Compact spectral filters, source conditions, and qualification

Assume Countable Choice. Let $X,Y$ be real or complex Hilbert spaces and let $A:X\to Y$ be compact and bounded. Put $S=A^{\ast }A$. By A compact Hilbert-space Gram operator is positive and has the forward kernel, $S$ is compact, positive, and self-adjoint, with $\ker S=\ker A$. Let $\mathrm{\Sigma }$ be the set of nonzero eigenvalues of $S$ and $P_t$ the orthogonal projection onto the eigenspace of $t\in \mathrm{\Sigma }$; let $P_0$ be the orthogonal projection onto $\ker S$. The compact spectral multiplier of a bounded scalar function $q:\{0\}\cup \mathrm{\Sigma }\to \mathbb{F}$ is $q(S)x:=q(0)P_{0}x+{\sum }_{t\in \mathrm{\Sigma }}q(t)P_{t}x,$ where the finite-subset net converges in norm by Bounded multipliers on compact self-adjoint eigenspaces define bounded operators. For $\nu >0$, define $S^{\nu }$ by the multiplier $q_{\nu }(0)=0$ and $q_{\nu }(t)=t^{\nu }$ for $t>0$. This multiplier is bounded: if $t\in \mathrm{\Sigma }$ and $u\in E_t$ is a unit vector, then $0<t=\mathrm{\parallel }Su\mathrm{\parallel }\le \mathrm{\parallel }S\mathrm{\parallel }$, so $\mathrm{\mid }{q}_{\nu }(t)\mathrm{\mid }\le \mathrm{\parallel }S{\mathrm{\parallel }}^{\nu }$.

For $\lambda >0$, the Tikhonov filter and residual filter are $g_{\lambda }(t)=\frac{1}{t+\lambda },r_{\lambda }(t)=1-t{g}_{\lambda }(t)=\frac{\lambda }{t+\lambda }\mathrm{.}$ The Tikhonov solution for data $y_{\delta }\in Y$ is $x_{\lambda }^{\delta }=g_{\lambda }(S)A^{\ast }{y}_{\delta }=(A^{\ast }A+\lambda I_X{)}^{-1}{A}^{\ast }{y}_{\delta }\mathrm{.}$ Indeed, the spectral decomposition gives $S+\lambda I_X$ the multiplier $t+\lambda$ on each $E_t$ and $\lambda$ on $\ker S$, so $g_{\lambda }(S)$ is its inverse, whose existence is also supplied by The shifted Hilbert-space normal operator is invertible with inverse norm at most $1\mathrm{/}\lambda$. The source condition of order $\nu >0$ and radius $R\ge 0$ means that $x^{†}=S^{\nu }w$ for some $w\in X$ with $\mathrm{\parallel }w\mathrm{\parallel }\le R$. A family of filters $g_{\alpha }$ with residual filters $r_{\alpha }(t)=1-t{g}_{\alpha }(t)$ has qualification at least $p>0$ when, for each $0<\nu \le p$, some finite constant $C_{\nu }\ge 0$ satisfies $t^{\nu }\mathrm{\mid }{r}_{\alpha }(t)\mathrm{\mid }\le C_{\nu }{\alpha }^{\nu }\text{for every }\alpha >0\text{ and }t\in \mathrm{\Sigma }\mathrm{.}$ When $\mathrm{\Sigma }=\mathrm{\varnothing }$, this condition is vacuous.

The parameter $\lambda$ here is the coefficient of $\mathrm{\parallel }x{\mathrm{\parallel }}^2$ in $\mathrm{\parallel }Ax-y_{\delta }{\mathrm{\parallel }}^2+\lambda \mathrm{\parallel }x{\mathrm{\parallel }}^2$. Hansen's discrete slides write this coefficient as ${\lambda }_H^2$, so $\lambda ={\lambda }_H^2$. Clason writes the source condition as $x^{†}=\mathrm{\mid }A{\mathrm{\mid }}^{\mu }w$ with $\mathrm{\mid }A\mathrm{\mid }=(A^{\ast }A{)}^{1\mathrm{/}2}$; here $\mu =2\nu$. Thus Tikhonov's qualification-one range in the present convention is $0<\nu \le 1$.