Investigating the Fourier Transform of Continuous Functions

Abstract

This study rigorously investigates the Fourier transform of continuous functions, delving into its mathematical foundations, properties, and practical applications. The exploration covers the analysis of specific classes of continuous functions, including piecewise continuous and differentiable functions, employing a detailed mathematical approach for Fourier transform calculations. Advanced topics such as generalized functions, distributions, and the Dirac delta function are examined, emphasizing their significance. The study also explores the sampling theorem and its connection to the Fourier transform, showcasing its relevance in signal processing. Applications in diverse fields, such as image processing, communications, and quantum mechanics, are discussed. The study concludes with a summary of key findings, potential avenues for future research, and reflections on the broader significance of Fourier transforms in mathematical analysis and applied sciences.