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Applications of Lucas sequences in convergence and signal processing
Majeed Ahmad Yousif1, Ibrahim Sulaiman Ibrahim1, Meraa Arab2
1Department of Mathematics, College of Education, University of Zakho, Zakho, 42002, Iraq.
Scientific Reports
|April 6, 2026
Summary
This study introduces new methods for Lucas sequence analysis, enhancing signal processing with statistical and summability techniques for improved noise immunity and data distinction.
Area of Science:
- Number Theory
- Signal Processing
- Mathematical Analysis
Background:
- Lucas sequences are fundamental in number theory.
- Statistical convergence and summability are crucial in mathematical analysis.
- Signal processing requires robust methods for noise reduction and feature extraction.
Purpose of the Study:
- To introduce novel statistical and summability concepts for Lucas sequences.
- To explore the application of these concepts in signal processing.
- To generalize existing convergence theories and enhance signal analysis.
Main Methods:
- Definition of λ-Lucas statistical convergence and strong λ-Lucas summability using modulus functions.
- Investigation of the Lucas transform and its embedding within Lucas numbers.
- Development of inclusion principles and equivalence theorems.
Main Results:
- Demonstration of the proposed approach's flexibility through numerical simulations on noisy signals and blurred images.
- Evidence of considerable noise immunity due to a steady decay to zero.
- Establishment of conditions for uniqueness and generalization of statistical convergence theory.
Conclusions:
- The proposed methods offer new perspectives for signal quality assessment, compression, and filtering.
- The study bridges theoretical contributions in summability theory with practical signal processing procedures.
- The developed techniques effectively distinguish relevant signal information from background noise.
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