Exploring Hyers-Ulam Stability in Additive Functional Equations: Applications in Signal and Image Processing

Abstract: The concept of Hyers-Ulam stability, introduced in response to Ulam's question on the stability of group homomorphisms, has evolved into a pivotal framework in modern mathematics. This study investigates the stability of additive functional equations, focusing on their practical applications in signal and image processing. By examining the behaviour of approximately satisfied equations under small perturbations, Hyers-Ulam stability provides robust methodologies for noise-resilient signal reconstruction and image denoising. Computational experiments demonstrate the practical utility of this concept in real-world scenarios, such as speech-to-text systems and digital image enhancement, where deviations caused by noise or system imperfections are systematically bounded. The results highlight how Hyers-Ulam stability ensures reliable performance, making it an essential tool for advancing technologies in communication and computational imaging.