An optical vortex (phase singularity) with a high topological strength resides on the axis of a high-order light beam. The breakup of this vortex under elliptic perturbation into a straight row of unit-strength vortices is described. This behavior is studied in helical Ince-Gauss beams and astigmatic, generalized Hermite-Laguerre-Gauss beams, which are perturbations of Laguerre-Gauss beams. App...

It is shown that as m tends to infinity, the error in the integration of the Chebyshev polynomial of the first kind, T{im+2)j±2^x), by an /n-point Gauss integration rule approaches (-!> • 2/(4/2 1), / = 0, 1, ■ • • , m 1, and (-!>' • tt/2, / = m, for all J. 1. Knowledge of the errors in the numerical integration of Chebyshev polynomials of the first kind, Tn(x), by given integration rules has p...

We provide a definition of Multidimensional Chebyshev Systems of order N which is satisfied by the solutions of a wide class of elliptic equations of order 2N . This definition generalizes a very large class of Extended Complete Chebyshev systems in the one-dimensional case. This is the first of a series of papers in this area, which solves the longstanding problem of finding a satisfactory mul...

We study the kernel Kn,s(z) of the remainder term Rn,s( f ) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod,...

This paper deals with Vandermonde matrices Vn whose nodes are the Gauss–Lobatto Chebyshev nodes, also called extrema Chebyshev nodes. We give an analytic factorization and explicit formula for the entries of their inverse, and explore its computational issues. We also give asymptotic estimates of the Frobenius norm of both Vn and its inverse and present an explicit formula for the determinant o...

A form of this was conjectured by Gauss about 1800, [Chebyshev 1848/52] and [Chebyshev 1850/52] made notable progress with essentially elementary methods. The landmark paper Riemann 1859] made clear the intimate connection between prime numbers and the behavior of ζ(s) as a function of a complex variable. The theorem was proven independently by [Hadamard 1896] and [de la Vallée Poussin 1896] by...

It has recently been shown that the Lebesgue constant for Berrut’s rational interpolant at equidistant nodes grows logarithmically in the number of interpolation nodes. In this paper we show that the same holds for a very general class of well-spaced nodes and essentially any distribution of nodes that satisfy a certain regularity condition, including Chebyshev–Gauss–Lobatto nodes as well as ex...

This paper establishes a direct method for solving variational problems via a set of Radial basis functions (RBFs) with Gauss-Chebyshev collocation centers. The method consist of reducing a variational problem into a mathematical programming problem. The authors use some optimization techniques to solve the reduced problem. Accuracy and stability of the multiquadric, Gaussian and inverse multiq...

In this article, it is shown that under certain conditions, the spectral difference (SD) method is independent of the position of the solution points. This greatly simplifies the design of such schemes, and it also offers the possibility of a significant increase in the efficiency of the method. Furthermore, an accuracy and stability study, based on wave propagation analysis, is presented for s...

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian...