Fischer Decomposition in Generalized Fractional Ternary Clifford Analysis

As an analogue of the massless field equations in Euclidean space, we consider the so-called generalized Cauchy–Riemann equations introduced by E. Stein and G. Weiss. In the spin 1/2 case these equations reduce to the Dirac equation for spin 1/2 fields, which was thoroughly and intensively studied in Clifford analysis. For general spin it was recently shown that, in dimension 4, homogenous solutions form irreducible Spin modules. The next step then is to describe the corresponding Fischer decomposition, i.e. an irreducible decomposition of the space of spinor fields, which is well-known for spin 1/2 and for spin 1. The main aim of the present paper is to describe, still in dimension 4, the Fischer decomposition for spin 3/2.

This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators, backward and forward partial difference, one needs to modify the main tools for a development of a discrete function theory, such as the replacement of a real Clifford algebra by a complexified Clifford algebra or of the classic Weyl relations by so-called S-Weyl relations. The main results, like Cauchy integral formula, Fischer decomposition, CK-extension, and Taylor series, will be derived. To give a better idea of the differences between the discrete and continuous case, this chapter contains the problem of discrete Hardy spaces as well as some discrete objects which do not have an equivalent object in continuous Clifford analysis, such as the CK-extension of a discrete Delta function.

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Orsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kahler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

In this note, we describe the Gel’fand‐Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy‐Kovalevskaya extension theorem and the Fischer decomposition in Hermitean Clifford analysis.

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function \({e^{\langle \underline{x}, \underline{u} \rangle}}\) where \({\langle . , . \rangle}\) denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.

Our main result shows that every Montel Kothe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space ((mE), τ0) of m-homegeneus polynomials on E endowed with the compact-open topology τ0 has an unconditional basis if and only if the space (ℋ(E), τδ) of holomorphic functions on E endowed with the bornological topology τδ associated to τ0 has an unconditional basis (for coechelon spaces τδ equals τ0). The main idea is to extend the concept of the Gordon-Lewis property from Banach to Frechet and (DF) spaces. In this way we obtain techniques which are used to characterize the existence of unconditional basis in spaces of m-th (symmetric) tensor products and, as a consequence, in spaces of polynomials and holomorphic functions.

Let [Formula: see text] be the complex vector space of homogeneous polynomials of degree [Formula: see text] with the independent variables [Formula: see text]. Let [Formula: see text] be the complex vector space of homogeneous linear polynomials in the variables [Formula: see text]. For any linear operator [Formula: see text] acting on [Formula: see text], there is a (unique) induced operator [Formula: see text] acting on [Formula: see text] satisfying [Formula: see text] In this paper, we study some algebraic and geometric properties of induced operator [Formula: see text]. Also, we obtain the norm of the derivative of the map [Formula: see text] in terms of the norm of [Formula: see text].

Clifford analysis is a higher-dimensional function theory offering a refinement of classical harmonic analysis. It is centred around monogenic functions, i.e. null solutions of the rotation invariant vector-valued Dirac operator. In this context, generalizations of classical orthogonal polynomials on the real line have been introduced in order to use them as building blocks for wavelets, as has been the case, e.g. for the generalized Clifford–Hermite polynomials. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focuses on simultaneous null solutions of two Hermitean Dirac operators which are invariant under the action of the unitary group. In this Hermitean setting, generalized Hermitean Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Hermitean Dirac operators. In this paper, we establish their mutual orthogonality relations w.r.t. a well-chosen weight function.

In this chapter we consider ridge functions that are algebraic polynomials. While some of the material detailed here is a consequence of more general results, we also present results that are particular to polynomials. We start with a review of some of the basic notions. In Section 5.1 we consider homogeneous polynomials and present results on spanning, linear independence and interpolation by linear combinations of the ( a · x ) m for fixed m ∊ Z + , as we vary over a subset Ω of directions in R n . In Section 5.2 we translate many of these results to the space of algebraic polynomials. Section 5.3 is concerned with what is called Waring's Problem for real polynomials (real linear forms). We consider the minimal number of linear combinations of ridge polynomials needed to represent any algebraic or homogeneous polynomial. Finally, in Section 5.4 we discuss generalized ridge functions that are polynomials. That is, we consider linear combinations of functions of the form p ( A x ), where the A are fixed d × n matrices, and the p are d -variate polynomials. Homogeneous Polynomials As a matter of convenience we will, for a fixed direction a ∊ R n \{ 0 } and univariate polynomial p of degree m , term p ( a · x ) a ridge polynomial of degree m with direction a , and the polynomial ( a · x ) m will be called a ridge monomial of degree m with direction a . For a given set Ω ⊆ R n let L (Ω) := {ƛ a : a ∊ Ω, ƛ ∊ R}. The set of ridge polynomials with direction a is the same as the set of ridge polynomials with direction ƛ a for any ƛ ∊ R, ƛ ≠ 0. Thus L (Ω) is the set of directions we should or could be considering. That is, P (Ω) is the set of all polynomials that vanish on L (Ω). Note that a homogeneous polynomial vanishes on L (Ω) if and only if it vanishes on Ω.

We investigate projection constants within classes of multivariate polynomials over finite‐dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted ‐norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions, and asymptotically accurate estimates for them.

In 1988 Lachaud introduced the class of projective Reed–Muller codes, defined by evaluating the space of homogeneous polynomials of a fixed degree d on the points of $$\mathbb {P}^n(\mathbb {F}_q)$$ . In this paper we evaluate the same space of polynomials on the points of a higher dimensional scroll, defined from a set of rational normal curves contained in complementary linear subspaces of a projective space. We determine a formula for the dimension of the codes, and the exact value of the dimension and the minimum distance in some special cases.

To generalize the Fisher Discriminant Analysis (FDA) algorithm to the case of discriminant functions belonging to a nonlinear, finite dimensional function space F (Nonlinear FDA or NFDA), it is sufficient to expand the input data by computing the output of a basis of F when applied to it [1,2,3,4]. The solution to NFDA can then be found like in the linear case by solving a generalized eigenvalue problem on the between- and within-classes covariance matrices (see e.g. [5]). The goal of NFDA is to find linear projections of the expanded data (i.e., nonlinear transformations of the original data) that minimize the variance within a class and maximize the variance between different classes. Such a representation is of course ideal to perform classification. The application of NFDA to pattern recognition is particularly appealing, because for a given input signal and a fixed function space it has no parameters and it is easy to implement and apply. Moreover, given C classes only C - 1 projections are relevant [5]. As a consequence, the feature space is very small and the algorithm has low memory requirements and high speed during recognition. Here we apply NFDA to a handwritten digit recognition problem using the MNIST database, a standard and freely available set of 70,000 handwritten digits (28 × 28 pixels large), divided into a training set (60,000 digits) and a test set (10,000 digits). Several established pattern recognition methods have been applied to this database by Le Cun et al. [6]. Their paper provides a standard reference work to benchmark new algorithms. We perform NFDA on spaces of polynomials of a given degree d, whose corresponding basis functions include all monomials up to order d in all input variables. It is clear that the problem quickly becomes intractable because of the high memory requirements. For this reason, the input dimensionality is first reduced by principal component analysis. On the preprocessed data we then apply NFDA by expanding the training patterns in the polynomial space and solving the linear FDA eigenvalue problem. As mentioned above, since we have 10 classes we only need to compute the first 9 eigenvectors. Since the within-class variance is minimized, the patterns belonging to different classes tend to cluster in the feature space when projected on the eigenvectors. For this reason we classify the digits with a simple method such as Gaussian classifiers. We perform simulations with polynomials of degree 2 to 5. With polynomials of degree 2 the explosion in the dimensionality of the expanded space with increasing number of input dimensions is relatively restricted, so that it is possible to use up to 140 dimensions. With higher order polynomials one has to rely on a smaller number of input dimensions, but since the function space gets larger and includes new nonlinearities, one obtains a remarkable improvement in performance. The best performance is achieved with polynomials of degree 3 and 35 input dimensions, with an error rate of 1.5% on test data. This error rate is comparable to but does not outperform that of the most elaborate algorithms (Table 1). The performance of NFDA is however remarkable considering the simplicity of the method and the fact that it has no a priori knowledge on the problem, in contrast for example to the LeNet-5 algorithm [6] which has been designed specifically for handwritten character recognition. In addition, for recognition, NFDA has to store and compute only 9 functions and has thus small memory requirements and a high recognition speed. It is also possible to formulate NFDA using the kernel trick, in which case one can in principle use function spaces of infinite dimensionality [1,2,3,4]. However, the limiting factor in that formulation is the number of training patterns, which makes it not realistic for this application. The performance of NFDA could be further improved using for example a more problem-specific preprocessing of the patterns (e.g., by increasing the size of the training set with new patterns generated by artificial distortion of the original one), boosting techniques, or mixture of experts with other algorithms [5,6].

We describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

In this article, we bring a new set of Bernoulli and Euler special monogenic polynomials. It is based on the umbral calculus adapted to the Clifford setting, which is equivalent to a polynomial representation using a special set of Appell polynomials. Also, we study the convergence properties (the effectiveness) for the Bernoulli and Euler polynomials in Clifford analysis. Finally, some results concerning the mode of increase (the order and type) for these sets are obtained.

The problem of determining whether two programs compute the same function is undecidable. We restrict the class of functions that a program can compute to be a finite dimensional vector space, enabling the problem to become tractable. We prove a theorem showing that the number of test points needed to distinguish any such programs is equal to the dimension of the vector space of functions which they compute, thus generalizing previous results of Howden. Our result unifies certain existing software testing theory and explains the successes and failures of one existing method for selecting test data called perturbation testing. We apply the theorem to obtain three results on the number of test points required to test in vector spaces of Boolean functions and polynomials. First, the Vandermonde matrix used together with our theorem yields the classical result that a polynomial of degree at most n is determined by n+1 points. Second, in vector spaces of Boolean functions we obtain useful results when the degree of the Boolean function is less than the number of inputs to the function. Third, we apply our theorem to describe a technique for testing nonpolynomial, real valued functions.