The Banach Algebra A* and Its Problems.
The Bohr-Pál theorem and the Sobolev space
The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if α < 1/2,...
The construction of a function given by moments. (La construction de quelque fonction par des moments donnés.)
The Euler Maclaurin Expansion for the Cauchy Principal Value Integral.
The Schrödinger density and the Talbot effect
We study the local properties of the time-dependent probability density function for the free quantum particle in a box, i.e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. √δ-families of initial functions are considered whose squared magnitudes approximate the periodic Dirac δ-function. The focus is on the set of rectilinear domains where the density has a special character, in particular,...
Théorie de la série de Fourier
Time-frequency analysis of Sjöstrand's class.
We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand's class, with methods of time-frequency analysis (phase space analysis). Compared to the classical treatment, the time-frequency approach leads to striklingly simple proofs of Sjöstrand's fundamental results and to far-reaching generalizations.
Topological results on sequences and their applications in the theory of trigonometric series
Two problems related to the non-vanishing of
We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or , where denotes the “centered” fractional part of . These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet -functions at .