Trigonometric Case Research Articles

A Family of 4-Point <i>n</i>-Ary Interpolating Scheme Reproducing Conics

The n-ary subdivision schemes contrast favorably with their binary analogues because they are capable to produce limit functions with the same (or higher) smoothness but smaller support. We present an algorithm to generate the 4-point n-ary non-stationary scheme for trigonometric, hyperbolic and polynomial case with the parameter for describing curves. The performance, analysis and comparison of the 4-point ternary scheme are also presented.

Transition between the Lagrange and Hermite–Fejér interpolations. I (Trigonometric case)

We construct a special class of discrete φ-summations to describe the transition between the well known (trigonometric) Lagrange and Hermite–Fejer interpolations. We give general properties, operator norm and convergence order for these cases.

Convergence of series of Fourier coefficients for multiplicative convolutions

We study the convergence of series of the Fourier-Vilenkin coefficients of functions represented as multiplicative convolutions. In the trigonometric case similar results are obtained by C. Onneweer, M. Izumi, and S. Izumi. Moreover, we consider certain analogs of I. Hirshman and W. Rudin transforms of Fourier coefficients. Some results are proved to be unimprovable in a certain sense.

Parametric Centers for Trigonometric Abel Equations

This article is devoted to one-dimensional perturbative theory on R × S 1. There is a recursive formula for the successive obstructions to parametric center at any order of the perturbation parameter. The first obstruction is studied by means of complex analysis techniques. This extends to the trigonometric case what was done previously for the polynomial case (Israel J. Math. 142, 273–283, 2004).

(Quantum) twisted Yangians: Symmetry, Baxterisation, and centralizers

Based on the (quantum) twisted Yangians, integrable systems with special boundary conditions, called soliton nonpreserving (SNP), may be constructed. In the present article we focus on the study of subalgebras of the (quantum) twisted Yangians, and we show that such a subalgebra provides an exact symmetry of the rational transfer matrix. We discuss how the spectrum of a generic transfer matrix may be obtained by focusing only on two types of special boundaries. It is also shown that the subalgebras, emerging from the asymptotics of tensor product representations of the (quantum) twisted Yangian, turn out to be dual to the (quantum) Brauer algebra. To deal with general boundaries in the trigonometric case we propose a new algebra, which also provides the appropriate framework for the Baxterisation procedure in the SNP case.

Solvable scalar and spin models with near-neighbors interactions

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials.

Interpolating polynomial wavelets on [?1,1

The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallee Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.

On Warnaar's elliptic matrix inversion and Karlsson–Minton-type elliptic hypergeometric series

Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton type. These latter identities seem quite unusual and appear to be new already in the trigonometric (i.e., p = 0 ) case.

Quantum Integrable Multiatom Matter-Radiation Models With and Without the Rotating-Wave Approximation

New integrable multi-atom matter-radiation models with and without rotating wave approximation (RWA) are constructed and exactly solved through algebraic Bethe ansatz. The models with RWA are generated through ancestor model approach in an unified way. The rational case yields the standard type of matter-radiaton models, while the trigonometric case corresponds to their q-deformations. The models without RWA are obtained from the elliptic case at the Gaudin and high spin limit.

Degenerations and Representations of Twisted Shibukawa–Ueno R-Operators

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa-Ueno. We define a two-parameter family of generalizations of the Shibukawa-Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer-Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer-Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix.

Topics in quantum integrable systems

Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H 1 and L 1 which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L 1 to L 1, and from H 1 to H 1. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.

Trigonometric osp(1|2) Gaudin model

The problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(1|2) classical r-matrix. The eigenvectors of the trigonometric osp(1|2) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik–Zamolodchikov system is discussed both in the rational and in the trigonometric case.

Approximation of nonnegative functions by means of exponentiated trigonometric polynomials

We consider the problem of approximating a nonnegative function from the knowledge of its first Fourier coefficients. Here, we analyze a method introduced heuristically in a paper by Borwein and Huang (SIAM J. Opt. 5 (1995) 68–99), where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function. In this note, we prove that approximations given by Borwein and Huang's method, in the trigonometric case, can be related to a nonlinear constrained optimization problem, and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of Toeplitz matrices. Moreover, they allow to obtain an improved convergence theorem for best entropy approximations.

An elementary construction of lowering and raising operators for the trigonometric Calogero-Sutherland model

Quantum Calogero-Sutherland model of $A_n$ type is completely integrable. Using this fact, we give an elementary construction of lowering an raising operators for the trigonometric case. This is similar, but more complicated (due to the fact that the energy spectrum is not equidistant) than the construction for the rational case.

On factorizing twists for the Yangian and the quantum affine algebra

We calculate explicit expressions for factorizing Drinfel’d twists in evaluation representations of the YangianY(sl2) and of the quantum affine algebraUq\(\left( {\widehat{\mathfrak{s}\mathfrak{l}}2} \right)\). From the twists we derive a closed and representation-independent form of theR-matrices in these representations. Employing a carefully chosen basis it is possible to recover the results for the Yangian (rational case) as theq → 1 limit of the expressions for the quantum affine algebra (trigonometric case).

INTEGRABLE XYZ HAMILTONIAN WITH NEXT-NEAREST-NEIGHBOR INTERACTION

In this paper, we construct a Hamiltonian of the impurity model with next-nearest-neighbor interaction within the framework of the open boundary Heisenberg XYZ spin chain. This impurity model is an exactly solved one and it degenerates to the integrable XXZ impurity model under the triangular limit. It is the first approach to add the impurities and next-nearest-neighbor interaction to the integrable completely anisotropic Heisenberg spin chain. We find also that the impurity parameters in the bulk are real when the cross parameter is imaginary for the Hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interaction is introduced. The eigenvalue of the Hamiltonian and the Bethe ansatz equations for the trigonometric limit case are derived also.

Generalized Lamé functions. II. Hyperbolic and trigonometric specializations

In Part I [J. Math. Phys. 40, 1595 (1999)] we studied eigenfunctions of the quantum dynamics that defines the two-particle relativistic Calogero–Moser system with elliptic interaction. In the present paper we consider the same system with hyperbolic and trigonometric interactions. In these special regimes the eigenfunctions are shown to admit an elementary representation that is far more explicit than the “zero representation” of Part I. In particular, the new representation can be exploited to prove that the hyperbolic eigenfunctions can be chosen to be symmetric under interchanging position and momentum variables (self-duality). In the trigonometric case duality properties are derived, too, and several orthogonality and completeness results are obtained.

On explicit parametrisation of spectral curves for Moser-Calogero particles and its applications

The system of $N$ classical particles on the line with the Weierstrass $\wp$ function as potential is known to be completely integrable. Recently D'Hoker and Phong found a beautiful parameterization by the polynomial of degree $N$ of the space of Riemann surfaces associated with this system. In the trigonometric limit of the elliptic potential these Riemann surfaces degenerate into rational curves. The D'Hoker-Phong polynomial in the limit describes the intersection points of the rational curves. We found an explicit determinant representation of the polynomial in the trigonometric case. We consider applications of this result to the theory of Toeplitz determinants and to geometry of the spectral curves. We also prove our earlier conjecture on the asymptotic behavior of the ratio of two symplectic volumes when the number of particles tends to infinity.

On the Hamiltonian structure of the spin Ruijsenaars-Schneider model

The Hamiltonian structure of the spin generalization of the rational Ruijsenaars-Schneider model is found by using the Hamiltonian reduction technique. It is shown that the model possesses current algebra symmetry. The possibility of generalizing the obtained Poisson structure to the trigonometric case is discussed and degeneration to the Euler-Calogero-Moser system is examined.