661 research outputs found
Characterizations of Hankel multipliers
We give characterizations of radial Fourier multipliers as acting on radial
L^p-functions, 1<p<2d/(d+1), in terms of Lebesgue space norms for Fourier
localized pieces of the convolution kernel. This is a special case of
corresponding results for general Hankel multipliers. Besides L^p-L^q bounds we
also characterize weak type inequalities and intermediate inequalities
involving Lorentz spaces. Applications include results on interpolation of
multiplier spaces.Comment: Final revised version to appear in Mathematische Annale
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
Potential operators associated with Jacobi and Fourier-Bessel expansions
We study potential operators (Riesz and Bessel potentials) associated with
classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for
the corresponding potential kernels. Then we characterize those , for which the potential operators are of strong type , of weak
type and of restricted weak type . These results may be thought
of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional
integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient
of our line of reasoning, we also obtain sharp estimates of the Poisson kernel
related to Fourier-Bessel expansions.Comment: 28 pages, 4 figures; v2 (some comments on Bessel potentials added
On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length
Solving inverse scattering problem for a discrete Sturm-Liouville operator
with the fast decreasing potential one gets reflection coefficients and
invertible operators , where is the Hankel operator
related to the symbol . The Marchenko-Fadeev theorem (in the continuous
case) and the Guseinov theorem (in the discrete case), guarantees the
uniqueness of solution of the inverse scattering problem. In this article we
asks the following natural question --- can one find a precise condition
guaranteeing that the inverse scattering problem is uniquely solvable and that
operators are invertible? Can one claim that uniqueness implies
invertibility or vise versa?
Moreover we are interested here not only in the case of decreasing potential
but also in the case of asymptotically almost periodic potentials. So we merege
here two mostly developed cases of inverse problem for Sturm-Liouville
operators: the inverse problem with (almost) periodic potential and the inverse
problem with the fast decreasing potential.Comment: 38 pages, AMS-Te
Riesz transforms on compact Riemannian symmetric spaces of rank one
In this paper we prove mixed norm estimates for Riesz transforms related to
Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one.
These operators are closely related to the Riesz transforms for Jacobi
polynomials expansions. The key point is to obtain sharp estimates for the
kernel of the Jacobi--Riesz transforms with uniform control on the parameters,
together with an adaptation of Rubio de Francia's extrapolation theorem. The
latter results are of independent interest.Comment: 19 pages. To appear in Milan Journal of Mathematic
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