Hypercomplex derivative bases of polynomials in Clifford analysis

Journal of Complex Analysis

Aloui

Bakali

2017

This article is concerned with the study of the theory of basic sets in Fréchet modules in Clifford analysis. The main aim of this account, which is based on functional analysis consideration, is to formulate criteria of general type for the effectiveness (convergence properties) of basic sets either in the space itself or in a subspace of finer topology. By attributing particular forms for the Fréchet module of different classes of functions, conditions are derived from the general criteria for the convergence properties in open and closed balls. Our results improve and generalize some known results in complex and Clifford setting concerning the effectiveness of basic sets.

“…We first show that is a seminorm on . Let , ∈ and ∈ A ; it follows from (27) and the linearity of Π that…”

Section: Effectiveness Of Basic Setsmentioning

confidence: 99%

“…Sufficiency. Multiplying the basic coefficient Π ( ) of (20) by and using (11), (27), and (36), we obtain…”

Section: Theorem 15 Suppose That ( ) ≥0 Is An Absolute Basis Formentioning

confidence: 99%

See 1 more Smart Citation

Basic Sets of Special Monogenic Polynomials in Fréchet Modules

Journal of Complex Analysis

Aloui

Bakali

2017

“…For more information about the study of bases of special monogenic polynomials, we refer to [13,18,19,[25][26][27][28][29].…”

Section: Remark 21mentioning

confidence: 99%

A note on the growth order of the inverse and product bases of special monogenic polynomials

Math Methods in App Sciences

2011

Self Cite

The purpose of this paper is to generalize some theorems on the representation of certain classes of the entire special monogenic functions by inverse and product bases of special monogenic polynomials. We derive bounds for the order of inverse and product bases. Some examples are given showing that the resulting bounds are attainable. Our results improve and generalize some known results in Clifford setting concerning the order of inverse and product bases. Copyright © 2011 John Wiley & Sons, Ltd.

“…This theory found a lot of applications, mainly in the theory of functions depending on one or several complex variables as well as in the approximation of solutions of differential equations or matrix functions.The topic of derivative BPs in one complex variable has been studied early (see [26,27,28]), the searchers considered the disks in the complex plane C. For several complex variables (see [16,17,19,24,25]), the representation domains are polycyclinderical, hyperspherical and hyperelliptical regions. Recently,in [12,35] the…”

mentioning

confidence: 99%

Approximation of analytic functions of exponential derived and integral bases in Fréchet spaces

Assiut University Journal of Multidisciplinary Scientific Resea

Atta²

2022

The theory of basic sets (bases) of polynomials has a significant role in mathematics and its applications, e.g., approximation theory, mathematical physics, Geometry, and partial differential equations. The interest of the present work focused on the expansion of analytic functions into BPs. Given a sequence of a base of polynomialsThe expansion of an analytic function as a basic series ∑ began with the papers by Whittaker and Cannon [14,15,30,31] about 90 years ago. Basic series generalize Taylor series, where can be Legendre, Laguerre, Chebyshev, Hermite, Bessel, Bernoulli and Euler polynomials (see [1,4,5,10,11,21]). This theory found a lot of applications, mainly in the theory of functions depending on one or several complex variables as well as in the approximation of solutions of differential equations or matrix functions.The topic of derivative BPs in one complex variable has been studied early (see [26,27,28]), the searchers considered the disks in the complex plane C. For several complex variables (see [16,17,19,24,25]), the representation domains are polycyclinderical, hyperspherical and hyperelliptical regions. Recently,in [12,35] the