#### Abstract

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.

#### 1. Introduction

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 [1, 2] who laid down the definition of basic sets, basic series, and effectiveness of basic sets. Many well-known polynomials such as Laguerre, Legendre, Hermite, Bernoulli, Euler, and Bessel polynomials form simple bases of polynomials (see [3–7]). A significant advance was contributed to the subject by Cannon [8, 9] who obtained necessary and sufficient conditions for the effectiveness of basic sets for classes of functions with finite radius of regularity and entire functions.

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 [10] and later by Nassif [11] and was studied in more detail afterwards by others (c.f [12–15]). Also, the representation of matrix functions by bases of polynomials has been studied by Makar and Fawzy [16]. For more information about the study of basic sets of polynomials in complex analysis, we refer to [17–20].

In theory of basic sets of polynomials in hypercomplex analysis, Abul-Ez and Constales gave in [21, 22] the extension of the theory of bases of polynomials in one complex variable to the setting of Clifford analysis. This is the natural generalization of complex analysis to Euclidean space of dimension larger than two, where the holomorphic functions have values in Clifford algebra and are null solutions of a linear differential operator. An important subclass of the Clifford holomorphic functions called special monogenic functions is considered, for which a Cannon theorem on the effectiveness in closed and open ball [21, 23] was established. Many authors studied the basic sets of polynomials in Clifford analysis [24–30].

In [31], Adepoju laid down a treatment of the subject of basic sets of polynomials of a single complex variable in Banach space which is based on functional analysis considerations. Also, the authors in [12, 32] studied the basic sets of polynomials of several complex variables in Banach space.

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(1)the expansion of Clifford valued functions in closed and open ball by infinite series in a given sequence of basic sets,(2)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 [12, 21, 23, 31, 32].

#### 2. Notation and Preliminaries

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

Let us denote by the canonical basis of the Euclidean vector space and by the associated real Clifford algebra in which one has the multiplication rules , , where denotes the Kronecker symbol.

A vector space basis for the Clifford algebra is given by the set with , , and . Every can be written in the form with . The conjugate element of is defined by , where , , and .

We denote also by the space of paravectors . In this notation, the paravector will be represented in the form with being the scalar part and being the vector part of . The induced Clifford norm of arbitrary is given 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 in (for more details, see [33, 34]).

*Definition 1 (unitary right -module). *A unitary right -module is an abelian group with a mapping ; such that for all and :(i).(ii).(iii).(iv).

*Remark 2. *Notice that becomes a real vector space if is identified with .

In the following, all -modules will be right -modules.

*Definition 3 (-linear operator). *Let and be two unitary -modules. Then a function is said to be an -linear operator if, for all and , The set of all -linear operators from into is denoted by .

*Definition 4 (proper system of seminorms). *Let be a unitary -module. Then a family of functions is said to be a proper system of seminorms on if the following conditions are fulfilled:There exists a constant such that, for all , and :(i).(ii), and if For any finite number , there exist and such that, for all , If for all then

*Definition 5 (Fréchet module). *A Fréchet module over is a Hausdorff space with a countable proper system of seminorms satisfying(i); ,(ii)a subset of is open if, , there exist and such that (iii) is complete with respect to this topology. We denote by the topology defined by the family of seminorms on .

*Definition 6 (convergent sequences in the topology ). *The sequence of elements of converges in the topology to the element of , if and only if, for all , we haveWe may also equivalently say that the sequence converges in to with respect to .

It is a familiar property for the Fréchet module that a seminorm on is -continuous, if and only if there is a seminorm and a positive finite constant such that

It is also known that a linear operator on is continuous if and only if there is a seminorm and a constant such that

#### 3. Basis and Absolute Basis

*Definition 7 (basis for Fréchet module ). *Let be a Fréchet module over . A sequence of nonzero elements of is called a basis for if, for each element , there is one and only one sequence of the Clifford algebra , such that

*Definition 8 (Cauchy’s inequality). *We shall assume that Cauchy’s inequality holds for the basis in the form that for each there is a positive finite constant such thatfor all integers and for all .

Also, the nature of the problems considered here necessitates that whenever , there is a finite positive constant , such that

*Definition 9 (absolute basis for Fréchet module ). *The basis is called an absolute basis for if the series is convergent in for all integers and for all . Thus, in this case, we can write

We start with the following introductory theorems.

Theorem 10. *If is a basis for and if Cauchy’s inequality (8) is satisfied, then is a continuous linear operator on , orthonormal to .*

*Proof. *It easily follows from the uniqueness of representation (7) that if and , then so that is a linear operator on . Also, putting in (7), it can be verified thatand is orthonormal to .

We deduce the continuity of from (6) and (8).

Theorem 11. *Let be an absolute basis for and let be given by (11). Then the family forms a proper system of continuous seminorms. Moreover, for , there exists a constant such thatfor all .*

*Proof. *Firstly, we prove that the family is a proper system of seminorms as follows.

We observe, from the linearity of and properties (i) and (ii) of seminorms, that whenever and

Let be defined by (11). Since is a proper system of seminorms, then there exist and such that, for all , Suppose that ; then . Since is a proper system of seminorms and is a basis, for each there exists such that . So for all and hence (7) implies that

Finally, when , inequality (14) can be obtained from (8), (9), and (11) as follows:It follows from condition (5) that the seminorm is continuous on . Therefore the family forms a proper system of continuous seminorms, as required.

#### 4. Basic Sets

In this section, we lay down the definition of basic sets, basic coefficients, and basic series and show (in Theorem 13) that when the basic series converges, it will converge to the element with which it is associated.

Let be a sequence of nonzero elements of , and suppose that is a matrix of coefficients in the Clifford algebra such that, for each , we have the unique representationIn this case, we shall call the sequence a basic set on .

Let be any element of and substitute (18) in (7) to obtain the formal serieswhereWhen series (20) converges in , exists and is called the th basic coefficient of relative to the set . When the basic coefficient exists for all , series (19) is called the basic series associated with .

The following theorem is concerned with the basic coefficients

Theorem 12. *If is defined for all in , the map is a continuous linear operator on .*

*Proof. *Let It is clear that is a continuous linear operator on as a finite sum of continuous linear functional .

Now, if is defined for all in , the sequence converges pointwise to in . Therefore, by the Banach-Steinhaus theorem for Fréchet space [35], we deduce that is equally a continuous linear operator on , and the theorem is established.

We now writefor the th partial sum of basic series (19). The following theorem establishes the required conformity of the limit of with the space .

Theorem 13. *If, for every , is defined for all and if the sequence converges in to some limit , then for all elements .*

*Proof. *We prove that is a continuous linear operator on as a limit of finite sum of continuous linear operators as in Theorem 12.

Now, it can be proven, from (13) and (20), that Hence, (18) and (22) together yieldLet be any element of and writeThen, in view of (24) and (25), we haveand, by continuity of , we deduce that and Theorem 13 is therefore established.

#### 5. Effectiveness of Basic Sets

We have seen that when converges for each of , converges to . This means that basic series (19) associated with the element converges to itself for all . In this case, we say that the set is effective for . To find a necessary and sufficient condition for the effectiveness of a basic set for the space , we consider, for each seminorm , the mapping defined bySuppose that is finite for all . We first show that is a seminorm on .

Let and ; it follows from (27) and the linearity of that

The first result concerning the effectiveness of basic set is the following theorem.

Theorem 14. *For to be effective for , it is necessary and sufficient that, for each , the seminorm exist and be continuous.*

*Proof. * *Necessity*. When the basic set is effective for , basic series (19) associated with each element converges to , and it follows that is a continuous linear operator on .

Therefore, if we write then from (27) we shall have If , then Then for every there exists a natural number , such that if , then Hence, we have If with fixed , then Hence, for some , there is such that, for all and , This shows that the seminorm exists.

Now, to prove the continuity of , let be a sequence in which converges to an element . By this hypothesis, if , there exists a natural number such that if , then

Hence, if , it follows that if , then Thus and is a continuous seminorm on .*Sufficiency*. We observe from (22) and (27) that Since is continuous on , we deduce that the sequence is an equicontinuous sequence on .

Now define the subspace of by It follows from the equicontinuity of that the set is closed. Hence, the set is everywhere dense and is closed, so that . Therefore is a Cauchy sequence on and since is complete, the sequence converges for all and hence it converges to in . Thus, the set is effective for and Theorem 14 is established.

Theorem 15. *Suppose that is an absolute basis for . Then the basic set will be effective for if and only if, for any continuous seminorm , there is a continuous seminorm and a positive finite number such that*

*Proof. * *Necessity*. If the basic set is effective for , then, by Theorem 14, the application is a continuous seminorm on . Hence, by (5), there is a seminorm and a positive number such that Putting , , we obtain (36).*Sufficiency*. Multiplying the basic coefficient of (20) by and using (11), (27), and (36), we obtain where is defined by (11). According to inequality (14), there is a seminorm and a positive number such that It follows from condition (5) that is continuous on . Then, by Theorem 14, we deduce that the set is effective for and the proof of Theorem 15 is therefore terminated.

#### 6. Alternative Treatment of the Problem

In this treatment, we consider the Fréchet module as a subspace of a Banach module with a continuous norm such thatwhere is the family of seminorms defined, as before, in the space . Thus, the topology induced in by the topology of determined by the norm is coarser than the topology defined on by the family of seminorms.

Let be a basis for and let be a sequence of nonzero elements of . We suppose that is a matrix of such that, for each , we have the unique representationand the convergence is in . In this case, we call the sequence a basic set on . Let be an element of and substitute (41) in (7) to obtain the formal serieswhere

When series (43) converges in , we call the basic coefficient of , and when exists for each , series (42) is called the basic series of . Recall that the partial sum is defined (see (22)) byTheorem 12 remains unchanged, while the alternative form of Theorem 13 is the following.

Theorem 16. *If, for every , is defined for all and if the sequence converges in to some element , then for all elements .*

*Proof. *We prove, as before, that, for each integer , is a linear operator from to . We show now that is continuous. In fact, if is a sequence of elements of converging to an element of , it can be deduced from (40) and (44) and Theorem 12 that Hence converges to in and hence is a continuous linear operator. Proceeding exactly as in the proof of Theorem 13, we can deduce that is a continuous linear operator.

Now, set . It is clear that in and ; hence , as required.

We see that when converges in , for every element , converges in to . This means that basic series (42) associated with the element converges in to the element , for all . In this case, we say that the basic set is effective for in .

The necessary and sufficient condition for effectiveness of for in is obtained through the expressionIt can be proven in exactly the same way as before that is a seminorm on . The revised version of Theorem 14 is as follows.

Theorem 17. *For the basic set to be effective for in , it is necessary and sufficient that exist and be continuous on .*

*Proof. * *Necessity*. Since is continuous linear operator from to , we apply the same method as in the proof of Theorem 14.*Sufficiency*. We see here, from (22) and (46), thatSince is continuous on , then the sequence will be equicontinuous from to . The proof is then completed in exactly the same way as in the proof of Theorem 14.

Now, if is an absolute basis for , the effectiveness of the set for in will be estimated through the expression as it is seen from the following Theorem which is the alternative form of Theorem 15.

Theorem 18. *Suppose that is an absolute basis for . Then the basic set will be effective for in if and only if there is a seminorm and a constant such that*

*Proof. * *Necessity*. If is effective for in , then, by Theorem 17, is a continuous norm on . Hence, by (5), there is a seminorm and a constant such that Putting , we obtain (48).*Sufficiency*. As in the proof of Theorem 15, we deduce from (11), (43), (46), and (48) that Applying inequality (14), we obtain So we deduce that is continuous on and hence Theorem 17 implies that is effective for in . Theorem 18 is therefore established.

#### 7. Applications

We need to mention some definitions and notations in Clifford analysis [21, 22, 33, 34].

*Definition 19 (monogenic function). *Let be an open set; then an -valued function is called left (resp. right) monogenic in if it satisfies (resp. ) in . Here, , defined in Section 2, is the generalized Cauchy-Riemann operator.

*Definition 20 (special monogenic polynomial). *A polynomial is special monogenic if and only if (so is monogenic) and there exists for which

*Definition 21 (special monogenic function). *Let be a connected open subset of containing and let be monogenic in . Then is called special monogenic in if and only if its Taylor series near zero (which exists) has the form for certain special monogenic polynomials .

The fundamental references for special monogenic function are [36, 37].

*Remark 22. *Note that if is a homogeneous special monogenic polynomial of degree , then (see [21, 29]) : is some constant in andwhere, for , .

It is well known that is an Appell sequence with respect to or (which represent the same operator for monogenic functions): and in (see [38–40]).

The maximum value of in is given by (see [21])

An open ball is usually denoted by , and closed ball is denoted by , where Also, the class of special monogenic functions in an open ball is written as and denotes the class of special monogenic functions in closed ball .

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

##### 7.1. Effectiveness in Open Balls

We propose to derive in the present section, from the results of Section 5, conditions for effectiveness of basic sets in open balls. For this case, we take the Fréchet module to be the class , , of special monogenic functions in the open ball .

Let be a certain positive number less than and construct the sequence as follows: SoThe countable family of seminorms on the Fréchet module is defined as follows.

For , we setThus, when , . Soand, therefore, condition (i) of Definition 5 is satisfied.

The topology on defined by the family is the topology of normal convergence over the compact sets . It is easy to show that the -module is complete for this topology; that is to say, is a Fréchet module.

We shall take a basis for , the Appell sequence . In fact every function has the unique expansionThus (7) is true. In this case, Cauchy’s inequality (8) takes the form (see [22])

It can be verified also that is an absolute basis for in the sense that series (11), which is rewritten here asis convergent .

Finally, when , , soand then relation (9) holds.

Now, let be a basic set. Expression (18) is the unique representationand if , then by substituting (63) in (59) we obtain the basic series of :whereis the basic coefficient of .

In this case, the expression is called the Cannon sum for the set and is denoted by :The Cannon function for the same set in isIt should be observed that (63), (66), and (67) together yieldand it is easily seen that is a monotonic increasing function of .

The fundamental theorem for effectiveness for is deducible from Theorem 15. It is stated in the following form.

Theorem 23. *The necessary and sufficient condition for the basic set to be effective for is that*

*Proof. *Suppose that is any positive number less than ; then there exists a number such thatIf the set is effective for , then, by Theorem 15, there exist and a constant such that Hence, (66) gives Using (67) and (70), we can deduce thatand condition (69) follows. Thus (69) is necessary.

On the other hand, suppose that condition (69) is satisfied and let be any element of sequence (56). So we have Since the sequence converges to as tends to infinity, then there exists an integer such thatThen, by definition (67) of , there exists such that Applying (53), (57), and (66), it follows thatHence, by Theorem 15, the set is effective for , and the theorem is satisfied.

##### 7.2. Effectiveness in Closed Balls

Let be any fixed positive number and take the number to be any finite number greater than . The -module of Section 6 will be taken as the class of special monogenic functions in the closed ball , with the norm defined byThus, the topology determined by the norm is the topology of normal convergence on . It is well known that is complete for this topology; that is to say, is a Banach module. The subspace of will be taken as the Fréchet module (see Section 7.1) which will be equipped with the family of seminorms defined bywhere It is clear that So condition (40) is satisfied.

The basis for is taken, as before, to be the Appell sequence which accords to Cauchy’s inequality and it is also an absolute basis. Moreover, condition (9) is satisfied.

Now, let be a basic set of and suppose that admits the representation where the convergence is in , so that Write the basic coefficient and the basic series as in (64) and (65). We are concerned with the convergence in ; that is to say, basic series (64) converges to in if The basic set will be effective for in if the basic series of each special monogenic function in converges to normally in . In this case, we say that the basic set is effective for in , .

The theorem about such effectiveness is deducible from Theorem 18.

As in (66), we can see, by using (78), thatWe shall establish the following Theorem.

Theorem 24. *The necessary and sufficient condition for the basic set to be effective for in , , is that*

*Proof. *Suppose that is effective for in . Hence, according to Theorem 18, there exist a norm and a constant such that Hence, (67), (79), and (85) together yield and condition (86) is necessary.

On the other hand, suppose that condition (86) is satisfied. Since as , there is a number such that . Then, by definition (67) of , there exists such that Hence, from (53), (57), and, (85) it follows that By Theorem 18, it follows that the basic set is effective for in as required.

Taking sufficiently near to , Theorem 24 leads to the following corollary.

Corollary 25. *The necessary and sufficient condition for the basic set to be effective in is that*

##### 7.3. Cannon Sets

When is a basic set of special monogenic polynomials, representation (63) is finite. Thus, if is the number of nonzero coefficients in (63), then is finite. If this number accords further to the restriction thatthe corresponding basic set of polynomials is called Cannon set of special monogenic polynomials (see [21]). WriteThen, in view of (66), we shall have Hence, if we putthen, for Cannon sets, relation (67) implies that Therefore, for Cannon sets, the Cannon sum and the Cannon function and are given by (93) and (96), respectively.

For Cannon sets, the Cannon function will replace in all the concerned relations: (69), (86), and (91). Hence, concerning the effectiveness of Cannon sets of special monogenic polynomials (see [21, 23]), we have the following results which are special cases of our results.

Corollary 26. *The necessary and sufficient condition for the Cannon sets of special monogenic polynomials to be effective for is that *

Corollary 27. *The necessary and sufficient condition for the Cannon sets of special monogenic polynomials to be effective for in , , is that*

Corollary 28. *The necessary and sufficient condition for the Cannon sets of special monogenic polynomials to be effective for in is that*

#### Disclosure

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

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

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).