Abstract:
The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.

Abstract:
Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called unified method in the interior of a convex polygon provides an example of the applicability of these analytic expressions.

Abstract:
The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points $x=0,1,\cdots, N-1$, $N$ being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for $t_{n}(aN,N+1)$ in the double scaling limit, namely, $N\rightarrow\infty$ and $n/N\rightarrow b$, where $b\in (0,1)$ and $a\in(-\infty,\infty)$; see [Studies in Appl. Math. \textbf{128} (2012), 337-384]. In the present paper, we continue to investigate the behaviour of these polynomials when the parameter $b$ approaches the endpoints of the interval $(0,1)$. While the case $b\rightarrow 1$ is relatively simple (since it is very much like the case when $b$ is fixed), the case $b\rightarrow 0$ is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities $n$, $x$ and $xN/n^2$, and different special functions have been used as approximants, including Airy, Bessel and Kummer functions.

Abstract:
In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn (z, N) as the degree grows to infinity. Global asymptotic formulas are obtained as n \rightarrow \infty, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

Abstract:
The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points $x=0,1,..., N-1$, N being a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions for $t_{n}(aN,N+1)$ in the double scaling limit, namely, $N\rightarrow\infty$ and $n/N\rightarrow b$, where $b\in(0,1)$ and $a\in(-\infty,\infty)$. One expansion involves the confluent hypergeometric function and holds uniformly for $a\in[0,1/2]$, and the other involves the Gamma function and holds uniformly for $a\in(-\infty, 0)$. Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for $a\geq1/2$ can be obtained via a symmetry relation of $t_{n}(aN,N+1)$ with respect to $a=1/2$. Asymptotic formulas for small and large zeros of $t_{n}(x,N+1)$ are also given.

Abstract:
There is presented an approach to find an approximation polynomial of a function with two variables based on the two dimensional discrete Fourier transform. The approximation polynomial is expressed through Chebyshev polynomials. There is given an uniform convergence result.

Abstract:
Spin-tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labelled by spin projections and 2-sphere coordinates. The star-product kernel for such functions is obtained in explicit form and connected with Fourier transform of characters of SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch-Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin-tomographic symbols and dual tomographic symbols.

Abstract:
By combining Chebyshev polynomials with modulus compute,extending Chebyshev polynomials' definition domain to real number,some conclusions were drawn by theoretic verification and data analysis.Making use of the framework of the traditional public-key algorithm RSA and ElGamal,proposed a chaotic public-key encryption algorithm based on extending discrete Chebyshev polynomials' definition domain to Real number.Its security is based on the intractability of the integer factorization problem as RSA,and it is a...

Abstract:
We expand some, mostly nonnegative rational functions of one or two variables in the series of Chebyshev polynomials. If such a series is multiplied by the density that makes Chebyshev polynomials orthogonal then we obtain Fourier series expansion of certain probabilistic density in the case of rational function of one variable or Lancaster expansion in the case of function of two variables. We study also the general case of rational function of order n depending (symmetrically) on n parameters and find (at least theoretical) expansion

Abstract:
A discrete Fourier analysis on the fundamental domain $\Omega_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $\Omega_2$ is the regular hexagon and $\Omega_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.