Analysis is a key area of mathematics, having enormous applications and relations with other areas of mathematics, physics, engineering, etc. The main strands of the work developed within the analysis group which completely described by the keywords are creation and developments of methods of integral transforms and integral equations with special functions as the kernels, methods of special functions and orthogonal polynomials to solve different analysis problems, numerical analysis and computation. Part of the group also contribute with applications to Engineering and Economics, in the scope of the Mathematical Models and Applications line of research. The group has a long tradition in collaborating with prestigious foreign institutions and researchers, namely from Europe and USA. Some participate in the Editorial Board of various international peer review journals, in the scientific committee of regular international meetings and others participate in international research projects. The group also hosts a number of postdocs, being very active in supervising PhD students.
*Problems of pure Analysis*
In this strand new methods are developed to solve integral equations of different classes in the closed form, to invert and study mapping properties of different integral transformations with special functions in the kernel, to develop method of operational and fractional calculus. One of the main topics also the use of transform methods to obtain new system of polynomials and to study classical systems of orthogonal polynomials. We are also developing analytic methods of number theory, involving the theory of Riemann's zeta-function and proving equivalences to the Riemann hypothesis.
*Problems of Numerical Analysis*
In the Numerical Analysis strand, new numerical algorithms as well as adaptation of classical ones tuned to novel computer architectures have been developed. In-depth research to efficiently solve certain problems lead us to explore in detail numerical linear algebra kernels, such those related with large dimensional eigenvalue computations. Nonlinear eigenvalue problems has been also under study and development.
New methodologies were developed, which are based on symbolic computations that are able to derive new closed formulas for connection coefficients for several orthogonal polynomials and to study deeply the cubic decomposition of polynomials by explaining their component sequences. These topics has produced the corresponding software written in the Mathematica language.