln

Integrationskugeln are a concept in analysis describing a family of averaging operators obtained by integrating a function over balls in Euclidean space. For a function f in a suitable function space, and for x in R^n and r > 0, define B(x, r) as the ball of radius r centered at x. The integrational averaging operator is then f_r(x) = (1/|B(0, r)|) ∫_{B(x, r)} f(y) dy, where |B(0, r)| is the volume of the ball. The mapping r ↦ f_r is used to smooth functions and is closely related to mollification, since {f_r} constitutes an approximate identity as r → 0.

Key properties include radial symmetry of the averaging region and the smoothing effect on f. If f ∈ L^p(R^n) (1 ≤ p ≤ ∞), then f_r → f in L^p as r → 0, and under mild regularity assumptions one obtains pointwise convergence almost everywhere. For smooth f, f_r admits a controlled expansion in terms of derivatives of f, reflecting the local averaging.

Variants include averaging over spheres: the spherical mean over S(x, r) = {y : |y − x| = r}. Harmonic functions satisfy the mean value property: the value at x equals the average over any sphere centered at x. The term Integrationskugeln is commonly used in German-language texts to denote ball-based integration regions, while related concepts in other languages discuss spherical means and mollifiers.

Applications span mollification in partial differential equations, potential theory, numerical quadrature schemes leveraging radial symmetry, and image smoothing. See also: mollifier, approximate identity, Poisson kernel, mean value property.