Applications of Fourier Series on Signal processing and Heat conduction

This paper explores the application of the Fourier series in mathematical analysis and its significant impact on scientific fields such as signal processing and heat conduction. The study emphasizes the utility of Fourier series in decomposing complex functions into simpler trigonometric components, which facilitates the analysis of periodic functions. A primary focus is on the series’ role in approximating square waves in signal processing, demonstrating its effectiveness despite challenges like the Gibbs phenomenon. Additionally, the Fourier series is applied to solving the heat equation, where it models the evolution of temperature distribution over time in a medium. Techniques to improve convergence and mitigate oscillations, such as corrected Fourier series and summation methods like Cesàro and Fejér summation, are discussed to enhance the accuracy of approximations in cases with discontinuities. The results underline the Fourier series’ versatility in representing and analyzing both smooth and discontinuous functions, showcasing its importance in theoretical and applied mathematics.