A unified spectral filter framework for ill-posed linear operator equations in Hilbert spaces

Abstract

Regularization is useful for stable recovery in inverse problems with ill-posed linear operator equations in Hilbert spaces because small perturbations in data can make problems highly unstable. Tikhonov regularization, truncated singular value decomposition, and iterative polynomial filtering, classical methods, have been understood from singular value decay and the Picard condition. However, most literature analyses convergence, parameter choice, and saturation from separate perspectives. This study fills the gap by constructing a unified spectral filter framework that integrates bias–variance decomposition, polynomial and exponential decay, convergence rate analysis, and stabilityconsistent parameter choice frameworks, including the discrepancy principle and the Lcurve criterion. To enhance saturation control and qualification, we propose extensions to fractional and generalized spectral filters. In the severely ill-posed setting, we identify logarithmic convergence barriers with the inductive method, thereby exposing accuracy limits that exist independently from filter design. The findings are directly applicable to stable inversion and are operator theoretically sound for real-world applications, including medical imaging, geophysical reconstruction, signal processing, and data-driven recovery of ill-conditioned systems.