Fractional smoothness in LpL^p with Dunkl weight and its applications

We define fractional power of the Dunkl Laplacian, fractional modulus of smoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight. As application, we prove direct and inverse theorems of approximation theory, and some inequalities for entire functions of spherical exponential type in fractional settings.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1703.0683

On amplification of light in the continuous EPR state

Two schemes of amplification of two-mode squeezed light in the continuous variable EPR-state are considered. They are based on the integrals of motion, which allow conserving quantum correlations whereas the power of each mode may increase. One of these schemes involves a three-photon parametric process in a nonlinear transparent medium and second is a Raman type interaction of light with atomic ensemble.Comment: 3 pages, revtex4, no figure

Transfer formalism for quantum optics problems

Consistent quantum formalism based on the localized basis of the Wannirer functions in Heisenberg and Schrodinger pictures to describe propagation of electromagnetic field in a three dimensional media including diffraction is presented. In the Schrodinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasiprobability and corresponding Langevin equations are given. As result the space-time description is obtained by a simple changing variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of nondegenerated parametric and two-photon interactions results in amplification of modes when nonclassical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.Comment: LaTeX file, \documentstyle[11pt,fleqn]{article

Statistics of the single mode light in the transparent medium with cubic nonlinearity

The quantum statistics of the light in the transparent medium with cubic nonlinearity is considered. Two types of transparent media are treated, namely, the cold transparent medium with a ground working level and the inversion-free medium with the lasing levels of the same population. The spectra of light fluctuation are found on the basis of both Scully-Lamb and Haken theories. The conditions for the use of effective Hamiltonian are determined. Basing on the exact solution of the Fokker-Plank equation for the Glauber-Sudarshan P-function the inversion-free medium with cubic nonlinearity is shown to be the source of spontaneous radiation with non-Gaussian statistics.Comment: LaTeX file(18 pages), documentclass{article

Riesz potential and maximal function for Dunkl transform

We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1, L^q)$ estimate. We find a sharp constant in the weighted $L^p$-inequality, generalizing the results of W. Beckner and S. Samko.Comment: 25 page

Positive LpL^p-bounded Dunkl-type generalized translation operator and its applications

We prove that the spherical mean value of the Dunkl-type generalized translation operator $\tau^y$ is a positive $L^p$-bounded generalized translation operator $T^t$. As application, we prove the Young inequality for a convolution defined by $T^t$, the $L^p$-boundedness of $\tau^y$ on a radial functions for $p>2$, the $L^p$-boundedness of the Riesz potential for the Dunkl transform and direct and inverse theorems of approximation theory in $L^p$-spaces with the Dunkl weight.Comment: 44 page

Uncertainty principles for eventually constant sign bandlimited functions

We study the uncertainty principles related to the generalized Logan problem in $\mathbb{R}^{d}$. Our main result provides the complete solution of the following problem: for a fixed $m\in \mathbb{Z}_{+}$, find $$\sup\{|x|\colon (-1)^{m}f(x)>0\}\cdot \sup \{|x|\colon x\in \mathrm{supp}\,\widehat{f}\,\}\to \inf,$$ where the infimum is taken over all nontrivial positive definite bandlimited functions such that $\int_{\mathbb{R}^d}|x|^{2k}f(x)\,dx=0$ for $k=0,\dots,m-1$ if $m\ge 1$. We also obtain the uncertainty principle for bandlimited functions related to the recent result by Bourgain, Clozel, and Kahane.Comment: 30 page

The blind detection for palette image watermarking without changing the color

To hide a binary pattern in the palette image a steganographic scheme with blind detection is considered. The embedding algorithm uses the Lehmer code by palette color permutations for which the cover image palette is generally required. The found transformation between the palette and RGB images allows to extract the hidden data without any cover work.Comment: 6 pages, 3 figure

Teleportation of entanglement for continuous variables

Teleportation of a pure two particle entangled state of continuous variables by triplet of the Greenberger-Horne-Zeilinger form is considered. The three-particle basis needed for a joint measurement is found. It describes a measurement of momentum one of single particle and total moment and relative position of the two others. Optical realization using squeezed state of the light is discussed.Comment: 7 pages, latex, minor changes, some typos correcte

Teleportation of entangled states and dense coding using a multiparticle quantum channel

A set of protocols for teleportation and dense coding tasks with the use of a N particle quantum channel, presented by entangled states of the GHZ class, is introduced, when N>2. Using a found representation for the multiparticle entangled states of the GHZ class, it has shown, that for dense coding schemes enhancement of the classical capacity of the channel due from entanglement is N/N-1. If N>2 there is no one - to -one correspondence between teleportation and dense coding schemes in comparison with the EPR channel is exploited. A set of schemes, for which two additional operations as entanglement and disentanglement are permitted, is considered.Comment: 10 pages, latex, a topic on capacity is added, stylistic improvements, conclusions and results unchange