We evaluate explicitly the integrals ∫ 1 −1 πn(t)/(r ∓ t)dt, |r| = 1, with the πn being any one of the four Chebyshev polynomials of degree n. These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing [−1, 1] in its interior.

We give the necessary and su cient conditions guaranteeing the validity of Chebyshev type inequalities for generalized Sugeno integrals in the case of functions belonging to a much wider class than the comonotone functions. For several choices of operators, we characterize the classes of functions for which the Chebyshev type inequality for the classical Sugeno integral is satis ed.

We consider replacing the monomial xn with the Chebyshev polynomial Tn(x) in the Diffie-Hellman and RSA cryptography algorithms. We show that we can generalize the binary powering algorithm to compute Chebyshev polynomials, and that the inverse problem of computing the degree n, the discrete log problem for Tn(x) mod p, is as difficult as that for xn mod p.

We provide a fast algorithm to compute arbitrarily many nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1]. This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions.

The Chebyshev map is a typical chaotic map. We consider generalized Chebyshev maps Tk of C2. The support of the maximal entropy measure of Tk is connected. We perturb Tk in a certain direction. Then we can show the support of the maximal entropy measure of this map is a Cantor set.

We construct a simple closed-form representation of degree-ordered system of bivariate Chebyshev-I orthogonal polynomials Tn,r(u, v, w) on simplicial domains. We show that these polynomials Tn,r(u, v, w), r = 0, 1, . . . , n; n ≥ 0 form an orthogonal system with respect to the Chebyshev-I weight function.

In this paper we characterize rational Szegő quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegő quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experime...

We solve a recent conjecture, proving that the Lebesgue constant of Chebyshev-like angular nodes for trigonometric interpolation on a subinterval [−ω, ω] of the full period [−π, π] is attained at ±ω, its value is independent of ω and coincides with the Lebesgue constant of algebraic interpolation at the classical Chebyshev nodes in (−1, 1). 2000 AMS subject classification: 42A15, 65T40.