Approximation of a Twice Continuously Differentiable Function with Bounded First and Second Derivatives by Test Functions

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and fix a dimension $q$. The set $\mathbb{R}^{q}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and $C_{c}^{\infty}(\mathbb{R}^{q})$ is the set of test functions on $\mathbb{R}^{q}$; limits of sequences of real numbers are those of that definition. Let $f:\mathbb{R}^{q}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{q}$, and let $M$ be a nonnegative real number with $|\partial_{i}f(x)|\le M$ and $|\partial_{j}\partial_{i}f(x)|\le M$ for all $x\in\mathbb{R}^{q}$ and $i,j\in[q]$.

1. (Approximation)¶ There are a sequence $(\psi_{n})_{n\in\mathbb{N}}$ in $C_{c}^{\infty}(\mathbb{R}^{q})$ and a nonnegative real number $M'$ such that

for all $n\in\mathbb{N}$, $x\in\mathbb{R}^{q}$ and $i,j\in[q]$, and

for all $x\in\mathbb{R}^{q}$ and $i,j\in[q]$.