 Approximate identity

A standard approximate identity on ${\mathbb{R}}^n$ is a family $K_{\epsilon }\in L^1$ withThe absolute tail condition matters for signed or complex kernels: cancellation of the signed mass alone is insufficient. For $1\le p<\infty$ and $f\in L^p$, these conditions imply $\Vert K_{\epsilon }\ast f-f{\Vert }_p\to 0$.

Indeed, express the difference as $\int K_{\epsilon }(y)[f(\cdot -y)-f]dy$. Translation continuity in Lp makes the integrand norm small near zero, the uniform L1 norm controls that part, and the vanishing absolute tail controls the remainder. The same proof gives uniform convergence for bounded uniformly continuous functions. It does not give norm convergence for every essentially bounded function.

For nonnegative normalized mollifiers $K_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$ with smooth compactly supported $\rho$, convergence also holds at every Lebesgue point of a locally integrable function. Such pointwise conclusions use additional size control on the kernels; they should not be attributed to the three norm-approximation conditions alone. The Cauchy approximate identity and summability kernel on the circle are related constructions on their respective domains.