Sufficient Condition For Polynomial Research Articles

On Weighted Polynomial Approximation With Gaps

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

Reduction of matrix polynomials to a special quasitriangular form

We find necessary and sufficient conditions for polynomials with matrix coefficients over an arbitrary field to be reducible via a similarity transformation to block triangular form with regular diagonal blocks of maximum degree.

On Coefficients of Polynomials in Network Functions

This paper presents a study of the relationships between the missing powers of polynomials in network functions and the network geometry. The elementary transformation of trees and the 2-trees of a network are introduced to obtain the necessary and sufficient conditions for polynomials in network functions to have missing powers. It is shown that the polynomial in the numerator of the transfer function of a grounded two-terminal-pair RLC network cannot have two successive missing powers unless some common factors of the numerator and the denominator are cancelled. This result is useful in topological synthesis where one must usually restore all the necessary surplus factors before deciding on the minimum number of vertices and the geometry of the network.