The recently proposed Chebyshev-like lifting map for the zeros of a uni-variate polynomial was motivated by its applications to splitting a univariate polynomial p(x) numerically into factors, which is a major step of some most eeective algorithms for approximating polynomial zeros. We complement the Chebyshev-like lifting process by a descending process, decrease the estimated computational co...

A novel Gaussian state estimator named Chebyshev Polynomial Kalman Filter is proposed that exploits the exact and closed-form calculation of posterior moments for polynomial nonlinearities. An arbitrary nonlinear system is at first approximated via a Chebyshev polynomial series. By exploiting special properties of the Chebyshev polynomials, exact expressions for mean and variance are then provi...

in this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. the method was based on orthogonal polynomial approximation. this method consists of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as chebyshev or legendre polynomials with undetermined...

The kth Markoff-Duffin-Schaeffer inequality provides a bound for the maximum, over the interval -1 </= x </= 1, of the kth derivative of a normalized polynomial of degree n. The bound is the corresponding maximum of the Chebyshev polynomial of degree n, T = cos(n cos(-1)x). The requisite normalization is over the values of the polynomial at the n + 1 points where T achieves its extremal values....

ECG (Electrocardiogram) signals originating from heart muscles, generate massive volume of digital data. They need to be compressed or approximated for efficient transmission and storage. ECG signal compression is traditionally performed in three ways: direct, transform and parameter extraction. Polynomial approximation which is a form of parameter extraction method, is employed here. This pape...

In this contribution, we study multivariate polynomial interpolation and quadrature rules on non-tensor product node sets linked to Lissajous curves and Chebyshev varieties. After classifying multivariate Lissajous curves and the interpolation nodes related to these curves, we derive a discrete orthogonality structure on these node sets. Using this discrete orthogonality structure, we can deriv...

A Chebyshev knot C(a, b, c, φ) is a knot which has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + φ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ ∈ R. We show that any two-bridge knot is a Chebyshev knot with a = 3 and also with a = 4. For every a, b, c integers (a = 3, 4 and a, b coprime), we describe an algorithm that gives all Cheb...

Stable polynomial evaluation and interpolation at n Chebyshev or adjusted (expanded) Chebyshev points is performed using O(nlog’ n) arithmetic operations, to be compared with customary algorithms either using on the order of n* operations or being unstable. We also evaluate a polynomial of degree d at the sets of n Chebyshev or adjusted (expanded) Chebyshev points using O(dlog d log n) if n 5 d...

This chapter provides a brief literature review together with detailed descriptions of the authors’ work on the stability and control of systems represented by linear time-periodic delay-differential equations using the Chebyshev and temporal finite element analysis (TFEA) techniques. Here, the theory and examples assume that there is a single fixed discrete delay which is equal to the principa...

A recently proposed public key cryptosystem based on Chebyshev polynomials suggests a new approach to data encryption. But the security of the cryptosystem has not been investigated in depth, for lack of an appropriate analysis method. In this paper, a new representation of Chebyshev polynomial is introduced to study security issues of the cryptosystem. The properties of Chebyshev polynomial se...