Effectiveness of cannon and composite sets of polynomials of two complex variables in Faber regions

Adepoju, Jerome Ajayi; Mogbademu, Adesanmi Alao

doi:10.22199/issn.0717-6279-2020-03-0040

Cited by 3 publications

(4 citation statements)

References 3 publications

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“…A new extension of the well-known Ruscheweyh derivative operator was introduced in [16], where the representation of certain special monogenic functions in different regions of convergence was investigated in Fréchet modules. The previously mentioned treatment generalizes the results in the complex and Clifford settings given in [10,13,17]. In [18], the authors established an expansion of a particular monogenic function in terms of generalized monogenic Bessel polynomials (GMBPs).…”

Section: Introductionmentioning

confidence: 76%

Equivalent Base Expansions in the Space of Cliffordian Functions

Zayed

Hassan

2023

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Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, first, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases.

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Section: Introductionmentioning

confidence: 76%

Equivalent Base Expansions in the Space of Cliffordian Functions

Zayed

Hassan

2023

Axioms

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“…For any element ∑ of an F-space E, substitute from (2. Results on the effectiveness of bases (see [5,8,23,37]). The author in [19] defined the -property of BPs of one complex variable in a closed disk.…”

Section: Suppose That { } Is a Base Of An F-space E Andmentioning

confidence: 99%

“…1. Previous studies [8,21,32,37,38,42,43] Bessel, Hermite, and Gontcharoff polynomials to hypercomplex analysis is a potential avenue. Moreover, the investigation into the effectiveness, growth type, and order of the above sets could be extended to several complex variables, covering regions such as polycylindrical, hyperelliptical, spherical, and Faber regions, for all entire functions and at the origin.…”

Section: Applicationsmentioning

confidence: 99%

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Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

Hassan

2024

Assiut University Journal of Multidisciplinary Scientific Resea

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The approximation of an analytic function as a basic series of the following type where is a BPs, has been developed by the British Mathematician J. M. Whittaker in the early 1930 s [39]. J. M. Whittaker and B. Cannon [15,16,40,41] have obtained many results about approximation of analytic and entire functions by basic series (1.1). Numerous specific instances of polynomial series have undergone thorough examination. Taylor's series stands out as the simplest case, with other notable examples including expansions from interpolation theory and series involving polynomials such as Hermite, Legendre, Legendre, Euler, Bernoulli, Chebyshev, and Gontcharoff. Several scholars, including Makar [34], Mikhail [35], and Newns [37], have delved into the convergence properties of derivative and integral bases for a given set of BPs in a single complex variable within a disk centered at the origin. For multiple complex variables, as explored in [12,20,32,33], representation domains extend to polycylindrical, hyperspherical, and hyperelliptical regions.

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Criteria in Nuclear Fréchet Spaces and Silva Spaces with Refinement of the Cannon-Whittaker Theory

Abul-Ez

Zayed

2020

Journal of Function Spaces

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Along with the theory of bases in function spaces, the existence of a basis is not always guaranteed. The class of power series spaces contains many classical function spaces, and it is of interest to look for a criterion for this class to ensure the existence of bases which can be expressed in an easier form than in the classical case given by Cannon or even by Newns. In this article, a functional analytical method is provided to determine a criterion for basis transforms in nuclear Fréchet spaces ((NF)-spaces), which is indeed a refinement and a generalization of those given in this concern through the theory of Whittaker on polynomial bases. The provided results are supported by illustrative examples. Then, we give the necessary and sufficient conditions for the existence of bases in Silva spaces. Moreover, a nuclearity criterion is given for Silva spaces with bases. Subsequently, we show that the presented results refine and generalize the fundamental theory of Cannon-Whittaker on the effectiveness property in the sense of infinite matrices.

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Effectiveness of cannon and composite sets of polynomials of two complex variables in Faber regions

Cited by 3 publications

References 3 publications

Equivalent Base Expansions in the Space of Cliffordian Functions

Equivalent Base Expansions in the Space of Cliffordian Functions

Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

Criteria in Nuclear Fréchet Spaces and Silva Spaces with Refinement of the Cannon-Whittaker Theory

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