Time series approximation is a key tool for predicting the future dynamics of complex systems, including climate behavior, financial markets and industrial processes. In practice, many multi-frequency signals display pronounced frequency separation among their components, a characteristic that severely limits the performance of standard approximation approaches. To address the limitations of conventional approaches, this paper introduces a novel multi-band recursive least squares method aimed at improving approximation and predictive accuracy. The proposed framework decomposes the signal spectrum into multiple frequency bands and performs independent parameter estimation within each band using customized window sizes and forgetting factors, thereby enabling significantly enhanced trade-off management across frequency scales. The proposed framework surpasses full-band estimation, where the signal bandwidth is handled as a unified frequency range, by delivering improved approximation and predictive accuracy. In addition, it reduces matrix dimensions and condition numbers, thereby increasing numerical robustness and reducing parameter variance. The recursive structure relies on rank two updates, ensuring convergence of both the inverses of the information matrices and the parameter estimation errors for each band. The key contribution of this paper lies in a Lyapunov based convergence analysis formulated for simplified error dynamics that characterize the dominant transient error mode. This mode is obtained through decomposition of the rank two increment matrix and its representation in dyadic form, which provides a structured basis for convergence assessment. The effectiveness of the proposed method is demonstrated through temperature prediction using high resolution observational data. The results confirm that multi-band predictions, optimized independently across frequency bands, offer improved accuracy compared with the conventional full-band model.

Multi-band recursive least squares estimation with rank two updates simplified error models Lyapunov analysis Richardson correction algorithm & convergence rate improvement short-term temperature forecast for control