Spectral theory is a branch of mathematics that studies the spectrum of linear operators, particularly in the context of functional analysis. It focuses on understanding how operators can be analyzed through their eigenvalues and eigenvectors, linking these concepts to various mathematical frameworks like Fourier analysis. In this context, spectral theory plays a crucial role in establishing important results like the inversion formula and Plancherel's theorem, which are essential for working with Fourier transforms and understanding the properties of functions in different spaces.
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