SOLIT in the mean-zero 2D satellite-gradiometry model

Interleaving $(a_n,b_n{)}_{n\ge 1}$ as $(a_1,b_1,a_2,b_2,\dots )$ is a linear isometry onto real ${\mathrm{\ell }}^2({\mathbb{N}}_{>0})$, so [F7] gives the stated real Hilbert structure. In angular coordinates with normalized Haar measure $dx\mathrm{/}(2\pi )$, write the characters as $e_n(x)=e^{inx}$ and use $C_n=(e_n+e_{-n})\mathrm{/}\sqrt{2}=\sqrt{2}\cos (nx)$ and $S_n=(e_n-e_{-n})\mathrm{/}(i\sqrt{2})=\sqrt{2}\sin (nx)$ for $n\ge 1$. Their orthonormality follows from character orthonormality in [F1]. They are complete in mean-zero real $L^2$: a real mean-zero function orthogonal to all $C_n,S_n$ is orthogonal to every character, hence is zero by [F1]. Thus their coefficient map identifies this function space isometrically with $X$. The bounded truth belongs to $L^2$, and its integral over $[-\pi ,\pi ]$ is ${\pi }^2-{\pi }^2=0$, so its constant Fourier coefficient vanishes and its normalized real Fourier coefficient sequence lies in $X$.

For every $n\ge 1$, $t_n>0$ and $t_n\to 0$ exponentially since $R>1$. The ratio of consecutive squared multipliers is $((n+1)\mathrm{/}n{)}^2((n+2)\mathrm{/}(n+1){)}^2{R}^{-2}$, which tends to $R^{-2}<1$; choose $\rho$ strictly between $R^{-2}$ and $1$, then the ratio is at most $\rho$ for all sufficiently large $n$. Thus ${\sum }_{n\ge 1}2t_n^2<\mathrm{\infty }$. The operator is injective because all its diagonal multipliers are positive. Its truncation to frequencies $n\le N$ is finite rank and the operator-norm error is ${\sup }_{n>N}{t}_n\to 0$, so $T_R$ is compact and Hilbert–Schmidt.

The frequency multipliers satisfy $t_{n+1}\mathrm{/}{t}_n=(n+2)\mathrm{/}(nR)$, which can exceed $1$ for small $n$; thus the frequency order is not always the singular-value order. The multiset of modes has a nonincreasing enumeration: at every stage a remaining maximum exists because $t_n\to 0$ and only finitely many modes lie above any fixed positive threshold, and the fixed tie rule selects one uniquely. Every mode is eventually listed because only finitely many modes have multiplier at least its positive multiplier. The operator is diagonal on the orthonormal cosine–sine basis, so the frequency eigenvalue is ${\mu }_n=t_n^2=n^2(n+1{)}^2{R}^{-2n-4}$ twice at each frequency, and the enumerated singular system has these values in nonincreasing order. Since $(n+1{)}^2\asymp n^2$, ${\mu }_n\asymp n^4{e}^{-2n\log R}$, an exponential spectrum with exponent $\vartheta =1$.

For every $\lambda \ge 0$, $q_{\alpha }(\lambda )=(\lambda +\alpha {)}^{-1}$ is nonnegative, $\alpha q_{\alpha }(\lambda )\le 1$, and $\lambda q_{\alpha }(\lambda )\le 1$. If $\tilde{\alpha }<\alpha$, then $\lambda +\tilde{\alpha }<\lambda +\alpha$, so $q_{\alpha }(\lambda )\le q_{\tilde{\alpha }}(\lambda )$. Thus it is an ordered filter with $C_q^{\mathrm{\prime }}=1$ by [F6]. Its trace variance is

On every compact interval $[a,b]\subset (0,\mathrm{\infty })$, each summand is continuous and is bounded above by $2{\mu }_n\mathrm{/}{a}^2$. Since ${\sum }_{n}2{\mu }_n<\mathrm{\infty }$ by step 1.2, the series converges uniformly there, so $V$ is continuous. Each summand strictly decreases as $\alpha$ increases, hence so does $V$. Also $V(\alpha )\le 2{\alpha }^{-2}{\sum }_n{\mu }_n\to 0$ as $\alpha \to \mathrm{\infty }$. For every fixed $n$, $V(\alpha )\ge 2{\mu }_n\mathrm{/}({\mu }_n+\alpha {)}^2\to 2\mathrm{/}{\mu }_n$ as $\alpha \downarrow 0$; because ${\mu }_n\to 0$, these lower bounds imply $V(\alpha )\to \mathrm{\infty }$. Fix any $c>0$ and $\theta >1$. Continuity, strict decrease, and these endpoint limits give a unique ${\alpha }_m>0$ with $V({\alpha }_m)=c{\theta }^m$ for every $m\in {\mathbb{N}}_0$. Thus ${\alpha }_m$ decreases to zero and the variances $v_m={\sigma }^{2}V({\alpha }_m)$ have the constant consecutive ratio $\theta$. The resulting infinite grid meets [F2], and the SOLIT rule truncates it at its finite $m_{\max }$.

The truth lies in $L^2$ because it is bounded, and its Fourier coefficients are coordinates in the model by [F1]. The positive diagonal multipliers give the injective compact Hilbert–Schmidt operator verified in step 1.2; the Tikhonov filter satisfies [F6] and its candidates satisfy [F2], and the positive-noise experiment and finite selector are [F3]–[F4]. The well-definedness and moment properties of its estimators follow from [F5]. This example asserts no rate for the resulting estimator.