Continuous functions are dense in $L^p$ of finite tori and of bounded intervals

Let $G:=F\circ q_n$ on $[0,1{)}^n$, extended by $0$ to ${\mathbb{R}}^n$. Then $G\in L^p({\mathbb{R}}^n)$ and $\mathrm{\parallel }G{\mathrm{\parallel }}_p=\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbb{T}}^n)}$, and for a measurable $E\subseteq {\mathbb{R}}^n$ with ${\lambda }_n(E)$ small the integral of $\mathrm{\mid }G{\mathrm{\mid }}^p$ over $E$ is small by absolute continuity of the integral.

Given $\epsilon >0$ choose $\delta >0$ with ${\lambda }_n([0,1{)}^n\setminus [\delta ,1-\delta {]}^n)<{\delta }_0$, where ${\delta }_0$ is a threshold for $\mathrm{\mid }G{\mathrm{\mid }}^p$ and ${\epsilon }^p$ from [A3]; put $G_{\delta }:=G\cdot {\mathbf{1}}_{[\delta ,1-\delta {]}^n}$. Then $\mathrm{\parallel }G-G_{\delta }{\mathrm{\parallel }}_p^p={\int }_{[0,1{)}^n\setminus [\delta ,1-\delta {]}^n}\mathrm{\mid }G{\mathrm{\mid }}^p<{\epsilon }^p$, so $\mathrm{\parallel }G-G_{\delta }{\mathrm{\parallel }}_p<\epsilon$.

By density of $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ choose $H\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $\mathrm{\parallel }{G}_{\delta }-H{\mathrm{\parallel }}_p<\epsilon$, and let $\chi$ be the cutoff of [A4] for this $\delta$. Since $\chi =1$ on the support of $G_{\delta }$ and $\mathrm{\mid }\chi \mathrm{\mid }\le 1$, one has $\mathrm{\parallel }{G}_{\delta }-\chi H{\mathrm{\parallel }}_p=\mathrm{\parallel }\chi (G_{\delta }-H){\mathrm{\parallel }}_p\le \mathrm{\parallel }{G}_{\delta }-H{\mathrm{\parallel }}_p<\epsilon$, and $\chi H$ is continuous, compactly supported in $[0,1{]}^n$, and vanishes on the boundary of that cube.

Let $K$ be the continuous function on ${\mathbb{T}}^n$ obtained by periodising $\chi H$, as in [A4]; on the fundamental domain $K$ agrees with $\chi H$. Therefore, using that the torus $L^p$ integral is represented on $[0,1{)}^n$ and Minkowski's inequality, $\mathrm{\parallel }f-K{\mathrm{\parallel }}_{L^p({\mathbb{T}}^n)}=\mathrm{\parallel }G-\chi H{\mathrm{\parallel }}_{L^p([0,1{)}^n)}\le \mathrm{\parallel }G-G_{\delta }{\mathrm{\parallel }}_p+\mathrm{\parallel }{G}_{\delta }-\chi H{\mathrm{\parallel }}_p<2\epsilon$.

Since $\epsilon >0$ was arbitrary, claim 1 follows: every $f\in L^p({\mathbb{T}}^n;\mathbb{C})$ is approximated in $L^p$ by the continuous functions $K$. For claim 2, let $f\in L^p((a,b))$ and extend it by $0$ to $\mathbb{R}$; the density theorem [A1] gives $H\in C_c^{\mathrm{\infty }}(\mathbb{R})$ with $\mathrm{\parallel }f-H{\mathrm{\parallel }}_{L^p(\mathbb{R})}<\epsilon$, and restricting $H$ to $[a,b]$ gives a continuous function with $\mathrm{\parallel }f-H{\mathrm{\parallel }}_{L^p((a,b))}\le \mathrm{\parallel }f-H{\mathrm{\parallel }}_{L^p(\mathbb{R})}<\epsilon$.

For claim 3, let $f$ be real valued and let $H$ approximate it complexly within $\epsilon$; then $\mathrm{Re}H$ is real continuous and $\mathrm{\parallel }f-\mathrm{Re}H{\mathrm{\parallel }}_p=\mathrm{\parallel }\mathrm{Re}(f-H){\mathrm{\parallel }}_p\le \mathrm{\parallel }f-H{\mathrm{\parallel }}_p<\epsilon$, so the real continuous functions are dense in each of the two settings.

Steps 5.1 and 6.1 establish the complex and real density statements on finite tori and on bounded intervals.