Almost everywhere convergence of Fourier integrals in Rn
Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an L2-localisation principle.
An analogue of Gutzmer's formula for Hermite expansions
We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of L²(ℝⁿ) under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.
An uncertainty principle related to the Poisson summation formula
We prove a class of uncertainty principles of the form , where is the short time Fourier transform of f. We obtain a characterization of the range of parameters a,b,p,q for which such an uncertainty principle holds. Counter-examples are constructed using Gabor expansions and unimodular polynomials. These uncertainty principles relate the decay of f and f̂ to their behaviour in phase space. Two applications are given: (a) If such an inequality holds, then the Poisson summation formula is valid...
An uniform boundedness for Bochner-Riesz operators related to the Hankel transform.
Analyse de Fourier sur des ensembles non-convexes
Analysis on arithmetic schemes. I.
Approximation properties of partial sums of Fourier series.
Arithmetical functions with periodic zeros
Asymptotic Fourier and Laplace transformations for hyperfunctions
We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.
Asyptotic Behaviour of Multiperiodic Functions G(x) = ... g (x/2...).
Average decay of Fourier transforms and geometry of convex sets.
Let B be a convex body in R2, with piecewise smooth boundary and let ^χB denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical Lp averages of ^χB and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.
Averages of functions over hypersurfaces in Rn.