Smooth mollifiers approximate Bochner L² functions

Finite-valued simple functions are dense in Bochner $L^2$. Choose simple $s_n\to q$ almost everywhere as in [F1]. Replace $s_n$ by zero where $\mathrm{\parallel }{s}_n\mathrm{\parallel }>2\mathrm{\parallel }q\mathrm{\parallel }$. The clipped functions are still simple, converge to $q$ almost everywhere, and have squared error bounded by $9\mathrm{\parallel }q{\mathrm{\parallel }}^2$. Scalar dominated convergence [F7] gives convergence in $L^2(X)$. If $q=0$ almost everywhere, the zero simple sequence suffices.

For fixed $t$, the integrand in the definition of $q_{ϵ}(t)$ is strongly measurable and has integrable norm on the compact support of the kernel, so [F2] defines the Bochner integral. Differentiating the scalar kernel under this integral gives $q_{ϵ}^{(k)}(t)=\int {\eta }_{ϵ}^{(k)}(t-s)q(s)ds(k\ge 0)\mathrm{.}$ Indeed, for $t$ in a fixed compact interval the difference quotients of each kernel derivative converge uniformly and are bounded on one compact set; the $L^1(X)$ norm of $q$ on that set and [F5] let the limit pass through the integral. The same bound proves continuity of every displayed derivative, so $q_{ϵ}$ is smooth.

For a finite-valued simple function $s={\sum }_{j=1}^m{x}_j{\mathbf{1}}_{E_j}$, omit zero values; each remaining $E_j$ has finite measure. Then $\mathrm{\parallel }s(\cdot -h)-s{\mathrm{\parallel }}_{L^2(X)}\le {\sum }_j\mathrm{\parallel }{x}_j{\mathrm{\parallel }}_X\mathrm{\parallel }{\mathbf{1}}_{E_j}(\cdot -h)-{\mathbf{1}}_{E_j}{\mathrm{\parallel }}_{L^2}$, which tends to zero by [F3]. For general $q$, use the density from step 1.1 and translation's isometry [F8] to obtain $\mathrm{\parallel }q(\cdot -h)-q{\mathrm{\parallel }}_{L^2(X)}\to 0$.

Fix $R>0$ with $\mathrm{supp}\eta \subset [-R,R]$ for the generating bump, so $\mathrm{supp}{\eta }_{ϵ}\subset [-Rϵ,Rϵ]$ by [F4]. Since ${\eta }_{ϵ}(s)ds$ is a probability measure, Cauchy–Schwarz and [F5] give $\mathrm{\parallel }{q}_{ϵ}(t)-q(t){\mathrm{\parallel }}_X^2\le \int {\eta }_{ϵ}(s)\mathrm{\parallel }q(t-s)-q(t){\mathrm{\parallel }}_X^{2}ds\mathrm{.}$ Integrate in $t$ and apply [F6]. The result is bounded by ${\sup }_{\mathrm{\mid }s\mathrm{\mid }\le Rϵ}\mathrm{\parallel }q(\cdot -s)-q{\mathrm{\parallel }}_{L^2(X)}^2$, which tends to zero by step 2.1. This proves the asserted $L^2$ convergence.