Lower Growth of Generalized Hadamard Product Functions in Clifford Setting

Zayed, Mohra

doi:10.1007/s40840-020-00983-y

Cited by 8 publications

(6 citation statements)

References 25 publications

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“…Where and gives the order and type of the base and bear its importance due to the fact that the base represents in the whole plane every integer function of type less than and order less than in any finite disk (see [21,22,39,41).We refer to work by [10,43,44] regarding the type and order of BPs.…”

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

confidence: 99%

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

confidence: 99%

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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“…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%

“…If possible, we can generalize the theorems proven in Section 5 of this paper in several complex variables. In this case, we expect the convergence regions to be spherical regions or polycylinderical regions, or elliptical regions. In previous studies, 13,15,19,45–48,51,54,59 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? In the future it is possible to study the convergent properties of new bases of polynomials in different regions (e.g., Laguerre, Lagender, Hermit, and Gontcharoff polynomials).…”

Section: Open Problemsmentioning

confidence: 99%

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Approximation of functions by complex conformable derivative bases in Fréchet spaces

Hassan

Abdel-Salam

Rashwan

2022

Math Methods in App Sciences

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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.

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“…A new light has been shed upon the study of elementary functions, 12–18 the computation of combinatorial identities, 19–21 and the study of a generalized Joukowski transformation in Euclidean space of arbitrary higher dimension 22 . Earlier results in the theory of special polynomial bases in hypercomplex analysis and its counterpart in the function theory of several complex variables can be found in previous studies 12,23–34 and elsewhere. Related results have been generalized to the quaternionic setting 35–37 …”

Section: Introductionmentioning

confidence: 99%

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

Zayed

Morais

2022

Math Methods in App Sciences

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This paper shows a hypercomplex function theory emerging in the representation of paravector-valued monogenic functions over the (m + 1)-dimensional Euclidean space through a basic set (or basis) of hypercomplex monogenic polynomials. We derive the properties of the arising hypercomplex Cannon function and present an extension of the well-known Whittaker-Cannon theorem to special monogenic functions defined in an open hyperball in R m+1 . More precisely, we determine what conditions should be applied to a basic set of special monogenic polynomials to attain the effectiveness property in an open hyperball employing Hadamard's three-hyperballs theorem. We also provide a necessary and sufficient condition for a special monogenic Cannon series to represent every function near the origin that is special monogenic there. Additionally, we investigate the effectiveness of a non-Cannon basis and show that the underlying hypercomplex Cannon function maintains similar properties in both cases, the Cannon basis and the non-Cannon basis.

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Lower Growth of Generalized Hadamard Product Functions in Clifford Setting

Cited by 8 publications

References 25 publications

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

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

Approximation of functions by complex conformable derivative bases in Fréchet spaces

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

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