Basic Sets of Special Monogenic Polynomials in Fréchet Modules

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.


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][4][5][6][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][13][14][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][18][19][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][25][26][27][28][29][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 2 Journal of Complex Analysis 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 These new results extend and generalize the known results in complex and Clifford setting given in [12,21,23,31,32].

Notation and Preliminaries
In order to introduce our results, we give several notations and assumptions.Let us denote by { 1 , . . .,   } the canonical basis of the Euclidean vector space R  and by A  the associated real Clifford algebra in which one has the multiplication rules     +     = −2  , ,  = 1, . . ., , where   denotes the Kronecker symbol.
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  = ∑  =0   (/  ) in R +1 (for more details, see [33,34]).
In the following, all A  -modules will be right A modules.
Definition 3 (A  -linear operator).Let  and  be two unitary A  -modules.Then a function  :  →  is said to be an A  -linear operator if, for all ,  ∈  and ,  ∈ A  ,  ( + ) =  ()  +  () . ( The set of all A  -linear operators from  into  is denoted by (, ).
Definition 4 (proper system of seminorms).Let  be a unitary A  -module.Then a family P of functions  :  → [0, ∞) is said to be a proper system of seminorms on  if the following conditions are fulfilled: (iii)  is complete with respect to this topology.
We denote by T  the topology defined by the family P of seminorms on .
Definition 8 (Cauchy's inequality).We shall assume that Cauchy's inequality holds for the basis (  ) ≥0 in the form that for each   ∈ P there is a positive finite constant   such that for 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 (  ) ≥0 is called an absolute basis for  if the series is convergent in R for all integers  and for all  ∈ .Thus, in this case, we can write We start with the following introductory theorems.
Theorem 10.If (  ) ≥0 is a basis for  and if Cauchy's inequality ( 8) is satisfied, then   is a continuous linear operator on , orthonormal to (  ) ≥0 .
Proof.It easily follows from the uniqueness of representation (7) that if ,  ∈  and ,  ∈ A  , then so that   is a linear operator on .Also, putting  =   in (7), it can be verified that and   is orthonormal to (  ) ≥0 .We deduce the continuity of   from ( 6) and ( 8).
Theorem 11.Let (  ) ≥0 be an absolute basis for  and let   be given by (11).Then the family (  ) ≥0 forms a proper system of continuous seminorms.Moreover, for  < , there exists a constant  , such that for all  ∈ .
Proof.Firstly, we prove that the family (  ) ≥0 is a proper system of seminorms as follows.
[ 1 ] We observe, from the linearity of   and properties (i) and (ii) of seminorms, that whenever ,  ∈  and  ∈ A  .

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 (  ) ≥0 be a sequence of nonzero elements of , and suppose that (  (  )) ,≥0 is a matrix of coefficients in the Clifford algebra A  such that, for each  ≥ 0, we have the unique representation In this case, we shall call the sequence (  ) ≥0 a basic set on .
Let  be any element of  and substitute ( 18) in ( 7) to obtain the formal series where When series (20) converges in A  , Π  () exists and is called the th basic coefficient of  relative to the set (  ) ≥0 .When the basic coefficient Π  () exists for all , series ( 19) is called the basic series associated with .
The following theorem is concerned with the basic coefficients (Π  ()) ≥0 .
Theorem 12.If Π  () is defined for all  in , the map Π  :  → A  is a continuous linear operator on .
It is clear that   is a continuous linear operator on  as a finite sum of continuous linear functional   .Now, if Π  () is defined for all  in , the sequence (  ()) ≥0 converges pointwise to Π  () in .Therefore, by the Banach-Steinhaus theorem for Fréchet space [35], we deduce that Π  is equally a continuous linear operator on , and the theorem is established.

We now write
for 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 (  ()) ≥0 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 yield Let  be any element of  and write Then, in view of ( 24) and ( 25), we have and, by continuity of , we deduce that () =  and Theorem 13 is therefore established.

Effectiveness of Basic Sets
We have seen that when (  ()) ≥0 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 (  ) ≥0 is effective for .To find a necessary and sufficient condition for the effectiveness of a basic set (  ) ≥0 for the space , we consider, for each seminorm   ∈ P, the mapping   :  → R defined by Suppose that   () is finite for all  ∈ .We first show that   is a seminorm on .
Let ,  ∈  and  ∈ A  ; it follows from (27) and the linearity of Π  that The first result concerning the effectiveness of basic set (  ) ≥0 is the following theorem.Theorem 14.For (  ) ≥0 to be effective for , it is necessary and sufficient that, for each   ∈ P, the seminorm   exist and be continuous.

Proof.
Necessity.When the basic set (  ) ≥0 is effective for , basic series (19) associated with each element  ∈  converges to , and it follows that   is a continuous linear operator on .
Hence, we have If ] → ∞ with fixed , then  ], () →  −  −1 ().Hence, for some  > 0, there is  such that, for all ] >  and  ≤ , This shows that the seminorm   exists.Now, to prove the continuity of   , let (  ) ≥0 be a sequence in  which converges to an element  ∈ .By this hypothesis, if  > 0, there exists a natural number  such that if  > , then   (  − ) < .
Sufficiency.We observe from ( 22) and ( 27) that Since   is continuous on , we deduce that the sequence (  ) ≥0 is an equicontinuous sequence on .Now define the subspace  of  by  fl { ∈ , the sequence (  ()) ≥0 is a Cauchy sequence} .
It follows from the equicontinuity of (  ) ≥0 that the set  is closed.Hence, the set  ⊂  is everywhere dense and is closed, so that  =  = .Therefore (  ()) ≥0 is a Cauchy sequence on  and since  is complete, the sequence (  ()) ≥0 converges for all  ∈  and hence it converges to  in .Thus, the set (  ) ≥0 is effective for  and Theorem 14 is established.
Sufficiency.Multiplying the basic coefficient Π  () of ( 20) by   and using ( 11), (27), and (36), we obtain where   () is defined by (11).According to inequality (14), there is a seminorm   ∈ P and a positive number   such that It follows from condition (5) that   is continuous on .Then, by Theorem 14, we deduce that the set (  ) ≥0 is effective for  and the proof of Theorem 15 is therefore terminated.

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 that where P = (  ) ≥0 is the family of seminorms defined, as before, in the space .Thus, the topology induced in  by the topology T  of  determined by the norm  is coarser than the topology T  defined on  by the family P of seminorms.Let (  ) ≥0 be a basis for  and let (  ) ≥0 be a sequence of nonzero elements of .We suppose that (  (  )) ,≥0 is a matrix of A  such that, for each  ≥ 0, we have the unique representation and the convergence is in .In this case, we call the sequence (  ) ≥0 a basic set on .Let  be an element of  and substitute (41) in ( 7) to obtain the formal series where When series (43) converges in A  , we call Π  () the basic coefficient of , and when Π  () exists for each , series (42) is called the basic series of .Recall that the partial sum   :  →  is defined (see (22)) by Theorem 12 remains unchanged, while the alternative form of Theorem 13 is the following.
Proof.We prove, as before, that, for each integer ,   is a linear operator from  to .We show now that   is continuous.In fact, if (  ) ≥0 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 (  ()) ≥0 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 (  ) ≥0 is effective for  in .
The necessary and sufficient condition for effectiveness of (  ) ≥0 for  in  is obtained through the expression It 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 (  ) ≥0 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), that Since   is continuous on , then the sequence (  ) ≥0 will be equicontinuous from  to .The proof is then completed in exactly the same way as in the proof of Theorem 14.Now, if (  ) ≥0 is an absolute basis for , the effectiveness of the set (  ) ≥0 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 (  ) ≥0 is an absolute basis for .Then the basic set (  ) ≥0 will be effective for  in  if and only if there is a seminorm   ∈ P and a constant   such that   (  ) ≤     (  ) ; ∀ ≥ 0. (48) Proof.
Necessity.If (  ) ≥0 is effective for  in , then, by Theorem 17,   is a continuous norm on .Hence, by (5), there is a seminorm   ∈ P and a constant   such that Putting  =   , we obtain (48).
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 (),  > 0, of special monogenic functions in the open ball ().
Let  0 be a certain positive number less than  and construct the sequence (  ) ≥0 as follows: So The countable family P of seminorms (  ) ≥0 on the Fréchet module () is defined as follows.
Thus, when  < ,   <   .So and, therefore, condition (i) of Definition 5 is satisfied.The topology on () defined by the family (  ) ≥0 is the topology of normal convergence over the compact sets (  ),  ≥ 0. It is easy to show that the A  -module () is complete for this topology; that is to say, () is a Fréchet module.
We shall take a basis for (), the Appell sequence (P  ()) ≥0 .In fact every function  ∈ () has the unique expansion Thus ( 7) is true.In this case, Cauchy's inequality (8) takes the form (see [22]) It can be verified also that (P  ()) ≥0 is an absolute basis for () in the sense that series (11), which is rewritten here as is convergent ∀ ≥ 0. Finally, when  < ,   <   , so and then relation ( 9) holds.Now, let (  ) ≥0 = (  ()) ≥0 be a basic set.Expression (18) is the unique representation and if  ∈ (), then by substituting (63) in (59) we obtain the basic series of : where is the basic coefficient of .

( 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.

3. Basis and Absolute Basis Definition
We may also equivalently say that the sequence (  ) ≥0 converges in  to  with respect to T  .It is a familiar property for the Fréchet module  that a seminorm  on  is T  -continuous, if and only if there is a seminorm   ∈ P and a positive finite constant   such that  () ≤     () ; (∀ ∈ ) .7 (basis for Fréchet module ).Let  be a Fréchet module over A  .A sequence (  ) ≥0 of nonzero elements of  is called a basis for  if, for each element  ∈ , there is one and only one sequence (  ()) ≥0 of the Clifford algebra A  , such that

Theorem 15 .
Suppose that (  ) ≥0 is an absolute basis for .Then the basic set (  ) ≥0 will be effective for  if and only if, for any continuous seminorm   ∈ P, there is a continuous seminorm   ∈ P and a positive finite number  , such that   (  ) ≤  ,   (  ) ; ∀ ∈ N. If the basic set (  ) ≥0 is effective for , then, by Theorem 14, the application is a continuous seminorm on .Hence, by (5), there is a seminorm   ∈ P and a positive number  , such that   () ≤  ,   () ; ∀ ∈ .
In this case, the expression   (P  ()) is called the Cannon sum for the set (  ) and is denoted by   (  ):   (P  ()) = sup =   ()  ,            =   (  ) .The fundamental theorem for effectiveness for () is deducible from Theorem 15.It is stated in the following form.The necessary and sufficient condition for the basic set (  ()) ≥0 to be effective for () is that  () < , ∀ < .(69)Proof.Suppose that  is any positive number less than ; then there exists a number   such that  ≤   < .(70)If the set (  ()) ≥0 is effective for (), then, by Theorem 15, there exist   <  and a constant  , such that   (P  ()) ≤  ,   (P  ()) ; ∀ ∈ N. On the other hand, suppose that condition (69) is satisfied and let   be any element of sequence (56).So we have  (  ) < .(74)Since the sequence (  ) ≥0 converges to  as  tends to infinity, then there exists an integer  >  such that  (  ) <   < .(75)Then,bydefinition(67) of (), there exists  , such that   (  ) ≤  ,    .(76)Applying(53),(57), and (66), it follows that   (P  ()) ≤  ,   (P  ()) ; ∀ ∈ N.(77)Hence, by Theorem 15, the set (  ) ≥0 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 A  -module  of Section 6 will be taken as the class () of special monogenic functions in the closed ball (), with the norm  defined by Thus, the topology T () 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 (  ) ≥0 defined by   () =  (,   ) ;  ∈ N,  ∈  () , ] ∑