A finite dimensional realization of the mollifier method for compact operator equations

We introduce and analyze a stable procedure for the approximation of $\langle f^\dagger, \varphi\rangle$ where $f^\dagger$ is the least residual norm solution of the minimal norm of the ill-posed equation $Af=g$, with compact operator $A:X\to Y$ between Hilbert spaces, and $\varphi\in X$ has some smoothness assumption. Our method is based on a finite number of singular values of $A$ and some finite rank operators. Our results are in a more general setting than the one considered by Rieder and Schuster (2000) and Nair and Lal (2003) with special reference to the mollifier method, and it is also applicable under fewer smoothness assumptions on $\varphi$.