On Simple Exponential Sets of Polynomials

In this paper, it is shown that certain classes of regular functions of several complex variables be represented by exponential sets of polynomials in hyperelliptical regions. Moreover, an upper bound for the order of exponential set is given.


Introduction
One of the fundamental problems in the classical analytic function theory is the existence of bases. In complex analysis, the local behavior of an analytic function in an open disc C(a, r) centered at a and with radius r is completely determined by means of a power series in (z − a) for a fixed complex number a. The question whether the basic set (basis) of polynomials {(z − a) n : n ∈ N} can be replaced by another set of polynomials (basis of polynomials) {P n (z) : n ∈ N}, representing each f (z) analytic in C(a, r) has been a subject of intensive theoretical research in classical function theory and its applications. The main development in the subject towards its present state was due to Whittaker [19,20] and Cannon [7,8]. Whittaker [20] laid down the minimum requirements for a set of polynomials to be suitable to represent a function and thus introduced the idea of "basic set" (or a basis) of polynomials. Consequently the expansion of a function by such series of polynomials is derived through an algebraic relation and not through a contour integral as in the previous trials. Also, considering the term function to refer to the Taylor expansion of the function about the origin, and thus reducing the domain of representation to a circle round the origin, Whittaker was able to introduce the idea of "effectiveness" of the set of polynomials and to obtain a criterion for effectiveness, i.e., a sufficient condition for effectiveness which does not depend on the particular function to be represented. This was proved to be necessary by Cannon [7] and thus the fundamental problem of the subject was solved. For sake of representation of entire (integral) functions, Whittaker introduced the idea of order and type of basic sets 338 Z. G. Kishka et al. MJOM (or bases) of polynomials, by giving these sufficient conditions for the representation of integral (entire) functions of certain class. These conditions were also proved to be necessary by Cannon [8]. Furthermore, many properties of the "criterion" of effectiveness were discovered and the idea of representation near the origin was introduced and developed by Whittaker [20]. The study of the basic sets of polynomials of several complex variables was initiated by Mursi and Makar [15,16], Nassif [17], Kishka and others [9][10][11][12][13][14], where the representation in polycylindrical and hyperspherical regions was considered. Also, there are studies on basic sets of polynomials such as in Clifford analysis [1][2][3][4][5][6] and in Faber regions [18]. In this work, we aim to establish certain convergence properties of a new basic set of polynomials of several complex variables in hyperelliptical regions. This set will be called (exponential set) in which its matrices of coefficients and operators are e p and e −p , respectively, with the properly that e p e −p = I. Also, effectiveness of this set is obtained. Moreover, the mode of increase of exponential set is given. For the purpose of this work, we give a different less elaborate exposition of the properties of theory polynomial bases. We refer the reader to [11,12,14,17].
In these notations, m 1 , m 2 , ..., m k and h 1 , h 2 , ..., h k are non-negative integers while t 1 , t 2 , ..., t k are non-negative numbers. Also, square brackets are used here in functional notation to express the fact that the function is either a function of several complex variables or one related to such function. In the space of several complex variables C k , an open hyperelliptical region k s=1 |zs| 2 r 2 s < 1 is here denoted by E [r] and its closure k s=1 , where r s , s ∈ I are positive numbers. In terms of the introduced notations, these regions satisfy the following inequalities: Vol. 11 (2014) On Simple Exponential Sets of Polynomials 339 Suppose now that the function f (z), is given by is regular in E [r] and s (see [10,13]), (2.4) Thus, it follows that Since ρ s can be chosen arbitrarily near to r s , s ∈ I, we conclude that Then, it can be easily proved that the function f (z) is regular in the open hyperelliptical E [r] . The numbers r s , given by (2.6), is thus conveniently called the radii of regularity of the function f (z).

Definition 2.1. ([9,10]) A set of polynomials
Thus, for the function f (z) given in (2.2) we get is the associated basic series of f (z). Now, let D m = D m1,m2,...,mn be the degree of the polynomial of the highest degree n represented in (2.9), that is to say, if D h = D h1,h2,...,hn is the degree of the polynomial P m , then D h < D m ∀ h s < m s . Since the elements of the basic set are linearly independent, then N m ≤ 1 + 2 + 3 + · · · + (D m+1 ) ≤ λ 1 D 2 m , where λ 1 is a constant. Therefore, the conditions (2.14) for a basic set to be a Cannon set implies the following condition (see [10,11]):

Definition 2.2. ([10,11]) The associated basic series
For any function f (z) of several complex variables, there is formally an associated basic series . When this associated series converges uniformity to f (z) in some domain it is said to represent f (z) in that domain. In other words, as in the classical terminology of Whittaker (see [19,20]), the basic set {P m [z]} will be effective in that domain. The convergence properties of basic sets of polynomials are classified according to the classes of functions represented by their associated basic series and also to the domain in which they are represented.
To study the convergence properties of such basic sets of polynomials in hyperelliptical regions (c.f. [10,11]), we consider the following notations for Cannon sums:

Effectiveness of Exponential Sets
Let { P E m (z)} be the exponential set of polynomials of the several complex variables z 1 , z 2 , ..., z s , whose matrices of coefficients and operators are e P and e -P , respectively. Thus e P e -P = I, and the exponential set First put j = 2 in (3.2) and apply (3.1) to get we shall assume that Hence, the assumption (3.3) is justified by mathematical induction. Since e P = E m;h is the matrix of coefficients of the exponential set { P E m (z)}, then it follows that We can write Since { P E m (z)} is basic, then there exists a unique representation in the form from this, we obtain that is to say It follows from (3.7) and (3.8), we see that Ω P E; E [r] = Π k s=1 r s , for all r s ≥ a s ; s ∈ I. Therefore, we infer that the exponential set { P E m (z)} is effective in the closed hyperellipse E [r] , for all r s ≥ a s ; s ∈ I.
Remark 3.1. When r s < a s , s ∈ I, the exponential set { P E m (z)} may not be effective in E [r] . This fact is shown by the following example: Thus, the Cannon sum of the the exponential set { P E m (z)} for the equihyperellipse E [r] * will be such that (3.11) Thus, Ω[ P E; E [r] * ] ≥ r (k−1) a for all r < a, and the exponential set { P E m (z)} is not effective in E [r] * for r < a as required.
Therefore, the following theorem is completely established.

Mode of Increase of Exponential Set
The mode of increase of a base P n (z) is determined by the order and type of the base. For a simple base {P n (z)}, the order ρ is defined by [20] ρ = lim r→∞ lim sup n→∞ log ω n (r) n log n , where ω n (r) stands as usual for the Cannon sum, given above. If 0 < ρ < 1, the type τ is given by It has been shown in [17] that the upper bound of the class of entire functions of several complex variables represented by a given base is determined by the mode of increase of the base. The significance of the order and type of a base lies in the fact that they define the class of entire functions represented by the base. Thus, the problem to be investigated in this section is that the mode of increase of the exponential set { P E m (z)} in the equi-hyperellipse E [r] * .
Suppose that {P m (z)} is a simple monic set for which where λ is positive constant and M ≥ 1, is finite number. Applying (3.1) in (4.1), it can be easily seen by induction that Thus, for the maximum modulus M( P E m ; E [r] * ) of the exponential set { P E m (z)} we have The fact that the bound λ is attainable is illustrated by the following example:  As in (4.5) we can apply (3.10) to prove by induction that Therefore, from (4.4) and (4.6), we conclude that Γ = λ and the bound is thus attainable. This completes the proof of the following theorem: Theorem 4.1. When condition (4.1) is satisfied, then the exponential set P E m (z) will be of order Γ ≤ λ. Moreover, the value λ is attainable.