On the occasion of S. Yakubovich's 50th birthday

The aim of this conference is to bring together mathematicians and physicists interested in the field of special functions, transform and operator theory, orthogonal polynomials and related topics (research areas of S. Yakubovich), with a central goal of increasing the quality of research, promoting interaction among researchers and discussing new directions for the future.

In this single-day conference there will be a set of eight plenary talks and a discussion session.

The venue will be the University of Porto.
The host of the event is the Department of Mathematics of the Faculty of Sciences of the University of Porto (DM-FCUP), drawing on the support of Centro de Matemática da Universidade do Porto (CMUP).

Lectures

A. BRANQUINHO (University of Coimbra)
"Interpretation of some full Kostant-Toda lattice via multiple orthogonal polynomials"

Abstract. Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based on the analysis of the matrical moments
for the main operator of the pair. The solutions of these systems are studied in terms of properties of this operator, giving, under some conditions, explicit
expressions for the resolvent function.

J.L. CARDOSO (University of Trás-os-Montes e Alto Douro)
"q-Integral of Jackson: associated orthogonality"

Abstract. q-Integral of Jackson: associated orthogonality. Some examples of orthogonal systems. Related questions. Corresponding basic Fourier expansions.

L. CASTRO (University of Aveiro)
"On the Fredholm characterization of Wiener-Hopf and Wiener-Hopf-Hankel integral operators"

Abstract. We will consider Wiener-Hopf, and Wiener-Hopf plus/minus Hankel integral operators in the framework of Lebesgue spaces. Several classes of Fourier symbols for these operators will be considered but special attention will be given to matrix Fourier symbols in the piecewise almost periodic algebra. Conditions which characterize the Fredholm property of those operators will be presented. Such characterizations are based on certain factorizations of matrix functions and on spectral properties of new functions which arise from the initial Fourier symbols of the integral operators under study.

M. DALLA RIVA (University of Porto)
"On a Riemann-Liouville fractional analog of the Laplace operator
with positive energy integral"

Abstract. We define a particular fractional analog of the Laplace
operator in a rectangular domain in the plane by exploiting Riemann-Liouville fractional derivatives in direction $x$ and $y$. Such a definition allows the introduction of fractional boundary value problems which correspond to the classical Dirichlet, Neumann and mixed boundary value problems for the Laplace operator. By expoiting a suitable Integration by Parts Formula with positive energy integral we show some uniqueness results for such fractional boundary value problems. (Joint work with S. Yakubovich)

H. MALONEK (University of Aveiro)
"A new Glimpse of Clifford Analysis through Special Functions and
Combinatorial Identities"

Abstract. Clifford Analysis is a generalization of the theory of functions of one
complex variables to higher dimensions by using Clifford algebras instead of
several complex variables. Its relations to other branches of mathematics as
well as its applications are countless. The aim of the talk is to show some
interesting relations to Special Functions and Combinatorial Identities.

S. SAITOH (University of Aveiro)
"Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels"

Abstract. For the many concrete applications of the convolutions, I would like to introduce a general concept of convolutions by means of the theory of reproducing kernels. I will give a very general concept for the convolutions, concrete new examples and concrete applications containing non-linear partial differential equations and norm inequalities.

G. SMIRNOV (University of Minho)
"The Cauchy problem for a short-wave equation"

Abstract. We prove an existence and uniqueness of solution for the
Cauchy problem of the simplest nonlinear short-wave equation, $u_{tx}=u-3u^{2}$,
with periodic boundary condition.

N. VIEIRA (University of Porto)
"The use of Kontorovich-Lebedev's transform in an analysis of regularized Schr\"odinger equation"

Abstract. In this talk we introduce a notion of Schr\"odinger's kernel to the familiar Kontorovich-Lebedev transform. In order to control its singularity at infinity, we will need to implement the so-called regularization procedure. Hence we will obtain a sequence of regularized kernels which converge to the original kernel when a regularization parameter tends to zero. We study differential and semigroup properties of the regularized kernel and construct fundamental solutions of a regularized time-dependent Schr\"odinger's equation. The correspondent regularized integral transformation is presented. We also establish analogs of the classical Heisenberg inequality and uncertainty principle for this transformation. Finally, we exame a pointwise convergence of this family of regularized integral operators, when the regularization parameter tends to zero.
(Joint work with S. Yakubovich)

Schedule

13h30 **************************Opening session – President of the Departament
14h00 H. Malonek
14h35 G. Smirnov
15h10 L. Castro
15h45 A. Branquinho
************************************Coffee break
16h45 M. Dalla Riva
17h20 S. Saitoh
17h55 J.L. Cardoso
18h30 N. Vieira
************************************Closing session
20h00 DINNER

Proceedings

OPUSCULA MATHEMATICA ( http://www.opuscula.agh.edu.pl/ ) will publish a special issue dedicated to this meeting. All papers will be refereed according to the usual standards for this journal.

Scientific Comittee

Ana F. Loureiro
H. Malonek
S. Yakubovich

Organizing Comittee

Ana F. Loureiro
Daniela Oliveira (secretary)