Abstract
The generalized polynomials such as Chebyshev polynomial and Hermite polynomial are widely used in interpolations and numerical fittings and so on. Therefore, it is significant to study inclusion regions of the zeros for generalized polynomials. In this paper, several new inclusion sets of zeros for Chebyshev polynomials are presented by applying Brauer theorem about the eigenvalues of the comrade matrix of Chebyshev polynomial and applying the properties of ovals of Cassini. Some examples are given to show that the new inclusion sets are tighter than those provided by Melman (2014) in some cases.
Highlights
The inclusion region of polynomial zeros is widely used in the theory of differential equations, the complex functions and the numerical analysis
The structures of comrade matrix for generalized polynomials are different from this of polynomial in power basis [4], so it is difficult to use them to generalized polynomials such as Chebyshev polynomial and Hermite polynomial, which are widely used in interpolations and numerical fittings
Several new inclusion sets of zeros for Chebyshev polynomials are presented by applying Brauer theorem about the eigenvalues of the comrade matrix of Chebyshev polynomial and applying the properties of ovals of Cassini
Summary
The inclusion region of polynomial zeros is widely used in the theory of differential equations, the complex functions and the numerical analysis. The Chebyshev polynomials { Ti (z)} and {Ui (z)} of the first and second kind, respectively, are defined by the relation. We define the families of the polynomial {φi } (i = 0, 1, 2, 3, · · · ) satisfying three-term recurrence relation as following φ0 (z) = 1, φ1 (z) = α1 z + β 1 ,. Because Chebyshev polynomials satisfy three-term recurrence relation, we can obtain the following corollaries from Theorem 3. We give the Brauer theorem for the eigenvalues of a matrix and Descartes’ rule of signs of polynomial zeros for using in the later. The number of positive real zeros of P( x ) (counted with multiplicities) is either equal to the number of variations in sign in the sequence a0 , · · · , an of the coefficients or less than that by an even whole number
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