The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. In order to get an optimal in spaces of locally continuous functions equipped weighted uniform norms, the related Lebesgue constants have be uniformly bounded. previous works this has already been proved under different sufficient con...

In this paper we study the number of limit cycles bifurcating from isochronous surfaces of revolution contained in R, when we consider polynomial perturbations of arbitrary degree. The method for studying these limit cycles is based in the averaging theory and in the properties of Chebyshev systems. We present a new result on averaging theory and generalitzations of some classical Chebyshev sys...

A new algebraic cubature formula of degree 2n+1 for the product Chebyshev measure in the d-cube with ≈ n/2 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpo...

A new algebraic cubature formula of degree 2n + 1 for the product Chebyshev measure in the d-cube with ≈ nd/2d−1 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3dimensional FFT. Moreover, integration of the hyperin...

Absboer -Robust stability conditions for low-order Schw polynomials are obtained. In McuLu, conditions for dewe n = 2, 3, 4, and 5 are explicitly obtained. These conditions relate to &&ty of the corner points for n = 2.3 and for corner and possible supplementary points for n = 4 and 5. Two countere-ples given in the literature are fully discussed in relation to the obtained conditions. Futme re...

By considering a family of orthogonal polynomials generalizing the Tchebycheff polynomials of the second kind we refine the corresponding results of De Sainte-Catherine and Viennot on Tchebycheff polynomials of the second kind (Lecture Notes in Mathematics, vol. 1171, 1985, Springer-Verlag, 120). © 2003 Elsevier Science Ltd. All rights reserved.

We consider the differential equations y = λ0(x)y ′ + s0(x)y, where λ0(x), s0(x) are C −functions. We prove (i) if the differential equation, has a polynomial solution of degree n > 0, then δn = λnsn−1 − λn−1sn = 0, where λn = λ ′ n−1 + sn−1 + λ0λn−1 and sn = s ′ n−1 + s0λk−1, n = 1, 2, . . . . Conversely (ii) if λnλn−1 6= 0 and δn = 0, then the differential equation has a polynomial solution o...

We characterize polynomial decomposition fn = r ◦ q with r, q ∈ C[x] of perturbed Chebyshev polynomials defined by the recurrence f0(x) = b, f1(x) = x− c, fn+1(x) = (x− d)fn(x)− afn−1(x), n ≥ 1, where a, b, c, d ∈ R and a > 0. These polynomials generalize the Chebyshev polynomials, which are obtained by setting a = 1/4, c = d = 0 and b ∈ {1, 2}. At the core of the method, two algorithms for pol...

We study Lissajous curves in the 3-cube that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single 1-d Fast Chebyshev Transform (by the Chebfun package), and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Applications co...