Yamilet Quintana

Weierstrass' theorem with weights

We characterize the set of functions which can be approximated by continuous functions in the L∞ norm with respect to almost every weight. This allows to characterize the set of functions which can be approximated by polynomials or by smooth functions for a wide range of weights.

Interior Controllability of a Broad Class of Reaction Diffusion Equations

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces given by , , where is a domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control and is an unbounded linear operator with the following spectral decomposition: . The eigenvalues of have finite multiplicity equal to the dimension of the corresponding eigenspace, and is a complete orthonormal set of eigenvectors of . The operator generates a strongly continuous semigroup given by . Our result can be applied to the D heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.

Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm , where the matrix and the measure constitute a -admissible pair for . In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to , stating hypothesis on the matrix rather than on the diagonal matrix appearing in its unitary factorization.

Controllability of Laguerre and Jacobi equations

In this paper we study the controllability of the controlled Laguerre equation and the controlled Jacobi equation. For each case, we find conditions which guarantee when such systems are approximately controllable on the interval [0, t 1]. Moreover, we show that these systems can never be exactly controllable.

Full length article: Measurable diagonalization of positive definite matrices

In this paper we show that any positive definite matrix V with measurable entries can be written as V=U@LU^*, where the matrix @L is diagonal, the matrix U is unitary, and the entries of U and @L are measurable functions (U^* denotes the transpose conjugate of U). This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.