Applications of Lucas sequences in convergence and signal processing

In this paper we propose a new perspective related to Lucas sequence with statistical and summability, some considerations have been given in connection with their applications of signal processing. We define the notions of λ-Lucas statistically convergent and strongly λ-Lucas summability through modulus functions. The study of investigates on the Lucas transform and its embedding diagram into the representative form of Lucas numbers is studied in the context of convergence and summability problems. Furthermore, we provide inclusions as well as principles of equivalence, which lead to conditions guaranteeing the uniqueness of restrictions and generalize statistical convergence theory. Numerical simulations on a wide variety of signals such as noisy sinusoidal signals and blurred images demonstrate the flexibility of the proposed approach. These experiments show a steady decay to zero, which implies considerable noise immunity. In addition, by providing alternative perspectives of assessing signal quality, compression strategies and filtering techniques, those methods help in distinguishing the relevant signal information from background noise. Bringing together theoretical results and practical real-world signal analysis examples, the paper introduces classical and recent contributions to summability theory as well as new signal-processing procedures.