The classical Fourier transform Fcl is an isomorphism of the Schwartz space S(Rk) onto itself. The space C∞ c (Rk) of smooth functions with compact support is dense in S(Rk), and the classical Paley-Wiener theorem characterises the image of C∞ c (R k) under Fcl as rapidly decreasing functions having an holomorphic extension to Ck of exponential type. Since Rk is self-dual, the same theorem also...

The book under review is an excellent introduction to the group theoretical and analytic aspects of the field by one of its pioneers. Before reviewing the book, we will provide an overview of the field. Integral geometry draws together analysis, geometry, and numerical mathematics. It has direct applications in PDEs, group representations, and the applied mathematical field of tomography. The f...

In Monte Carlo simulations of lattice field theory with a θ term, one confronts the complex weight problem, or the sign problem. This is circumvented by performing the Fourier transform of the topological charge distribution P (Q). This procedure, however, causes flattening phenomenon of the free energy f(θ), which makes study of the phase structure unfeasible. In order to treat this problem, w...

We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced ...

The Hilbert transform is essentially the only singular operator in dimension 1. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on several theoretical and physical problems across a wide range of disciplines; this includes problems in Fourier convergence, complex analysis, potential theory, modulatio...

) ( ) ( 2 q q x x x Tr x f + + = and A = {ai} where ) ( i i f a α = . In reference [1], the sequence A is conjectured to have a two-level autocorrelation for all n > 1 and this conjecture has been verified for 23 3 ≤ ≤ n . In this note, we present an interesting formula for the Hadamard transform of the sequence A or, equivalently, for the function f(x). The Hadamard transform of f(x) is define...

A method for derivation of inversion formulas for the Radon transform of f (x) is given. This method allows one to construct functions ψ(x) which are easy to compute and which have the same wave fronts as f (x). This, in turn, allows one to calculate the singular support S of f (x) by computing ψ(x) given tomographic data. A simple geometrical relation between the singular supports of f andˆf ,...

We give an explicit formula (i.e., an explicit expression for the formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. Introduction We will denote by C((t)) (resp. C({t})) the field C[[t]][t] (resp. C{t}[t]) of formal (resp. convergent) Laurent series of the variable t, ...

It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f . However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set...