Improvement Taykov's lower bound in an inequation between C and L norms for trigonometric polynomials

We give new lower asymptotical estimate of constant $$C_n=\sup\biggl\{\frac{\|t_n\|_{C(\mathbb T)}}{\|t_n\|_{L(\mathbb T)}}:t_n\text{are real trigonometric polynomials}, \operatorname{deg}t_n<n\biggr\}$$ as $n\to\infty$. This estimate improves known bound of L.V.Taykov (1965).Comment: 3 page

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

A binary noisy channel to model errors in printing process

To model printing noise a binary noisy channel and a set of controlled gates are introduced. The channel input is an image created by a halftoning algorithm and its output is the printed picture. Using this channel robustness to noise between halftoning algorithms can be studied. We introduced relative entropy to describe immunity of the algorithm to noise and tested several halftoning algorithms.Comment: 5 pages 7 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

Relation between Tur\'an extremum problem and van der Corput sets

Let $K\subset\mathbb N$ and $\mathbf T(K)$ is a set of trigonometric polynomials $$T(x)=T_0+\sum_{k\in K, k\le H}T_k\cos(2\pi kx), \qquad H>1,$$ $T(x)\ge0$ for all $x$ and $T(0)=1$. Suppose that $0<h\le1/2$ and $K(h)$ is the class of functions $$f(x)=\sum_{n=0}^{\infty}a_n\cos(2\pi nx)$$ satisfying the following conditions: $a_n\ge0$ for all $n$, $f(0)=1$ and $f(x)=0$ for $h\le|x|\le1/2$. We consider an relation between extremum problem $$\delta(K)=\inf_{T\in\mathbf T(K)}T_0$$ and Tur\'an extremum problem $$A(h)=\sup_{f\in K(h)}a_0=\sup_{f\in K(h)}\int_{-h}^hf(x) dx$$ for rational numbers $h=p/q$ and set $K=\bigcup\limits_{\nu=0}^\infty\{q\nu+p,...,q\nu+q-p\}$. The problem $\delta(K)$ is connection with van der Korput sets. Van der Korput sets study in analytic number theory

Turan Extremum Problem for Periodic Function with Small Support

We consider an extremum problem posed by Turan. The aim of this problem is to find a maximum mean value of 1-periodic continuous even function such that sum of Fourier coefficient modules for this function is equal to 1 and support of this function lies in $[-h,h]$, $0<h\le 1/2$. We show that this extremum problem for rational $h=p/q$ is equivalent two finite-dimensional linear programming problems. Here there are exact results for rational $h=2/q$, $h=p/(2p+1)$, $h=3/q$, and asymptotic equalities.Comment: 8 page

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