The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.
Using Bochner-Riesz means we get a multidimensional sampling theorem for band-limited functions with polynomial growth, that is, for functions which are the Fourier transform of compactly supported distributions.
Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially lntegrable...
A general method is given for recovering a function , , knowing only an approximation of its Fourier transform.
Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.
This paper uses the theory of entire functions to study the linear quadratic optimization problem for a class of continuous 2D systems. We show that in some cases optimal control can be given by an analytical formula. A simple method is also proposed to find an approximate solution with preassigned accuracy. Some application to the 1D optimization problem is presented, too. The obtained results form a theoretical background for the design problem of optimal controllers for relevant processes.
The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from to L² for any s > 3/8. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of α = 2. The extension from α = 2 to general α > 1 faces three essential obstacles: the lack of Lee’s reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable...
2000 Mathematics Subject Classification: 42B10, 43A32.In this paper we take the strip KL = [0, +∞[×[−Lπ, Lπ], where L is a positive integer. We consider, for a nonnegative real number α, two partial differential operators D and Dα on ]0, +∞[×] − Lπ, Lπ[. We associate a generalized Fourier transform Fα to the operators D and Dα. For this transform Fα, we establish an Lp − Lq − version of the Morgan's theorem under the assumption 1 ≤ p, q ≤ +∞.