The Algebraic Structure Underlying Fourier Analysis

Abstract

In this paper, we begin with the concept of a projection in the familiar vector space R^n, and then extend it to an arbitrary inner product space. This foundational idea enables us to compute a line of best fit, approximate functions using polynomials, and decompose a function into an infinite series of sines and cosines, known as its Fourier series. We then extend the Fourier series to finite groups using representation and character theory. In the process, we will examine key results, including Maschke’s theorem and Schur’s orthogonality relations. Lastly, we provide a reasonably detailed guide to further extend this reasoning to compact groups using the Peter-Weyl theorem. For the most part, a background in lower-division linear algebra and introductory group theory will be sufficient, but towards the end, knowledge of measure theory and general topology will be necessary.

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