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

Bimodal Resonance Phenomena. Part III: High-Contrast Grating Reflectors

The extraordinary broadband high-reflectivity features of high-contrast gratings are stimulating great interest in many opto-electronic applications. In view of obtaining a simple simulation framework, the analogy of high-contrast grating reflectors with bimodal Fabry-Pérot interferometers is proposed. The closed-form expressions of the interferometer reflectivity, obtained starting from a novel parametrization of the scattering matrices characterizing the bar-air interface, allow a complete exploration of the device parameter space, explaining and predicting the phenomenon of ultra-broadband quasi-100% reflectivity. In this paper an optimized and numerically efficient design procedure is described and compared with the standard rigorous coupled wave analysis, both for the classical "bar-in-air" configuration and for a more robust and practical one, with bars lying on a dielectric support. It is shown that the model can be applied also in the more realistic case of lossy gratings