Isabel Cação

On Complete Sets of Hypercomplex Appell Polynomials

We construct a family of hypercomplex Appell polynomials and it corresponding generating function of exponential type. An extension of this approach leads to the construction of a complete set of monogenic Appell polynomials. Moreover, it is shown how the obtained Appell family is connected with complete sets of solutions of the Riesz system.

Special monogenic polynomials andL2-approximation

The paper features some observations about the behaviour of special monogenic polynomials in the unit ball of ℝn which are relevant to the development of general approximation methods in the framework of Clifford Analysis. It focuses on an estimate for the best approximation of monogenic functions by a set of polynomials in Sobolev spaces. We illustrate some rather surprising orthogonality properties between symmetric homogeneous polynomials of the same degree and also an elementary method of obtaining a variety of linearly independent homogeneous polynomials of such type for the case of functions defined in ℝ3 and with values in the algebra of quaternions. The proof of their generalizations to higher dimensions in a suitable way is possible but needs more complicated combinatorical calculations which exceed the scope of this communication.

Laguerre derivative and monogenic Laguerre polynomials: An operational approach

Hypercomplex function theory generalizes the theory of holomorphic functions of one complex variable by using Clifford Algebras and provides the fundamentals of Clifford Analysis as a refinement of Harmonic Analysis in higher dimensions. We define the Laguerre derivative operator in hypercomplex context and by using operational techniques we construct generalized hypercomplex monogenic Laguerre polynomials. Moreover, Laguerre-type exponentials of order m are defined.

Complete orthonormal sets of polynomial solutions of the Riesz and Moisil-Teodorescu systems in R 3

As it is well-known, the generalization of the classical Cauchy-Riemann system to higher dimensions leads to the so-called Riesz and Moisil-Teodorescu systems. Rewriting these systems in quaternionic language and taking advantage of the underlying algebra, we construct complete sets of polynomials solutions of both systems that are orthonormal with respect to a certain inner product. The restrictions of those polynomials to the unit sphere can be viewed as analogues to the complex case of the Fourier exponential functions

On generalized hypercomplex laguerre-type exponentials and applications

\{e^{i n \theta}\}_{n\geq 0}

Matrix Representations of a Special Polynomial Sequence in Arbitrary Dimension

on the unit circle and constitute a refinement of the well-known spherical harmonics.

On an hypercomplex generalization of Gould-Hopper and related Chebyshev polynomials

In hypercomplex context, we have recently constructed Appell sequences with respect to a generalized Laguerre derivative operator. This construction is based on the use of a basic set of monogenic polynomials which is particularly easy to handle and can play an important role in applications. Here we consider Laguerre-type exponentials of order m and introduce Laguerre-type circular and hyperbolic functions.

Quaternion Zernike spherical polynomials

This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows for their recursive construction in the same way as for complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.

An Orthogonal Set of Weighted Quaternionic Zernike Spherical Functions

An operational approach introduced by Gould and Hopper to the construction of generalized Hermite polynomials is followed in the hypercomplex context to build multidimensional generalized Hermite polynomials by the consideration of an appropriate basic set of monogenic polynomials. Directly related functions, like Chebyshev polynomials of first and second kind are constructed.

An introduction to the quaternionic Zernike spherical polynomials

Abstract. Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many di↵erent fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this development. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quaternionic analysis (R(Q)ZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). R(Q)ZSPs are complete and orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second order homogeneous di↵erential equation are also discussed. As an application, we provide the reader with 3D plot simulations that demonstrate the e↵ectiveness of our approach. R(Q)ZSPs are new in literature and have some consequences that are now under investigation.