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

In this


INTRODUCTION
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.
In [11,46], the authors address this matter within the Clifford setting, known as hypercomplex derivative bases of Cliffordian polynomials, where approximation occurs in closed balls.Recently, Hassan et al. [23] pioneered an exploration into the concept of BPs), primarily rooted in Clifford analysis and functional analysis.They formulated a comprehensive criterion to gauge the effectiveness, specifically the convergence properties, of BPs within Fréchet modules.Additionally, they provided practical applications showcasing the convergence properties of BPs in approximation theory.This pertains to the representation of special monogenic functions through an infinite series involving a sequence of special monogenic polynomials, within both closed and open ball contexts.In a recent paper [24], the authors delved into the representation of analytic functions using complex conformable fractional derivative and integral bases within the framework of Fréchet space.These bases exhibit connections with special functions such as Bernoulli, Euler, Bessel, and Chebyshev polynomials.Furthermore, a separate investigation was conducted in [28], focusing on the approximation properties of monogenic functions through hypercomplex Ruscheweyh derivative bases in the Fréchet module.This exploration established links with Bernoulli special monogenic polynomials, Euler special monogenic polynomials, and Bessel special monogenic polynomials.An intriguing research discovery is highlighted in [45], where the authors extended the well-known Whittaker-Cannon theorem in open hyperballs in by employing Hadamard's three-hyperballs theorem [7].Specifically, the hypercomplex Cannon functions were proven to preserve the effectiveness properties of both Cannon and non-Cannon bases.In [4], the authors presented an expansion of a specific monogenic function using generalized monogenic Bessel polynomials (GMBPs).Additionally, they demonstrated that the GMBPs serve as solutions to second-order homogeneous differential equations.However, several fundamental questions pertaining to the convergence properties of complex derivative and integral bases in Fréchet space still remain open.Similarly, crucial queries regarding the convergence properties of other hypercomplex derivative bases in Fréchet modules within Clifford analysis also persist.Motivated by the aforementioned gaps in the literature, we introduce a novel definition of the Hyperbolic Theta operator and Hyperbolic integral operator within the realm of complex calculus.Subsequently, we apply these operators to a set of BPs to derive the HTOBPs and HIOBPs.The majority of the theorems established in this study revolve around the representation of analytic and entire functions through infinite series composed of HTOBPs and HIOBPs.
The structure of the current work is outlined as follows.Section 2 is dedicated to recalling the most commonly used notations and results.In Section 3, we introduce a new operator, the Hyperbolic Theta operator, and the Hyperbolic integral operator to derive the HTOBPs and HIOBPs.Section 4 presents the results indicating the effectiveness properties of the HTOBPs and HTOBPs.This section also includes the demonstration of some bases of special classes of polynomials that fulfill the established results.Section 6 contains the proofs for all the theorems.In Section 7, future work is discussed as an integral part of this paper.This section offers an overview of key definitions, notations, and results essential for providing background information in this paper.The concepts covered include: bases, basic series, , effectiveness, type, order, and Fréchet space (refer to [8,14,17,18,23]).Definition 2.1.Let be a vector space over .A real-valued function is called a semi-norm if it satisfies the following conditions: Where any open disk enclosing the closed disk ̅ .Also, the class of entire functions on with the family of semi-norm make into an F-space E (see [ 23,24,28]).
Suppose that { } is a base of an F-space E and The matrices and are called the matrix of coefficients and the matrix of operators of the base { }.The set { } is base (see [23,37]), iff Cauchy's inequality [23,37] of a base (2.1) takes the form 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.
For The author in [19] defined the -property of BPs of one complex variable in a closed disk.In addition, the -property is defined in polycylinderical regions [31].-property is also known in the case of Clifford setting in [3].

Definition 2.4. A base { } has the property in ̅
, if the basic series (2.10) represents in ̅ every integer function of order less than ρ.

Let
In fact [19] proved the following result on the -property in ̅ .

HTOBs and HIOBs
The hyperbolic Theta operator (HTOC) is a differential operator defined by the power series ∑ , where is called the Theta operator, also called the homogeneity operator, because its eigenfunctions are the monomials in : Where Now, we would like to know its effect on the function .First let's see its effect in the function .

∑ ∑
Now, we can use this result to see the effect on a function .If the function has a power series at , then The HTOC of certain functions (3) (4) .
The HTOC is linear i.e. .
By applying the operator into (2.1),we define the HTOBs as follows: Definition 3.1.Suppose that { } is a base.The HTOBs is defined as: The following definition for hyperbolic integral operator (HIOC) of a function , where the integral operator is defined by ∫ and ( ) where .
Let us elaborate on the HIOC.It is defined by the power series ∑ .First let's see its effect in the function .
Now, we would like to know its effect on the function , where where the series is uniformly convergent.
By applying the operator into (2.1),we define the HIOBs as follows: (3) The authors [1], considered the base of Gontcharaff polynomials { } (B Ps), associated with a given set of points, are the polynomials defined by Where and are given complex numbers.
When | | , The authors [1] has shown that the B Ps { } is effective for  There exists a number such that In view of (6.3) and (6.4), we have and { } is effective for .We can proceed very similar as above to prove that { } is effective for or or (see [24,28]).Also, using the same steps, we can show that { } is effective for the same spaces.
Proof of Theorem 4.3.The maximum moduli of { } on the closed disk ̅ , is given by Now, using (2.3) and (6.7), we get A combination between (2.5), (6.8) and condition (4.1), yields But .Therefore and the HTOBP { } is effective for ̅ .Similarly, we can also prove that the base { } is effective for ̅ .
The following example show that the condition (4.1) cannot be dropped.We can show that it is of order and type .
The following example show that the condition (4.2) cannot be dropped.This base is of order (see [24]).For the HTOBP { } such that:

{
It is easily seen that and .Therefore Theorem 4.4 is not verified.

Proof of Theorem 4.5. Let given by
In view of (4.3), (6.8) and (6.10) we have In this conclusion, we will discuss some ideas for future works that contain new research ideas for the reader.The future works are divided into seven categories: 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.
3. Exploring new types of bases like q-Bernoulli and q-Euler in both Complex and Clifford analysis is a prospective area of study.
4. Generalizing the theory of BPs in Clifford analysis to regions of representation such as polycylindrical, spherical, hyperelliptical, and Faber regions is an avenue worth exploring.
5. The extension of basic sets of polynomials from complex matrices to Fréchet spaces or Fréchet modules is another potential research direction.
6. Investigating the idea of fractional derivatives of basic sets in several complex variables and Clifford analysis is an intriguing prospect.
7. Considering the potential generalization of HTOBs and HIOBs to the case of several complex variables and Clifford analysis would be a significant contribution.If feasible, generalizing the theorems proven in this paper to accommodate this extension is an important consideration.

Corollary 5 . 1 .Corollary 5 . 2 .Corollary 5 . 3 .Corollary 5 . 4 .
According to Theorem 4.3, the following results follows: The HTOC and HIOC of Bessel polynomials { } and { } are effective in the same space of effectiveness of the original { }.The HTOC and HIOC of general Bessel polynomials { } and { } are effective in the same space of effectiveness of the original { }.The HTOC and HIOC of Chebyshev polynomials { } and { } } are effective in the same space of effectiveness of the original { }.The HTOC and HIOC of the Gontcharaff polynomials { } and { } are effective in the same space of effectiveness of the original

Example 6 . 3 .
Consider the BPs { } given by { ( ) have extensively explored the convergence properties of associated bases, such as product base, the inverse base, transposed inverse base, transpose base, similar base, square root base, and Hadamard product base.These bases are comprised of simple monic bases .. Future endeavors may involve generalizing the theorems for the aforementioned associated bases when their constituents are non-simple monic bases.2. The forthcoming research may focus on the convergence properties of bases, specifically the generalization of Kronecker product bases of polynomials to higher dimensions in Clifford analysis.Additionally, extending the representation of Cliffordvalued functions in open and closed balls by infinite series with Legendre, Laguerre, The main results of the present work are formulated in the following theorems.
The fact that the condition (4.1) cannot be dropped is illustrated below by Example 6.1 in Section 6.