Regularization of Inverse Problems by Filtered Diagonal Frame Decomposition under general source

Let $X$ and $Y$ be Hilbert spaces, and $\mathbf{K}:\text{dom}\mathbf{K}\subset X\to Y$ a bounded linear operator. This paper addresses the inverse problem $\mathbf{K}x=y$, where exact data $y$ is replaced by noisy data $y^{\delta }$ satisfying $\mathrm{\parallel }{y}^{\delta }-y{\mathrm{\parallel }}_Y\le \delta$. Due to the ill-posedness of such problems, we employ regularization methods to stabilize solutions. While singular value decomposition (SVD) provides a classical approach, its computation can be costly and impractical for certain operators. We explore alternatives via Diagonal Frame Decomposition (DFD), generalizing SVD-based techniques, and introduce a regularized solution $x_{\alpha }^{\delta }={\sum }_{\lambda \in \mathrm{\Lambda }}{\kappa }_{\lambda }{g}_{\alpha }({\kappa }_{\lambda }^2)⟨y^{\delta },v_{\lambda }⟩{\bar{u}}_{\lambda }$. Convergence rates and optimality are analyzed under a generalized source condition ${\mathbf{M}}_{\phi ,E}=\{x\in \text{dom}\mathbf{K}:{\sum }_{\lambda \in \mathrm{\Lambda }}[\phi ({\kappa }_{\lambda }^2){]}^{-1}\mathrm{\mid }⟨x,u_{\lambda }⟩{\mathrm{\mid }}^2\le E^2\}$. Key questions include constructing DFD systems, relating DFD and SVD singular values, and extending source conditions. We present theoretical results, including modulus of continuity bounds and convergence rates for a priori and a posteriori parameter choices, with applications to polynomial and exponentially ill-posed problems.