On the growth type of entire monogenic functions

In this paper we establish an explicit relation between the growth type of general entire solutions to the generalized Cauchy-Riemann system in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} and their Taylor coefficients. This formula then enables us to compute the growth type of some higher dimensional generalizations of the trigonometric and special functions that are null-solutions to this system.     

The square root base of polynomials in Clifford analysis

In this paper the problem of taking the square root of bases of special monogenic polynomials is studied, thus leading to a number of results under some additional conditions of associated infinite matrices, related essentially to the so-called algebraicness of these matrices.     

Universality properties of the quaternionic power series and entire functions

In this paper we establish the existence of “almost universal” quaternionic power series and entire functions. Denoting by B(0, 1) the open unit ball in , this means that there exists a quaternionic power series with radius of convergence 1 such that, denoting by the n‐th partial sum of S, for every , for every axially symmetric open subset Ω of containing K and every f slice regular on Ω, there exists a subsequence of the partial sums of S such that uniformly on K, as . The symbol denotes the set of axially symmetric compact sets in such that is connected for some . This is a slightly weaker property than the classical universal power series phenomenon obtained for analytic only on the interior of K and continuous on K. We also generalize a result originally proven by Birkhoff and finally we show that there exists an entire quaternionic function whose set of derivatives is dense in the class of entire quaternionic functions.     

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let be the Euclidean Dirac operator in the n-dimensional flat space the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials. Received: 27 October 2005     

On the approximation of functions satisfying defective renewal equations

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions do not admit an explicit expression. In this work, we consider their approximation by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give the optimal order of uniform convergence of the approximation. We apply our results to ruin probabilities in the classical risk model, paying special attention to mixtures of gamma claim amounts.     

ON THE RATES OF CONVERGENCE FOR PROBABILITY TYPE OPERATORS SEQUENCE

Abstract. In this paper, the rates of convergence for some probability type operators sequence are obtained. The quantitative Poisson type limit theorem is established as an application.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

On the Structure of Harmonic Multi-Vector Functions

Let F_r (0 < r < m + 1) be a smooth r-vector valued function in a suitable open domain of satisfying in Ω, where ∂ is the Dirac operator in . Then it is proved that there exists H _r , an r-vector valued harmonic function in Ω, such that F _r = . Two proofs of this structure theorem are given, one based on properties of harmonic differential forms and one relying upon primitivation of monogenic functions.     

On W-solvability for special vectorial Hamilton-Jacobi systems

We study the solvability of special vectorial Hamilton-Jacobi systems of the form F(Du(x))=0 in a Sobolev space. In this paper we establish the general existence theorems for certain Dirichlet problems using suitable approximation schemes called W^1,p-reduction principles that generalize the similar reduction principle for Lipschitz solutions. Our approach, to a large extent, unifies the existing methods for the existence results of the special Hamilton-Jacobi systems under study. The method relies on a new Baire's category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for W^1,p-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.     

On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we derive the bounds on the magnitude of l  th (l=2,3)$(l=2,3)$ order derivatives of rational Bézier curves, estimate the error, in the _L∞$L_{\infty }$ norm sense, for the hybrid polynomial approximation of the l  th (l=1,2,3)$(l=1,2,3)$ order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the l  th (l=1,2,3)$(l=1,2,3)$ derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.     

Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.     

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

On integral bernstein operators in some classes of measurable bivariate functions

The two main theorems are concerned with the approximations of (complex-valued) functions on the real plane by sums of Bernstein pseudoentire functions. They are formulated and proved in Section 4, after prior determination of the suitable integral operators. Analogous results for pseudopolynomial approximations were obtained by Brudnyî, Gonska, and Jetter ([2],[3]).Research supported by KBN grant 2 1079 91 01.     

Almost everywhere convergence of Ciesielski-Fourier series of <Emphasis Type="Italic">H</Emphasis><Subscript>1</Subscript>functions

It is proved that a lacunary sequence of the Ciesielski-Fourier series of converges almost everywhere to f. Received: 8 September 2003     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

Maximal polynomial subordination to univalent functions in the unit disk

Let C be a simply connected domain, 0, and let _n,nN, be the set of all polynomials of degree at mostn. By _n() we denote the subset of polynomials p _n withp(0)=0 andp(D), whereD stands for the unit disk {z: |z|<1}, and=" by=">we denote the maximal range of these polynomials. Letf be a conformal mapping fromD onto ,f(0)=0. The main theme of this note is to relate _n (or some important aspects of it) to the imagesf _s (D), wheref _s (z):=f[(1–s)z], 0s<1. for=" instance=" we=" prove=" the=" existence=" of=" a=" universal=">c ₀ such that, forn2c ₀,     

On the iterates of Jackson type operator GS,N in Hilbert space

In this paper we study the limit of the iterates of Jackson type operator. Our results continue the works of Badea [2] and Nagler et al. [9, 10]. The proofs are based on spectral theory of linear operators and are performed at first for Hilbert space and then are extended for some Banach spaces.