Fourier Analysis on Groups

By W. Rudin. 1962

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Bayesian Nonparametric Kernel-Learning (2015, methodology): “er Rd: K(x,y) = k(x− y) where x,y ∈ Rd and k is a positive deﬁnite function. The use of random Fourier features for kernel approximation is a result of Monte Carlo integration using Bochner’stheorem [Rudin, 1990]. Bochner’stheorem states that a continuous shift-invariant kernel K(x,y) = k(x−y) is a ”
Pointwise ergodic theorems for non-conventional bilinear polynomial averages (2020, background): “2L2(G). If ˆG is measurable, we say that f2L2(G) is Fourier supported in if F Gfvanishes outside of (modulo null sets). The space of such functions will be denoted L2(G) . As is well known (see e.g., [70]), every LCA group G has a Pontryagin dual G, and the inverse Fourier transform F 2 1 G : L(G ) ”
Codes on graphs: normal realizations (2000, methodology): “Pontryagin duality theory applies to locally compact Abelian (LCA) groups such as finite Abelian groups and real Euclidean -space , and underlies all Fourier analysis [35], [ 40 ], [23],”