Paley Johnson (1917–2011)

Generalized Paley-Wiener Theorems

Non-harmonic Fourier transform is useful for the analysis of transient signals, where the integral kernel is from the boundary value of Möbius transform. In this note, we study the Paley–Wiener type extension theorems for the non-harmonic Fourier transform. Two extension theorems are established by using real variable techniques.

Paley triple arrays

Paley-wiener-type Theorems

The Fourier transforms of functions with compact and convex supports in R are described. The Fourier transforms of functions with nonconvex and unbounded supports are also considered.

Trace Paley - Wiener Theorem

w Statement of the theorem 1.1. Let G be a reductive p-adic group. A smooth representation (~', E) of the group G on a complex vector space E is called a G-module. Usually we shorten the notation and write w or E. Let d~(G) be the category of G-modules, Irr G the set of equivalence classes of irreducible G-modules, and R (G) the Grothendieck group of G-modules of fnite length; R(G) is a free ab...

Littlewood–Paley Inequailty: A Survey

Let Sωf = ∫ ω f̂(ξ)e ixξ dξ be the Fourier projection operator to an interval ω in the real line. Rubio de Francia’s Littlewood Paley inequality [28] states that for any collection of disjoint intervals Ω, we have ∥∥ [∑ ω∈Ω |Sωf | 1/2∥∥ p . ‖f‖p, 2 ≤ p < ∞. We survey developments related to this inequality, including the higher dimensional case, and consequences for multipliers.