Clifford algebra-valued orthogonal polynomials in Euclidean space

Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space

A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into ...

A matrix recurrence for systems of Clifford algebra-valued orthogonal polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory (Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of o...

Clifford Algebra of the Vector Space of Conics for decision boundary Hyperplanes in m-Euclidean Space

In this paper we embed m-dimensional Euclidean space in the geometric algebra Clm to extend the operators of incidence in R to operators of incidence in the geometric algebra to generalize the notion of separator to a decision boundary hyperconic in the Clifford algebra of hyperconic sections denoted as Cl(V2). This allows us to extend the concept of a linear perceptron or the spherical percept...

Eecient Computation of Orthogonal Polynomials in Computer Algebra Eecient Computation of Orthogonal Polynomials in Computer Algebra

Clifford Algebra Cl3, 0-Valued Wavelet Transformation, Clifford Wavelet Uncertainty inequality and Clifford Gabor Wavelets

In this paper, it is shown how continuous Clifford Cl3,0-valued admissible wavelets can be constructed using the similitude group SIM(3), a subgroup of the affine group of R3. We express the admissibility condition in terms of a Cl3,0 Clifford Fourier transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how...