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Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
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When is a collection of exponentials dense in $L^2(K), |K|<\infty$
Just a comment but I am not empowered. Firstly don't you mean something like the linear hull being dense? In general, the type of family you describe will not be dense. If we use this recasting of …