Encontrados 15 documentos, a visualizar página 1 de 2
Landau's necessary density conditions for the Hankel transform
Abreu, Luís Daniel; Bandeira, AfonsoWe will prove an analogue of Landau’s necessary conditions [Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967).] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier-Bessel frames are obtained. In the course of our pro...
A Paley-Wiener theorem for the Askey-Wilson function transform
Abreu, Luis Daniel; Bouzeffour, FethiWe define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type ln q−1, providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and g...
On the structure of Gabor and super Gabor spaces
Abreu, Luís DanielWe study the structure of Gabor and super Gabor spaces as subspaces of L2(R2d) and specialize the results to the case where the spaces are generated by vectors of Hermite functions. We then show that such spaces are isometrically isomorphic to Fock spaces of polyanalytic functions and obtain structure theorems and orthogonal projections for both spaces at once. In particular we recover a structure result obtain...
Sampling and interpolation in the Bargmann-Fock space of polyanalytic functions
Abreu, Luis DanielWe give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions (poly-Fock spaces), displaying a ”Nyquist rate” which increases with n, the degree of polyanaliticity of the space: A sequence of lattice points is sampling if and only if its density is strictly larger than n, and it is interpolating if and only if its density is strictly smaller ...
On Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg...
Abreu, Luís DanielGabor frames with Hermite functions are equivalent to Fock frames with monomials windows and to sampling sequences in true poly-Fock spaces. In the L2 case, such an equivalence results from the unitarity of the so-called true poly- Bargmann transform. We will extend the equivalence to Banach spaces, applying Feichtinger-Gr¨ochenig coorbit theory to the Fock representation of the Heisenberg group. This task requ...
Bilinear biorthogonal expansions and the spectrum of an integral operator
Abreu, Luís Daniel; Ciaurri, Óscar; Varona, Juan LuisWe study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier- Neumann type series as special cases. Concerning applications, several new results are obtained. From the Dunkl analogue of Gegenbauer’s expansion of the plane wave, we derive sampling...
A Paley-Wiener theorem for the Askey-Wilson function transform
Abreu, Luís Daniel; Bouzeffour, FethiWe define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type ln q−1, providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and g...
Sampling and interpolation in the Bargmann-Fock space of polyanalytic functions
Abreu, Luís DanielWe give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions (poly-Fock spaces), displaying a ”Nyquist rate” which increases with n, the degree of polyanaliticity of the space: A sequence of lattice points is sampling if and only if its density is strictly larger than n, and it is interpolating if and only if its density is strictly smaller ...
The Reproducing Kernel Structure Arising from a Combination of Continuous and D...
Abreu, Luís DanielAbstract We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier-type systems.We prove Ismail’s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh’s theory of linear transformations in Hilbert space. T...
Real Paley-Wiener theorems for the Koornwinder-Swarttouw q-Hankel transform
Abreu, Luís DanielWe derive two real Paley-Wiener theorems in the setting of quantum calculus. The first uses techniques due to Tuan and Zayed [21] in order to describe the image of the space L2 q(0,R) under Koornwinder and Swarttouw q-Hankel transform [14] and contains as a special case a description of the domain of the q-sampling theorem associated with the q-Hankel transform [1]. The second characterizes the image of compact...
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