Searcharxiv⌕ Search

arXiv subjects

Paula Cerejeiras

Publications and source records attributed to Paula Cerejeiras.

16 recordsLinked to original sources
arXiv2025

Generalized Fock space and fractional derivatives with Applications to Uniqueness of Sampling and Interpolation Sets

In this paper we introduce a Fock space related to derivatives of Gelfond-Leontiev type, a class of derivatives which includes many classic examples like fractional derivatives or Dunkl operators. For this space we establish a modified Bargmann transform as well as density theorems for sampling and interpolation. These density theorems allow us to establish lattice conditions for the construction of frames arising from integral transforms which are linked by the modified Bargmann transform with the Fock space.

math.FA↗
arXiv2024

Nuclearity and Grothendieck-Lidskii formula for quaternionic operators

We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula, relating the trace of an operator with its eigenvalues, holds. We then extend these results to the so-called $\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic locally convex spaces. While doing so, we develop some results in the theory of topological tensor products of noncommutative modules, and show that the trace defined ad hoc in terms of companion matrices, arises naturally as part of a canonical trace.

math.CA↗
arXiv2024

$q$-Rational functions and interpolation with complete Nevanlinna Pick Kernels

In this paper we introduce the concept of matrix-valued $q$-rational functions. In comparison to the classic case we give different characterizations with principal emphasise on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna Pick kernels in this context and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna Pick kernels.

math.CV↗
arXiv2023

Generalized $q$-Fock spaces and structural identities

Using $q$-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their $q$-calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.

math.FA↗
arXiv2023

Variational principles in quaternionic analysis with applications to the stationary MHD equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder's fixed point theorem. The advantage of the approach using Schauder's fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These consideration will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

math.AP↗
arXiv2022

Commutators on Fock spaces

Given a weighted $\ell^2$ space with weights associated to an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond-Leontiev derivatives. This general class of operators includes many known examples, like classic fractional derivatives and Dunkl operators. This allows us to establish a general framework which goes beyond the classic Weyl-Heisenberg algebra. Concrete examples for its application are provided.

math-ph↗
arXiv2021

Generalized Grassmann algebras and applications to stochastic processes

In this paper we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmentry with Z3-graded algebras. To this end we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.

math.FA↗
arXiv2020

Generalized Fock space and moments

In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-Itô expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of stochastic processes.

math.PR↗
arXiv2019

A primer on Script Geometry

This text is an exposition of a new approach into discrete differential geometry, called Script Geometry. In difference to classic approaches while scripts are based on complexes of cells we are not limited to simplicial complexes. One of the principal concepts of Script Geometry is the notion of tightness which is a minimality condition corresponds to the condition that the local homology at the level of cells is trivial and provides us the equivalence of a Poincare lemma. Basic concepts and properties of scripts are presented as well as operations on scripts. Also the necessary concepts of metrics, dual scripts, Dirac and Laplace operators are presented to provide the groundwork for a discrete function theory. To help understanding the concepts concrete and illustrative examples are provided.

math.DG↗
arXiv2018

Applications of parabolic Dirac operators to the instationary viscous MHD equations on conformally flat manifolds

In this paper we apply classical and recent techniques from quaternionic analysis using parabolic Dirac type operators and related Teodorescu and Cauchy-Bitzadse type operators to set up some analytic representation formulas for the solutions to the time depedendent incompressible viscous magnetohydrodynamic equations on some conformally flat manifolds, such as cylinders and tori associated with different spinor bundles. Also in this context a special variant of hypercomplex Eisenstein series related to the parabolic Dirac operator serve as kernel functions.

math.AP↗
arXiv2018

Some applications of parabolic Dirac operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori in $\mathbb{R}^3$

In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators involving explicit expressions for the Cauchy kernel that are associated to the parabolic Dirac operators acting on spinor sections of these manifolds.

math.AP↗
arXiv2017

Perturbation of normal quaternionic operators

The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two $S$-resolvent operators and the $S$-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the $S$-spectrum of a quaternionic linear operator $T$ is given by the non invertibility of a second order operator. This presents new challenges which makes our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of $S$-spectrum and of slice hyperholomorphicity of the $S$-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

math.FA↗
arXiv2014

Generating functions for spherical harmonics and spherical monogenics

In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in R^m. Here spherical monogenics are polynomial solutions of the Dirac equation in R^m. In particular, we obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m-1. Hence we can find closed formulae of generating functions in R^m by induction on the dimension m.

math.CV↗
arXiv2014

Lower bounds for the approximation with variation-diminishing splines

We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval $[0,1]$ in terms of classical moduli of smoothness depending on the degree of the spline basis using a functional analysis based framework. Thereby, we characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates. Finally, we prove the equivalence between the approximation error and the classical second order modulus of smoothness as an improved version of an open conjecture from 2002.

math.CA↗
arXiv2011

Hermite Polynomials in Dunkl-Clifford Analysis

In this paper we present a generalization of the classical Hermite polynomials to the framework of Clifford-Dunkl operators. Several basic properties, such as orthogonality relations, recurrence formulae and associated differential equations, are established. Finally, an orthonormal basis for the Hilbert modules arising from the corresponding weight measures is studied.

math.CV↗