New iterative algorithms for signal approximation by using frames in a Hilbert space

The aim of this paper is to improve the methods of reconstruction by frames. We present two iterative algorithms to approximate a signal ???? based on a given frame, by defining a recursive sequence that converges to the signal in a separable Hilbert space. To do so, we initiate the process by squaring the convergence rate of the classical frame algorithm. This, in turn, halves the number of required iterations while concurrently enhancing the overall speed of convergence. Then, by using the Chebyshev polynomials, we further improve the recently studied acceleration. Due to the special conditions of some signals, we may have to use frames with large condition numbers. The importance of these algorithms is better understood when the frame has a large condition number. Numerical results are presented to show that the established results are valid and the proposed algorithms are applicable. The results presented here represent a significant stride towards the advancement and acceleration of the existing frame algorithms. These findings introduce novel iterative techniques characterized by modified convergence rates, specifically designed for signal reconstruction and approximation.