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5.  Matrix biorthogonal polynomials: eigenvalue problems and non-Abelian discrete Painlevé equations: a Riemann–Hilbert problem perspective

Branquinho, Amílcar and Moreno, Ana Foulquié and Mañas, Manuel

Journal of Mathematical Analysis and Applications

Elsevier

In this paper we use the Riemann–Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials on the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first degree matrix polynomials, is given. All of these are applied to the discussion of possible scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. Nonlinear matrix difference equations are discussed next. Firstly, for the general Hermite situation a general non linear relation (non trivial because of the non commutativity features of the setting) for the recursion coefficients is gotten. In the next case of higher difficulty, degree two polynomials are allowed in the Pearson equation, but the discussion is simplified by considering only a left Pearson equation. In the case, the support of the measure is on an appropriate branch of a hyperbola. The recursion coefficients are shown to fulfill a non-Abelian extension of the alternate discrete Painlevé I equation. Finally, a discussion is given for the case of degree three polynomials as coefficients in the left Pearson equation characterizing the matrix of weights. However, for simplicity only odd polynomials are allowed. In this case, a new and more general matrix extension of the discrete Painlevé I equation is found.

ria.ua.pt | doi | Peer Reviewed

4.  Discrete Hardy spaces for bounded domains in Rn

Cerejeiras, Paula and Kähler, Uwe and Legatiuk, Anastasiia and Legatiuk, Dmitrii

Complex Analysis and Operator Theory

Springer; Birkhäuser

Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in R^n. On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.

ria.ua.pt | doi | Peer Reviewed

3.  A fractional analysis in higher dimensions for the Sturm-Liouville problem

Ferreira, M. and Rodrigues, M. M. and Vieira, N.

Fractional Calculus and Applied Analysis

De Gruyter

In this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.

ria.ua.pt | doi | Peer Reviewed

2.  Time-fractional telegraph equation of distributed order in higher dimensions

Vieira, N. and Rodrigues, M. M. and Ferreira, M.

Communications in Nonlinear Science and Numerical Simulation

Elsevier

In this work, the Cauchy problem for the time-fractional telegraph equation of distributed order in $BR^n times BR^+$ is considered. By employing the technique of the Fourier, Laplace and Mellin transforms, a representation of the fundamental solution of this equation in terms of convolutions involving the Fox H-function is obtained. Some particular choices of the density functions in the form of elementary functions are studied. Fractional moments of the fundamental solution are computed in the Laplace domain. Finally, by application of the Tauberian theorems, we study the asymptotic behaviour of the second-order moment (variance) in the time domain.

ria.ua.pt | doi | Peer Reviewed

1.  Krein reproducing kernel modules in Clifford analysis

Alpay, Daniel and Cerejeiras, Paula and Kähler, Uwe

Journal d'Analyse Mathématique

Springer

Classic hypercomplex analysis is intimately linked with elliptic operators, such as the Laplacian or the Dirac operator, and positive quadratic forms. But there are many applications like the crystallographic X-ray transform or the ultrahyperbolic Dirac operator which are closely connected with indefinite quadratic forms. Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry. In this paper we are going to show that Clifford–Krein modules are naturally appearing in this context. Even taking into account the difficulties, e.g., the existence of different inner products for duality and topology, we are going to demonstrate how one can work with them. Taking into account possible applications and the nature of hypercomplex analysis, special attention will be given to the study of Clifford–Krein modules with reproducing kernels. In the end we will discuss the interpolation problem in Clifford–Krein modules with reproducing kernel.

ria.ua.pt | doi | Peer Reviewed
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