L2 Fourier inversion

Apply substitution to $\mathrm{\mid }f{\mathrm{\mid }}^2$ with the C1 diffeomorphism $x\mapsto -x$, whose absolute Jacobian is one. It gives $\mathrm{\parallel }Rf{\mathrm{\parallel }}_2=\mathrm{\parallel }f{\mathrm{\parallel }}_2$ and preserves null equivalence, so $R$ is an isometry on classes with $R^2=I$. For Schwartz approximants $u_j\to f$ supplied in [F1], [F2] gives ${\mathcal{F}}_2^2{u}_j=R{u}_j$. Both sides converge in norm by [F1] and the reflection isometry, hence ${\mathcal{F}}_2^{2}f=Rf$. Associativity then gives ${\mathcal{F}}_{2}R=R{\mathcal{F}}_2$ and both inverse identities for $R{\mathcal{F}}_2$.

The closed ball $B_R=\{\mathrm{\mid }x\mathrm{\mid }\le R\}$ is measurable and finite-measure by [F4]. Thus for $f_R={\mathbf{1}}_{B_R}f$, [F5] gives $\mathrm{\parallel }{f}_R{\mathrm{\parallel }}_1\le \lambda (B_R{)}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_2$, and $f_R\in L^2$ as well. [F6] applied to the explicit integer tails of $\mathrm{\mid }f{\mathrm{\mid }}^2$ gives $\mathrm{\parallel }{f}_R-f{\mathrm{\parallel }}_2\to 0$ for all real $R\to \mathrm{\infty }$ by monotonicity between integers. By [F3], its integral transform is ${\mathcal{F}}_2{f}_R$, and [F1] gives error norm $\mathrm{\parallel }{\mathcal{F}}_2{f}_R-{\mathcal{F}}_{2}f{\mathrm{\parallel }}_2=\mathrm{\parallel }{f}_R-f{\mathrm{\parallel }}_2\to 0$. The positive-sign integral is the reflection of this integral transform; step 1.1 gives its limit $R{\mathcal{F}}_{2}f={\mathcal{F}}_2^{-1}f$.