We introduce fusion bialgebras and their duals systematically study Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of rings. prove Hausdorff-Young inequality, uncertainty principles for duals. show that Schur product property, Young's inequality sum-set estimate hold bialgebras, but not always If ring is Grothendieck a catego...

This note describes the pleasant features that accrue in weighted settings when the partial sums of the operator-valued Fourier series corresponding to a multiplier function ψ : T → C are uniformly bounded in operator norm. This circle of ideas also includes a Tauberiantype condition on the multiplier function ψ sufficient to insure such uniform boundedness of partial sums. These considerations...

We characterize in terms of Fourier spectrum the boundary values of functions in the complex Hardy spaces H(C±), 1 ≤ p ≤ ∞. As an application we extend the Bedrosian identity, originally stated for square-integrable functions, to the Lp(R) cases.

This is an expository paper on finite Fourier analysis and basic number theory. The focus of the paper is finite Abelian groups and the properties of these groups which are used in the proof of the Dirichlet prime number theorem. We then introduce the specific elements and basic structure of the proof of Dirichlet’s theorem.