Approximate identity

An approximate identity is a family of integrable kernels $K_{\epsilon }$ whose total mass is one and whose mass concentrates near zero as $\epsilon \to 0$. Under standard hypotheses, $f\ast K_{\epsilon }$ converges to $f$ in norm and at almost every Lebesgue point.

In the distributional sense,This kernel is not positive, but testing it against a smooth compactly supported function and using the Dirichlet integral proves convergence.

The Cauchy approximate identity isIt is a probability density of total mass one, and its mass outside every fixed neighbourhood of zero tends to zero as $\epsilon \to 0$. It therefore converges to the Dirac delta function as a distribution.