23,091 research outputs found
Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions
The Fourier transforms of Laguerre functions play the same canonical role in
wavelet analysis as do the Hermite functions in Gabor analysis. We will use
them as analyzing wavelets in a similar way the Hermite functions were recently
by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C.
R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K.
Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113,
21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we
find a sufficient density condition for constructing frames by translations and
dilations of the Fourier transform of the nth Laguerre function. As in
Groechenig-Lyubarskii theorem, the density increases with n, and in the special
case of the hyperbolic lattice in the upper half plane it is given by b\log
a<\frac{4\pi}{2n+\alpha}, where alpha is the parameter of the Laguerre
function.Comment: 15 page
The reproducing kernel structure arising from a combination of continuous and discrete orthogonal polynomials into Fourier systems
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
conjecture regarding the existence of a reproducing kernel structure behind
these kernels, by establishing a link with Saitoh theory of linear
transformations in Hilbert space. The results are illustrated with Fourier
kernels with ultraspherical weights, their continuous q-extensions and
generalizations. As a byproduct of this approach, a new class of sampling
theorems is obtained, as well as Neumann type expansions in Bessel and q-Bessel
functions.Comment: 16 pages; Title changed, major reformulations. To appear in Constr.
Appro
A planar large sieve and sparsity of time-frequency representations
With the aim of measuring the sparsity of a real signal, Donoho and Logan
introduced the concept of maximum Nyquist density, and used it to extend
Bombieri's principle of the large sieve to bandlimited functions. This led to
several recovery algorithms based on the minimization of the -norm. In
this paper we introduce the concept of {\ planar maximum} Nyquist density,
which measures the sparsity of the time-frequency distribution of a function.
We obtain a planar large sieve principle which applies to time-frequency
representations with a gaussian window, or equivalently, to Fock spaces,
, allowing for perfect recovery of
the short-Fourier transform (STFT) of functions in the modulation space
(also known as Feichtinger's algebra ) corrupted by sparse noise and for
approximation of missing STFT data in , by -minimization
Universal Landauer conductance in chiral symmetric 2d systems
We study transport properties of an arbitrarily shaped ultraclean graphene
sheet, adiabatically connected to leads,composed by the same material. If the
localized interactions do not destroy chiral symmetry, we show that the
conductance is quantized, since it is dominated by the quasi one-dimensional
leads. As an example, we show that smooth structural deformations of the
graphene plane do not modify the conductance quantization.Comment: 6 pages, no figure
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