On convergence rate of kernel approximation

Question

Let $\{K_\epsilon\}$ be a sequence of mollifiers, (or often take heat kernels). We have known from classical analysis that if $f$ is uniformly continuous then the error $$||f-f*K_\epsilon||_\infty\to 0$$ as $\epsilon\to 0$. I wish to know an explicit upper bound for that convergence rate as a function of $\epsilon,f$. Will the regularity of $f$ and $K$ influence the global sup-norm convergence rate in this context? Any comment shall be greatly appreciated.

Answer 1

Given $f:\mathbb{R}^n\rightarrow[0,+\infty)$ with $\int f=1$, $\forall \epsilon>0$ let $f_\epsilon(x)=\epsilon^{-n}f(\frac{x}{\epsilon})$ then $\parallel g-g\ast f_{\epsilon}\parallel_{\infty}\rightarrow0$ when $\epsilon\rightarrow0$ if $g\in C_0(\mathbb{R}^n)$. The proof can be found in Fourier Analysis - Duoandikoetxea j.