Forward (Δ) and Backward (∇) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains

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Received 7 July 2016; accepted 26 August 2016; published 29 August 2016

1. Introduction

Recently, there has been an upsurge of interest in the investigations of the basic sets of polynomials [1] - [27] . The inspiration has been the need to understand the common properties satisfied by these polynomials, crucial to gaining insights into the theory of polynomials. For instance, in numerical analysis, the knowledge of basic sets of polynomials gives information about the region of convergence of the series of these polynomials in a given domain. Namely, for a particular differential equation admitting a polynomial solution, one can deduce the range of convergence of the polynomials set. This is an advantage in numerical analysis which can be exploited to reduce the computational time. Besides, if the basic set of polynomials satisfies the Cannon condition, then their fast convergence is guaranteed. The problem of derived and integrated sets of basic sets of polynomials in several variables has been recently treated by A. El-Sayed Ahmed and Kishka [1] . In their work, complex variables in complete Reinhardt domains and hyperelliptical regions were considered for effectiveness of the basic set. Also, recently the problem of effectiveness of the difference sets of one and several variables in disc D(R) and polydisc has been treated by A. Anjorin and M.N Hounkonnou [27] .

In this paper, we investigate the effectiveness, in Reinhardt and hyperelliptic domains, of the set of polynomials generated by the forward (D) and backward (Ñ) difference operators on basic sets. These operators are very important as they involve the discrete scheme used in numerical analysis. Furthermore, their composition operators form the most of second order difference equations of Mathematical Physics, the solutions of which are orthogonal polynomials [25] [26] .

Let us first examine here some basic definitions and properties of basic sets, useful in the sequel.

Definition 1.1 Let be an element of the space of several complex variables . The hyperelliptic region of radii), is denoted by and its closure by where

And

Definition 1.2 An open complete Reinhardt domain of radii is denoted by and its closure by, where

The unspecified domains and are considered for both the Reinhardt and hyperelliptic domains. These domains are of radii. Making a contraction of this domain, we get the domain where stands for the right-limits of:

Thus, the function of the complex variables, which is regular in can be represented by the power series

(1)

where represents the mutli indicies of non-negative integers for the function F(z). We have [1]

(2)

where is the radius of the considered domain. Then for hyperelliptic domains [1]

t being the radius of convergence in the domain, assuming and, whenever. Since, we have (1)

where also, using the above function of the complex variables, which is regular in and can be represented by the power series above (1), then we obtain

and

(3)

Hence, we have for the series

. Since can be taken

arbitrary near to, we conclude that

With and.

Definition 1.3 A set of polynomials is said to be basic when every polynomial in the complex variables can be uniquely expressed as a finite linear combination of the elements of the basic set.

Thus, according to [4] , the set will be basic if and only if there exists a unique row-finite-matrix such that, where is a matrix of coefficients of the set; are multi indices of nonegative integers, is the matrix of operators deduced from the associated set of the set and is the infinite unit matrix of the basic set, the inverse of which is. We have

(4)

Thus, for the function given in (1), we get where

,. The series

is an associated basic series of F(z). Let be the number of non zero coefficients in the representation (4).

Definition 1.4 A basic set satisfying the condition

(5)

Is called a Cannon basic set. If

Then the set is called a general basic set.

Now, let be the degree of polynomials of the highest degree in the representation (4). That is to say is the degree of the polynomial; the and since the element of basic set are linearly independent [6] , then, where is a constant. Therefore the condition (5) for a basic set to be a Cannon set implies the following condition [6]

(6)

For any function of several complex variables there is formally an associated basic series. When the associated basic series converges uniformly to in some domain, in other words as in classical terminology of Whittaker (see [5] ) the basic set 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 property of such basic sets of polynomials in complete Reinhardt domains and in hyperelliptic regions, we consider the following notations for Cannon sum

(7)

For Reinhardt domains [24] ,

(8)

For hyperelliptic regions [1] .

2. Basic Sets of Polynomials in Generated by Ñ and D Operators

Now, we define the forward difference operator D acting on the monomial such that

where E is the shift operator and -the identity operator. Then

So, considering the monomial

Hence

Since,

Hence

where and by definition. Similarly, we define the backward difference operator

Ñ acting on the monomial such that

(9)

Equivalently, in terms of lag operator L defined as, we get. Remark that the advantage which comes from defining polynomials in the lag operator stems from the fact that they are isomorphic to the set of ordinary algebraic polynomials. Thus, we can rely upon what we know about ordinary polynomials to treat problems concerning lag-operator polynomials. So,

(10)

The Cannon functions for the basic sets of polynomils in complete Reinhardt domain and in hyperelliptical regions [1] , are defined as follows, respectively:

Concerning the effectiveness of the basic set in complete Reinhardt domain we have the following results:

Theorem 2.1 A necessary and sufficient condition [7] for a Cannon set to be:

1. effective in is that;

2. effective in is that.

Theorem 2.2 The necessary and sufficient condition for the Cannon basic set of polynomials of

several complex variables to be effective [1] in the closed hyperelliptic is that where.

The Cannon basic set of polynomials of several complex variables will be effective in

if and only if. See also [1] . We also get for a given polynomial set:

So, considering the monomial,

Let’s prove the following statement:

Theorem 2.3 The set of polynomials and

Are basic.

Proof: To prove the first part of this theorem, it is sufficient to to show that the initial sets of polynomials and, from which and are generated, are linearly independent. Suppose there exists a linear relation of the form

(11)

For at least one i,. Then

Hence, it follows that. This means that would not be linearly independent. Then the set would not be basic. Consequently (11) is impossible. Since are polynomials, each of them can be represented in the form. Hence, we write

In general, given any polynomial and using

Hence the representation is unique. So, the set is a basic set. Changing D to Ñ leads to the same conclusion. We obtain the following result.

Theorem 2.4 The Cannon set of polynomials in several complex variables is Effective

in the closed complete Reinhardt domain and in the closed Reinhardt region.

Proof: In a complete Reinhardt domain for the forward difference operator D, the Cannon sum of the monomial is given by

Then

where is a constant. Therefore,

which implies that

Then the Cannon function

But. Hence

Similarly, for the backward difference operator Ñ, the Cannon sum

Then

where as is bounded for the Reinhardt domain is complete. Thus,

But

Hence, we deduce that.

Theorem 2.5 If the Cannon basic set (resp.) of polynomials of the several complex variables for which the condition (5) is satisfied, is effective in, then the (D) and (Ñ)-set

(resp.) of polynomials associated with the set (resp.) will

be effective in.

The Cannon sum of the forward difference operator D of the set in will have the form

where and

where is a constant. Then

where

So, by similar argument as in the case of Reinhardt domain we obtain

where, since the Cannon function is such that [1] . Similarly, for the backward difference operator

Such that the Cannon function writes as

But

Since the Cannon function is non-negative. Hence

3. Examples

Let us illustrate the effectiveness in Reinhardt and hyperelliptic domains, taking some examples. First, suppose that the set of polynomials is given by

Then

Hence

which implies

for;.

Now consider the new polynomial from the polynomial defined above:

Hence by Theorem 2.4,

where

where is a constant. Hence,

The Cannon function

which implies

where

and

Hence

Similarly, for the operator Ñ, we have

Since

Then

Table 1. Region of effectiveness: (1) Disc; (2) Hyperelliptic; Reinhardt domain.

Implication: The new sets are nowhere effective since the parents sets are nowhere effective. By changing

in Reinhart domain to, where, we obtain the

same condition of effectiveness as in Reinhart domain for both operators D and Ñ in the hyperelliptic domain.

The following notations are relevant to the table below.

(12)

(13)

(14)

(15)

Finally, for the classical orthogonal polynomials, the explicit results of computation are given in a Table 1 below.

Thus, in this paper, we have provided new sets of polynomials in C, generated by Ñ and D operators, which satisfy all properties of basic sets related to their effectiveness in specified regions such as in hyperelliptic and Reinhardt domains. Namely, the new basic sets are effective in complete Reinhardt domain as well as in closed Reinhardt domain. Furthermore, we have proved that if the Cannon basic set is effective in hyperelliptic domain, then the new set is also effective in the hiperelliptic domain.

Appendix

Key Notations

1) = Cannon sum of the new D-set in Reinhardt domain.

2) = Cannon sum of the new Ñ-set in Reinhardt domain.

3) = Cannon sum of the new D-set in Hyperelliptic domain.

4) = Cannon sum of the new Ñ-set in Hyperelliptic domain.

5) = Cannon function of the new D-set in Reinhardt domain.

6) = Cannon function of the new Ñ-set in Reinhardt domain.

7) = Cannon sum of the new D-set in Hyperelliptic domain.

8) = Cannon sum of the new Ñ-set in Hyperelliptic domain.

9).

10).

11)

where is a constant. is a coefficient corresponding to polynomials set, is a coefficient corresponding to polynomial set. We should note that or.

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