Mollifiers কী?

Example 12.10 (Mollifying $\abs x$, with rates)

Take $f(x) = \abs x$ on $\R$ (locally $L^1$; the theorem applies on every bounded window) and a symmetric mollifier $\rho_\varepsilon$. Then

is $\mathcal C^\infty$; away from the kink, nothing happens: for $\abs x \geq \varepsilon$, $\abs{x - y}$ is linear in $x$ on the support of $\rho_\varepsilon$, so $f_\varepsilon(x) = \abs x$ exactly (symmetry kills the correction). Near $0$, smoothing costs precisely

the approximation error is confined to the $\varepsilon$-neighborhood of the singularity and is of its size. Meanwhile $f_\varepsilon'' \geq 0$ everywhere ($f$ is convex, and convolution against $\rho_\varepsilon \geq 0$ preserves convexity), with $\int f_\varepsilon'' = f_\varepsilon'(\infty) - f_\varepsilon'(-\infty) = 2$: the second derivative is a bump of mass $2$ squeezed into width $O(\varepsilon)$, so $\norm{f_\varepsilon''}_\infty \gtrsim \varepsilon^{-1}$. Smoothing is a trade: uniform error $O(\varepsilon)$ against derivative blow-up $O(\varepsilon^{-1})$ — the exact exchange rate that quantitative analysis (interpolation inequalities, Problem 12.1’s circle of ideas) formalizes.