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<p>We consider a novel setting for local harmonic analysis on reductive groups motivated by Langlands functoriality conjecture. To this end, we characterize certain non-linear Schwartz spaces on tori and reductive groups in spectral terms, and develop some of their structure in the unramified case, and we derive estimates of their moderate growth at infinity. We also consider non-linear Fourier transforms, and calculate their action on tame supercuspidal representations of $GL_2(F)$ in terms of inducing cuspidal data.</p>

Since Hecke first proved his correspondence between Dirichlet series with functional equations and automorphic forms, there have been a great number of generalizations. Of particular interest is a generalization due to Bochner that gives a correspondence between Dirichlet series with any finite number of poles that satisfy the classical functional equation and automorphic integrals with (finite) log-polynomial sum period functions. In this dissertation, we extend Bochner's result to Dirichlet series with finitely many essential singularities. With some restrictions on the underlying group and the weight, we also prove a correspondence for Dirichlet series with infinitely many poles. For this second correspondence, we provide a technique to approximate automorphic integrals with infinite log-polynomial sum period functions by automorphic integrals with finite log-polynomial period functions. / Mathematics