In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all “integer valued”. The equilibrium positions of Calogero and Sutherland systems for the classical root systems (Ar, Br, Cr an...

In this paper, a new efficient method is proposed to obtain the transient response of linear or piecewise linear dynamic systems with time delay and periodic coefficients under arbitrary control excitations via Chebyshev polynomial expansion. Since the time domain can be divided into intervals with length equal to the delay period, at each such interval the fundamental solution matrix for the c...

In the present paper we give a characterization of Chebyshev subspaces in the space of (real or complex) continuously-differentiable functions of two variables. We also discuss various applications of the characterization theorem. Introduction. One of the central problems in approximation theory consists in determining the best approximation; that is, in a normed linear space X with a prescribe...

It is well known that biomedical signals carry important information about the behavior of the living systems under study. With the analysis of the Electrocardiogram (ECG) signal it may be possible to predict heart problems or monitor patient recovery after a heart intervention. A proper processing of these signals enhances their physiological and clinical information. The quality of biomedical...

The design of partial-band linear periodically time-varying systems (PBTV) imitating linear time-invariant (LTI) systems is approached using a relative ` error criterion. This error criterion results from a natural extension of the Chebyshev error criterion for the design of LTI systems. A complete analysis of the error criterion is presented. An algorithm is introduced which nds a locally opti...

We construct an hyperinterpolation formula of degree n in the three-dimensional cube, by using the numerical cubature formula for the product Chebyshev measure given by the product of a (near) minimal formula in the square with Gauss-Chebyshev-Lobatto quadrature. The underlying function is sampled at N ∼ n/2 points, whereas the hyperinterpolation polynomial is determined by its (n + 1)(n + 2)(n...

In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by...

A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are appro...

We exhibit a numerical technique based on Newton’s method for finding all the roots of Legendre and Chebyshev polynomials, which execute less iterations than the standard Newton’s method and whose results can be compared with those for Chebyshev polynomials roots, for which exists a well known analytical formula. Our algorithm guarantees at least nine decimal correct ciphers in the worst case, ...