Hankel transforms of general monotone functions

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for real-valued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series

Maximal values for the simultaneous number of null components of a vector and its Fourier transform

The author acknowledges the support of prof. J. Soria de Diego (Universitat de Barcelona), who advised the Master thesis in which this research was carried out. The research was partially supported by an AGAUR master's grant (course 2013-14) and the grant MTM2014-59174-

Riesz bases of exponentials for convex polytopes with symmetric faces

We prove that for any convex polytope $\Omega\subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$. The result is new in all dimensions $d$ greater than one.Research supported by ISF Grants No. 447/16 and No. 227/17 and ERC Starting Grant No. 713927publishe

Riesz bases of exponentials for convex polytopes with symmetric faces

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$. The result is new in all dimensions $d$ greater than one.Comment: To appear in Journal of the European Mathematical Society (JEMS

Gabor orthonormal bases, tiling and periodicity

We show that if the Gabor system $\{g(x − t)e^{2\pi isx}, t\in T,s\in S\}$, is an orthonormal basis in $L^2(\mathbb{R})$ and if the window function $g$ is compactly supported, then both the time shift set $T$ and the frequency shift set $S$ must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.publishe