deltafunctionlike

Deltafunctionlike describes a family or sequence of functions or distributions that approximate the Dirac delta distribution as a parameter tends to zero. In the sense of distributions, a family fε is deltafunctionlike if, for every smooth test function φ with compact support, the integral ∫ fε(x) φ(x) dx converges to φ(0) as ε → 0. This captures the idea of concentrating mass at a single point while preserving total integral.

Typical properties of a deltafunctionlike family include nonnegativity, normalization (the integral over the whole space equals 1 for all ε), and a peak that becomes increasingly sharp near the origin as ε decreases. In multiple dimensions, the same ideas apply with integrals over R^n and concentration near the origin.

Common examples of deltafunctionlike families are mollifiers or approximate identities. A Gaussian example is fε(x) = (1/(ε√π)) exp(-(x/ε)^2) in one dimension, which integrates to 1 and concentrates at 0 as ε → 0. Other examples include a Lorentzian sequence fε(x) = (1/π) (ε/(x^2 + ε^2)) and compactly supported bumps obtained by scaling a fixed smooth function with integral 1. Each of these is deltafunctionlike in the distributional sense, though they differ in smoothness and decay.

Applications include regularization of singular objects, smoothing of functions, and analysis in partial differential equations and signal processing. They allow the delta distribution to be approximated by ordinary functions, enabling rigorous justification of calculations that involve point evaluations or impulse inputs. A key caveat is that convergence is typically in the sense of distributions, not pointwise, and a delta function is not itself a function.