Generalized Dual Wilson Frame Systems for Signal Representation: Stability, Reconstruction, and Functional Optimization Analysis

Abstract

Generalized dual Wilson frame systems constitute a powerful mathematical framework for stable signal representation in time–frequency analysis, particularly within Hilbert space settings. This paper develops a structured theoretical model for analyzing dual Wilson frame constructions with emphasis on stability, reconstruction accuracy, and functional optimization properties. Building upon classical Wilson bases and their extensions in modulation spaces and Gabor analysis, we examine generalized dual structures that allow redundancy-controlled signal decomposition while maintaining reconstruction fidelity. The study integrates foundational principles from time–frequency analysis (Gröchenig, 2001; Feichtinger & Strohmer, 2003) and frame theory (Christensen, 2008), extending them into a dual-frame optimization context suitable for adaptive signal representation. Special attention is given to stability conditions under perturbations, reconstruction operators, and optimization constraints that govern frame duality. The theoretical development is further grounded in polynomial reproduction and approximation properties of Wilson systems as established in prior research (Bittner, 2002). The results demonstrate that generalized dual Wilson frames provide enhanced robustness in signal decomposition while preserving computational tractability. The framework also highlights trade-offs between redundancy and numerical efficiency, offering insights into practical implementations in signal processing and functional analysis domains.