arXiv subjects

Kenier Castillo

Publications and source records attributed to Kenier Castillo.

3 recordsLinked to original sources

arXiv2025

A Note on the Paper "Localization of Zeros of Polar Polynomials on the Unit Disk"

We show that none of the principal claims of Costas-Santos and Rehouma, Rocky Mountain J. Math. 54 (2024), 995-1004, survives as a new and valid result. The main localisation theorem is false as stated; under the missing hypothesis of orthogonality on the unit circle, it is a classical exercise recorded by Marden and by Borwein and Erd\'elyi, while the published proof omits its decisive finite-degree estimate. The annular claim and Sendov application are unproved, and some reported numerical values contradict the latter under its required hypothesis. We establish instead a measure-independent Grace--Szego theorem with sharp affine and optimal disc bounds, and characterise exactly when the convolution method extends to a general factor. This provides the necessary correction to the published record.

math.CA↗

arXiv2021

On a formula of Thompson and McEnteggert for the adjugate matrix

For an eigenvalue $\lambda_0$ of a Hermitian matrix $A$, the formula of Thompson and McEnteggert gives an explicit expression of the adjoint of $\lambda_0 I-A$, $\mathrm{adj}(\lambda_0 I-A)$, in terms of eigenvectors of $A$ for $\lambda_0$ and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix $A$, a formula for the elementary divisors of $\mathrm{adj}(A)$ is provided in terms of those of $A$. Finally, a generalization of the eigenvalue-eigenvector identity and two applications of the Thompson-McEnteggert's formula are presented.

math.RA↗

arXiv2013

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula % \[ R_{n+1}(z) = \big[(1+ic_{n+1})z+(1-ic_{n+1})\big] R_{n}(z) - 4 d_{n+1} z R_{n-1}(z), \quad n \geq 1, \] % with $R_{0}(z) =1$ and $R_{1}(z) = (1+ic_{1})z+(1-ic_{1})$, where $\{c_n\}_{n=1}^{\infty}$ is a real sequence and $\{d_n\}_{n=1}^{\infty}$ is a positive chain sequence. We establish that there exists an unique nontrivial probability measure $μ$ on the unit circle for which $\{R_n(z) - 2(1-m_n)R_{n-1}(z)\}$ gives the sequence of orthogonal polynomials. Here, $\{m_n\}_{n=0}^{\infty}$ is the minimal parameter sequence of the positive chain sequence $\{d_n\}_{n=1}^{\infty}$. The element $d_1$ of the chain sequence, which does not effect the polynomials $R_n$, has an influence in the derived probability measure $μ$ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if $\{M_n\}_{n=0}^{\infty}$ is the maximal parameter sequence of the chain sequence, then the measure $μ$ is such that $M_0$ is the size of its mass at $z=1$. An example is also provided to completely illustrates the results obtained.

math.CA↗