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

Let X and Y be Hilbert spaces, and K: dom K ⊂ X → Y a bounded linear operator. This paper addresses the inverse problem Kx = y, where exact data y is replaced by noisy data y^δ satisfying \|y^δ- y\|Y ≤ δ. 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^δ_α= ∑λ∈ Λ κ_λg_α(κ_λ²) y^δ, v_λ ū_λ. Convergence rates and optimality are analyzed under a generalized source condition Mφ, E = \ x ∈ dom K : ∑λ∈ Λ [φ(κ_λ²)]⁻¹ | x, u_λ|² ≤ E² \. 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.