Research proof index

Let $T>0$, $1 \le p < \infty$, and let $f$ belong to $L^p((0,T])$. For every $\varepsilon>0$, there is a smooth $T$-periodic function $f_0 : \mathbb{R}\to\mathbb{C}$ such that

$\lVert f-f_0\rVert_{L^p((0,T])} \le \varepsilon.$

Global smooth compactly supported approximation of a compactly supported W^{1,p} function on ℝ^d, in the finite-p regime.

L² specialization of the unit-ball smooth approximation theorem.

Sequence form of full W^{1,p} smooth approximation on the unit ball.

Localization by compact support: a finite-p Sobolev function whose support is compactly contained in an open set belongs to W₀^{1,p}.

Two W^{1,2} witnesses on an open set have a.e.-equal gradients.