Fourier transforms of Lorentz invariant functions

Self Fourier functions and fractional Fourier transforms

It was shown [ 21 that any SFF can be decomposed in this manner. Thus, F(x) is an SFF if, and only if, it can be expressed as the sum of four functions in the form of the above equation. Additional SFF studies are reported in refs. [ 3-51. Another issue that has been recently investigated is the fractional Fourier transform [ 6-91. Two distinct definitions of the fractional Fourier transform ha...

Bounded Functions and Fourier Transforms

1. Statement of the principal result. The class of sequences obtainable as Fourier coefficients of measures on the circle group presents numerous structure problems of interest. Amongst the earliest results somewhat akin to that stated below appear Banach's theorems about lacunary coefficients; see e.g., [4, pp. 215-220]. More recently Helson [3] has studied analogous questions when the group c...

Nonsmooth Analysis of Lorentz Invariant Functions

A real valued function g(x, t) on Rn×R is called Lorentz invariant if g(x, t) = g(Ux, t) for all n×n orthogonal matrices U and all (x, t) in the domain of g. In other words, g is invariant under the linear orthogonal transformations preserving the Lorentz cone: {(x, t) ∈ Rn × R | t ≥ ‖x‖}. It is easy to see that every Lorentz invariant function can be decomposed as g = f ◦ β, where f : R2 → R i...

Surface Measures - Maximal Functions and Fourier Transforms

Discrete Morse Functions from Fourier Transforms

A discrete Morse function for a simplicial complex describes how to construct a homotopy equivalent CW-complex with hopefully fewer cells. We associate a boolean function with the simplicial complex and construct a discrete Morse function using its Fourier transform. Methods from theoretical computer science by O’Donnell, Saks, Schramm, and Servedio, together with experimental data on complexes...