C1 Changes of Variable: Beurling-Helson Type Theorem and Hörmander Conjecture on Fourier Multipliers.
Carleson's Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson's theorem. An outline of these variations is also given.
We determine the convolution operators on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).
In a recent paper [3] C. Baiocchi, V. Komornik and P. Loreti obtained a generalisation of Parseval's identity by means of divided differences. We give here a proof of the optimality of that theorem.
On connaît le lien intime qui existe entre les équations fonctionnelles des fonctions et les formules sommatoires dont le prototype est donné par celle de Poisson. Ce lien fait intervenir la transformation intégrale de Fourier et ses généralisations. Ici, nous réexaminons la signification harmonique (ainsi qu’hilbertienne et distributionnelle) des équations fonctionnelles ayant la forme la plus simple, à savoir, celle s’appliquant pour la fonction dzêta de Riemann et les séries de Dirichlet...
We study one-dimensional oscillator integrals which arise as Fourier-Stieltjes transforms of smooth, compactly supported measures on smooth curves in Euclidean spaces and determine their decay at infinity, provided the curves satisfy certain geometric conditions.