A conjecture of Schoenberg.
A Convergence Theorem for a Method for Simultaneous Determination of all Zeros of a Polynomial
A convergence theorem for a method for simultaneous determination of all zeros of a polynomial.
A Correction on a Theorem By Uspensky
A Discrepancy Theorem Concerning Polynomials of Best Approximation in L...[-1, 1].
A generalization of Lucas' theorem to vector spaces.
A generalization of the Gauss-Lucas theorem
Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.
A geometric maximization problem
A Modified Fast Fourier Transform for Polynomial Evaluation and the Jenkins-Traub Algorithm.
A necessary and sufficient condition for stability of the convex combination of polynomials
A note on Babylonian square-root algorithm and related variants.
A note on Rouché's theorem
A note on the number of zeros of polynomials in an annulus
Let p(z) be a polynomial of the form , . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.
A note on the zeros of exponential polynomials
A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix
It is shown that if A is a bounded linear operator on a complex Hilbert space, then , where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.
A Polya "shire" theorem for functions with algebraic singularities.
A Problem Concerning the Zeros of a Certain Kind of Holomorphic Function in the Unit Disk.
A proof of Ilieff's conjecture for polynomials with four zeros.
A pure smoothness condition for Radó’s theorem for -analytic functions
Let be a bounded, simply connected -convex domain. Let and let be a function on which is separately -smooth with respect to (by which we mean jointly -smooth with respect to , ). If is -analytic on , then is -analytic on . The result is well-known for the case , , even when a priori is only known to be continuous.
A relationship between a Hankel matrix of Markov parameters and the associated matrix polynomial with some applications