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This article is concerned with the study of the theory of basic sets in Fréchet modules in Clifford analysis. The main aim of this account, which is based on functional analysis consideration, is to formulate criteria of general type for the effectiveness (convergence properties) of basic sets either in the space itself or in a subspace of finer topology. By attributing particular forms for the Fréchet module of different classes of functions, conditions are derived from the general criteria for the convergence properties in open and closed balls. Our results improve and generalize some known results in complex and Clifford setting concerning the effectiveness of basic sets.

The theory of bases in function spaces plays an important role in mathematics and its applications, for example, in approximation theory, partial differential equations, geometry, and mathematical physics.

The subject of basic sets of polynomials in one complex variable, in its classical form, was introduced by Whittaker [

The theory of basic sets of polynomials can be generalized to higher dimensions in several different ways, for instance, to several complex variables or to hypercomplex analysis.

The theory of basic sets of polynomials in several complex variables was developed at the end of the 1950s by Mursi and Maker [

In theory of basic sets of polynomials in hypercomplex analysis, Abul-Ez and Constales gave in [

In [

We shall lay down in this paper a treatment of the subject of basic sets based primarily on functional analysis and Clifford analysis. The aim of this treatment is to construct a criterion, of general type, for effectiveness of basic sets in Fréchet modules. By attributing particular forms to these Fréchet modules, we derive, in the remaining articles of the present paper, from the general criterion of effectiveness already obtained, particular conditions for effectiveness in the different forms of the regions which are relevant to our subsequent work. Thus, effectiveness in open and closed balls is studied. In addition, we give some applications of the effectiveness of basic sets of polynomials in approximation theory concerned with

the expansion of Clifford valued functions in closed and open ball by infinite series in a given sequence of basic sets,

the expansion of Clifford valued functions in closed and open ball by infinite series in a given sequence of Cannon sets of special monogenic polynomials.

These new results extend and generalize the known results in complex and Clifford setting given in [

In order to introduce our results, we give several notations and assumptions.

Let us denote by

A vector space basis for the Clifford algebra

We denote also by

Some care must be taken when using this norm to estimate product. We will always use the formula

One useful approach to generalize complex analysis to higher dimensional spaces is the Cauchy-Riemann approach which is based on the consideration of functions that are in the kernel of the generalized Cauchy-Riemann operator

A unitary right

Notice that

In the following, all

Let

Let

There exists a constant

For any finite number

If

A Fréchet module

a subset

We denote by

The sequence

It is a familiar property for the Fréchet module

It is also known that a linear operator

Let

We shall assume that Cauchy’s inequality holds for the basis

Also, the nature of the problems considered here necessitates that whenever

The basis

We start with the following introductory theorems.

If

It easily follows from the uniqueness of representation (

We deduce the continuity of

Let

Firstly, we prove that the family

Finally, when

In this section, we lay down the definition of basic sets, basic coefficients, and basic series and show (in Theorem

Let

Let

The following theorem is concerned with the basic coefficients

If

Let

Now, if

We now write

If, for every

We prove that

Now, it can be proven, from (

We have seen that when

Let

The first result concerning the effectiveness of basic set

For

Therefore, if we write

Now, to prove the continuity of

Hence, if

Now define the subspace

Suppose that

In this treatment, we consider the Fréchet module

Let

When series (

If, for every

We prove, as before, that, for each integer

Now, set

We see that when

The necessary and sufficient condition for effectiveness of

For the basic set

Now, if

Suppose that

We need to mention some definitions and notations in Clifford analysis [

Let

A polynomial

Let

The fundamental references for special monogenic function are [

Note that if

It is well known that

The maximum value of

An open ball is usually denoted by

The first application of the above theory is to the effectiveness in an open ball.

We propose to derive in the present section, from the results of Section

Let

For

The topology on

We shall take a basis for

It can be verified also that

Finally, when

Now, let

In this case, the expression

The fundamental theorem for effectiveness for

The necessary and sufficient condition for the basic set

Suppose that

On the other hand, suppose that condition (

Let

The basis for

Now, let

The theorem about such effectiveness is deducible from Theorem

As in (

The necessary and sufficient condition for the basic set

Suppose that

On the other hand, suppose that condition (

Hence, from (

Taking

The necessary and sufficient condition for the basic set

When

For Cannon sets, the Cannon function

The necessary and sufficient condition for the Cannon sets of special monogenic polynomials

The necessary and sufficient condition for the Cannon sets of special monogenic polynomials

The necessary and sufficient condition for the Cannon sets of special monogenic polynomials

The current address of Gamal Farghaly Hassan is “Department of Mathematics, Faculty of Sciences, Northern Border University, P.O. Box

The authors declare that they have no competing interests.

The authors wish to acknowledge the approval and the support of this research study from the Deanship of Scientific Research in Northern Border University, Arar, Saudi Arabia (Grant no. 5-7-1436-5).