From a novel but incomplete idea that faced criticism from some of the greatest mathematicians of the early 19th century, Fourier series theory went on to be refined and developed by subsequent generations of mathematicians into a powerful mathematical tool with applications across many modern fields. The article is based on the method of reviewing and synthesizing academic literature, aiming to summarize the development of the Fourier series and highlight one of its concrete applications in modern life — specifically, its role in audio processing. The implications of this review point to how theoretical advancements have enabled efficient digital applications, especially through the Fast Fourier Transform (FFT), which plays an important part in audio technology.
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