Family of funcitons that approximates uniform density on an ellipsoid

Question

Given an nondegerate ellipsoid $E$ in $\mathbb{R}^d$, described as $E = \{x\in\mathbb{R}^d: (x-x_0)^TQ_0(x-x_0)\leq 1\}$ and let $\chi_E$ be the characteristic function supported on $E$. I am thinking about how to construct a family $\{f_n:\mathbb{R}^d\to\mathbb{R}\}_{n= 1}^\infty$, where each $f_n$ can be written as a smooth expression in terms of $Q_0, x_0$ and $n$, such that, as $n\to\infty$, $f_n\to\chi_E$ in $L^p$, for all $p\geq 1$. I have googled related key words but found nothing. Any advice or related reference would be appreciated. Thanks.

Answer 1

There are many good ways to accomplish this. One is by using the logistic function $L$, defined by the formula $$L(u):=\frac1{1+e^{-u}}.$$ Then $$g_n(u):=L(n(1-u)) \left\{ \begin{aligned} \uparrow1&\text{ if }0\le u<1,\\ \downarrow0&\text{ if }u>1 \end{aligned} \right. $$ as $n\to\infty$. So, assuming that $Q_0$ is positive definite, letting $$f_n(x):=g_n((x-x_0)^TQ_0(x-x_0)) $$ for all $x\in\mathbb R^n$, and using (say) the dominated convergence, we conclude that $f_n$ converges to the characteristic function of the set $E$ in $L^p$ for each $p\ge1$, as desired.