A smooth Bessel-blur truth with an explicit TV source certificate

The multiplier $m$ is real, even, and satisfies $0<m\le 1$, so [F1] makes $A$ a contraction on complex $L^2$. For real Schwartz $g$, the integral formula [F2] gives $\widehat{g}(-\xi )=\overline{\widehat{g}(\xi )}$, and multiplication by $m$ preserves this symmetry. Approximate $m\widehat{g}$ in $L^2$ by Schwartz functions using [F3], then replace each approximant $s$ by $\frac{1}{2}(s(\xi )+\overline{s(-\xi )})$. These symmetrized approximants remain Schwartz, converge to $m\widehat{g}$, and have real inverse Fourier transforms by [F2]; [F1] and the closedness observation above pass this property to $Ag$. Density of real Schwartz functions and boundedness of $A$ extend real-valuedness to real $L^2$. By [F7], the complex Plancherel pairing restricts to the real Hilbert pairing. Multiplication by the real function $m$ is self-adjoint, so $⟨Au,v⟩=⟨u,Av⟩$ for real $u,v$, and $A^{\ast }=A$. If $Au=0$, then $m\widehat{u}=0$ almost everywhere; strict positivity of $m$ and injectivity of the unitary Fourier transform give $u=0$. Thus $A$ is bounded, self-adjoint, and injective.

Because $\varphi$ is supported away from zero and from $b$, $u^{†}$ is constant near the origin, smooth across radii $a,b$, and zero for $\mathrm{\mid }x\mathrm{\mid }\ge b$; hence it is a compactly supported smooth real function. Its gradient is $\mathrm{\nabla }{u}^{†}(x)=-\varphi (\mathrm{\mid }x\mathrm{\mid })x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ wherever $\varphi (\mathrm{\mid }x\mathrm{\mid })\mathrm{\ne }0$ and is zero elsewhere. The cutoff in the Statement exists by [F6] and gives $z\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;{\mathbb{R}}^2)$ with $\mathrm{\parallel }z{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$. Integration by parts gives ${\int }_{{\mathbb{R}}^2}{u}^{†}\mathrm{div}zdx=-{\int }_{{\mathbb{R}}^2}z\cdot \mathrm{\nabla }{u}^{†}dx={\int }_{{\mathbb{R}}^2}\varphi (\mathrm{\mid }x\mathrm{\mid })dx={\int }_{{\mathbb{R}}^2}\mathrm{\mid }\mathrm{\nabla }{u}^{†}\mathrm{\mid }dx\mathrm{.}$ For every admissible test field $y$, the same integration by parts gives $\int u^{†}\mathrm{div}y\le \int \mathrm{\mid }\mathrm{\nabla }{u}^{†}\mathrm{\mid }$, while the field $z$ attains equality. Thus $T{V}_2(u^{†})=\int u^{†}\mathrm{div}z$. Setting $q=\mathrm{div}z$, [F4] gives $T{V}_2(v)\ge \int vq=T{V}_2(u^{†})+⟨q,v-u^{†}⟩$ for every $v$, so $q\in \mathrm{\partial }T{V}_2(u^{†})$.

The vector $q$ is smooth and compactly supported, so $q,{\mathrm{\partial }}_{1}q,{\mathrm{\partial }}_{2}q\in L^2$. Integration by parts using [F2] gives $\widehat{{\mathrm{\partial }}_{j}q}=2\pi i{\xi }_j\widehat{q}$, and Plancherel [F1] yields ${\int }_{{\mathbb{R}}^2}(1+4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2)\mathrm{\mid }\widehat{q}(\xi ){\mathrm{\mid }}^{2}d\xi =\mathrm{\parallel }q{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }q{\mathrm{\parallel }}_2^2<\mathrm{\infty }\mathrm{.}$ Therefore $\widehat{p}(\xi ):=(1+4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\widehat{q}(\xi )$ is in $L^2$, defining $p={\mathcal{F}}_2^{-1}\widehat{p}\in L^2$. The integral formula [F2] gives Hermitian symmetry of $\widehat{q}$, and the real even multiplier preserves it. As in step 1.1, approximate $\widehat{p}$ by Schwartz functions, symmetrize them, apply inverse Fourier transformation, and use Plancherel continuity and closedness of the real subspace to conclude that $p$ is real. By construction $A^{\ast }p=Ap=q$, so [F5] gives the required source certificate. Injectivity from step 1.1 makes $u^{†}$ the unique exact solution, hence the unique minimum-TV exact solution.

Step 2.1 provides a certificate at the unique minimum-TV exact truth. For every positive ${\alpha }_n$ and zero noise $w_n=0$, the parameter ratio is $\mathrm{\parallel }{A}^{\ast }\mathrm{\parallel }\mathrm{\parallel }{w}_n{\mathrm{\parallel }}_2\mathrm{/}{\alpha }_n=0$. Thus this example verifies the source condition and the zero-noise parameter side of the full-space deblurring setup without asserting existence of a regularized minimizer.