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

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

Section: Discussionmentioning

confidence: 99%

“…The importance of order and type lies in that if the BPs {Pn(z)} has finite order ω and finite type τ, it represent every entire function of order less than and type less than in any finite disk (c.f. [20,21,30]). Related results to order and type of the BPs can be found in [2,33,34].…”

Section: Preliminariesmentioning

confidence: 99%

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

Assiut University Journal of Multidisciplinary Scientific Resea

Atta²

2022

“…Related to order and type of the BP, we refer to previous studies. [46][47][48][49][50][51] Now, we recall the definition of the T 𝜌 -property as given by Abul-Ez and Constales 45 as follows: Definition 9. If 0 < 𝜌 < ∞, then a base is said to have property T 𝜌 in a closed disk D(R), if it represents all entire functions of order less than 𝜌 in D(R).…”

Section: Definition 1 (Seminorm) a Seminorm On A Vector Spacementioning

confidence: 99%

“…3. In previous studies, 13,15,19,[45][46][47][48]51,54,59 the convergence properties in different regions of associated BP (such as inverse base, product base, transpose base, transposed inverse base, square root base, similar base, Hadamard product base ) were studied. Is the CCFDB and CCFIB of these bases convergent in the same regions?…”

Section: Open Problemsmentioning

confidence: 99%

Approximation of functions by complex conformable derivative bases in Fréchet spaces

Math Methods in App Sciences

Abdel-Salam

Rashwan

2022

In the present paper, the representation, in different domains, of analytic functions by complex conformable fractional derivative bases (CCFDB) and complex conformable fractional integral bases (CCFIB) in Fréchet space are investigated.Results are proved to show that such representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin, and for all entire functions. Also, some results concerning the growth order and type of CCFDB and CCFIB are determined. Moreover, the T 𝜌 -property of CCFDB and CCFIB is discussed. The obtained results recover some known results when 𝛼 = 1.Finally, some applications to the CCFDB and CCFIB of Bernoulli, Euler, Bessel, and Chebyshev polynomials have been studied.

“…An important subclass of the Clifford holomorphic functions called special monogenic functions is considered, for which a Cannon theorem on the effectiveness in closed and open ball [21,23] was established. Many authors studied the basic sets of polynomials in Clifford analysis [24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Basic Sets of Special Monogenic Polynomials in Fréchet Modules

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.