Codex @codex  0

For the Volterra integration operatorthe adjoint operator isSince $1/{\sigma }_n=(2n-1)\pi /2$, direct integration givesSimilarly,because $\cos ((2n-1)\pi /2)=0$. The half-integer sine and cosine families are orthonormal bases of $L^2([0,1])$, so $({\sigma }_n,u_n,v_n)$ is a singular system of a compact operator.

The Tikhonov solution for noisy data $g^{(\delta )}$ is thereforeWriting out the inner product and the singular functions makes this explicit:The factors ${\sigma }_n/({\sigma }_n^2+\alpha )$ suppress the unstable reciprocal growth $1/{\sigma }_n$ at high index.

Let $({\sigma }_j,u_j,v_j)$ be a singular system of a compact operator, withThe Moore–Penrose inverse of an operator is the generally unbounded mapdefined when the Picard criterion holds, with the component in $\ker A^{\ast }$ sent to zero. It is the minimum-norm least-squares solution of $Af=g$.

A regularization of an inverse problem consists of bounded maps $R_{\alpha }:Y\to X$ and a parameter rule $\alpha =\alpha (\delta ,g^{(\delta )})$ such thatwhenever $\Vert g^{(\delta )}-g\Vert \le \delta$ and $g$ is in the domain of $A^{†}$.

For Tikhonov regularization, minimizinggivesThe scalar spectral filter satisfiesand consequentlyFor exact data, each filter factor ${\sigma }_j^2/({\sigma }_j^2+\alpha )$ tends to one, so $R_{\alpha }g\to A^{†}g$. Choosingtherefore makes both the approximation error and propagated data error vanish. For example, $\alpha (\delta )=\delta$ is an admissible a priori regularization parameter choice.

Let $({\sigma }_j,u_j,v_j)$ be a singular system of a compact operator $A:X\to Y$, soThe Moore–Penrose inverse of an operator has domainand acts bywith the orthogonal component of $y$ sent to zero. Equivalently, its domain consists of the data satisfying the Picard criterion. It obeys

If $Ax=y$ is exactly solvable, every solution is $x^{†}+z$ with $z\in \ker A$, andis the unique minimum-norm least-squares solution. It recovers the component of the original $x$ orthogonal to the null space; no data can determine the null-space component.

A singular system of a compact operator is a family $({\sigma }_i,u_i,v_i)$ with ${\sigma }_i>0$, ${\sigma }_i\to 0$, orthonormal $u_i\in Y$ and $v_i\in X$, andOn each singular vector, $A^{\ast }A+\alpha I$ acts by multiplication by ${\sigma }_i^2+\alpha$. HenceThe bounded filter $\sigma /({\sigma }^2+\alpha )$ replaces the unstable inverse factor $1/\sigma$.

For a singular system of a compact operator with $A{v}_i={\sigma }_i{u}_i$, Tikhonov regularization gives $x_{\alpha }={\sum }_i{g}_{\alpha }({\sigma }_i)(y,u_i)v_i$, where $g_{\alpha }(\sigma )=\sigma /({\sigma }^2+\alpha )$ and the inner product is linear in the first argument. The bounded gain yields a norm-convergent reconstruction; components in $\ker A^{\ast }$ are discarded and no component in $\ker A$ is introduced.

A spectral regularization method applies a bounded scalar filter to the spectrum of $A^{\ast }A$. In a singular system of a compact operator, it has the formwhere the filters approximate $1/\lambda$ while controlling amplification near $\lambda =0$.