Knowlpedia

Theorem

Locally finite sum of smooth functions

A family with locally finite supports may be summed and differentiated as a finite sum near each point.

Statement

Let (fj)j∈J(f_j)_{j\in J} be smooth functions on an open set U⊆RnU\subseteq\mathbb R^n. If their supports form a , then

f(x)=∑j∈Jfj(x)f(x)=\sum_{j\in J}f_j(x)

is smooth and ∂αf=∑j∂αfj\partial^\alpha f=\sum_j\partial^\alpha f_j. Each point has a neighborhood on which only finitely many terms can be nonzero, so both statements reduce there to finite sums.

Local finiteness versus bounded overlap

Bounded overlap is a pointwise multiplicity bound and does not supply such a neighborhood. For example, choose disjoint bump functions of height one in intervals (2−j,2−j+2−j−2)(2^{-j},2^{-j}+2^{-j-2}). Their supports have multiplicity at most one, but accumulate at zero. Their sum, with value zero at zero, is not continuous there.

If supports accumulate at a boundary, smoothness across that boundary instead requires control of derivative tails, as in .