**Transformadas integrales,
Análisis Numérico, Análisis de Señales y Modelado**

o By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, their asymptotic expressions and bilinear generating functions, we obtain quadrature formulas for the integral transforms generated by the classical orthogonal polynomials.

o We
have derived a Gaussian-like quadrature of the continuous fractional Fourier
transform. This quadrature is given in terms of the Hermite polynomials and
their zeros. By using some asymptotic formulas, we rewrite the quadrature as a
chirp-FFT-chirp transformation, yielding a fast discretization of the
fractional Fourier transform and its inverse in closed form. We find that, the
chirp-FFT-chirp transformation evaluated at *z=i*, becomes a more accurate
version of the FFT which can be used for non-periodic functions. This algorithm
can be extended to obtain a fast version of the Linear Canonical
Transform. This fast transforms have been used to study brain and
physiological signals.

o We have obtained an extension of the standard collocation method based on polynomial interpolation. The underlying differentiation matrix can be partitioned to yield a superconvergent implicit multistep-like method to solve the initial value problem numerically.

o Our lines of research includes:

· Numerical Grid Generation

· Numerical Solution of Differential Equations

· Numerical Optimization/Mathematical Programming

· Numerical Analysis/Scientific Computing

o Our interest also lies in practical algorithms and their application to problems like multimedia information retrieval, pattern recognition and ad hoc network communications. This research field lies between theory and practice, it is the application of algorithm engineering:

· Scalable pattern recognition

· Metrics databases

· Information retrieval

· Ad hoc networks

Multimedia information retrieval consists in querying object repositories using an example. The techniques are fundamentally different from those of exact searching which are applicable only to text-like queries. Multimedia databases encompass a large number of applications ranging from query by content in images, audio or video to scalable pattern recognition in general.

o Our lines of research also includes

· Mathematical modeling of complex natural systems. Continuum mechanics.

· Numerical solution of nonlinear partial differential equations. Finite element method. Finite volume method.

· Thermo-porous-elasticity and porous rock deformation. Non-isothermal fluids.

· Mass and energy flux in geothermal water reservoirs, water reservoirs and water oil reservoirs.