# Basic Sets of Special Monogenic Polynomials in Fréchet Modules

Authors: Gamal Farghaly Hassan; Lassaad Aloui; Allal Bakali
Journal: Journal of Complex Analysis
 (2017)

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## 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.


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## Body

## 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{e1,…,em} the canonical basis of the Euclidean vector space Rm and by Am the associated real Clifford algebra in which one has the multiplication rules eiej+ejei=-2δij,  i,j=1,…,m, where δij denotes the Kronecker symbol.A vector space basis for the Clifford algebraAm is given by the set {eA:A⊆{1,…,m}} with eA=eα1,…,αh=eα1⋯eαh,  1≤α1<α2<⋯<αh≤m, and eϕ=e0=1. Every a∈Am can be written in the form a=∑AaAeA with aA∈R. The conjugate element of a∈Am is defined by a¯=∑AaAe¯A, where e¯A=e¯αh⋯e¯α1, e¯j=-ej(j=1,…,m), and e¯0=e0=1.We denote also byRm+1=spanR{e0,e1,…,em}=R⊕Rm⊂Am the space of paravectors x=x0e0+∑j=1mxjej. In this notation, the paravector x will be represented in the form x=x0+x_ with Sc(x)=x0 being the scalar part and Vec(x)=x_ being the vector part of x. The induced Clifford norm of arbitrary a∈Am is given by a=(∑AaA2)1/2.Some care must be taken when using this norm to estimate product. We will always use the formulaab≤2m/2ab.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 operatorD=∑i=0mei(∂/∂xi) in Rm+1 (for more details, see [33, 34]).Definition 1 (unitary rightAm-module).
A unitary rightAm-module X is an abelian group (X,+) with a mapping X×Am→X; (f,λ)→fλ such that for all λ,μ∈Am and f,g∈X:(i)
f(λ+μ)=fλ+fμ.(ii)
f(λμ)=(fλ)μ.(iii)
(f+g)λ=fλ+gλ.(iv)
fe0=f.Remark 2.
Notice thatX becomes a real vector space if R is identified with Re0=A0⊂Am.
In the following, allAm-modules will be right Am-modules.Definition 3 (Am-linear operator).
LetX and Y be two unitary Am-modules. Then a function T:X→Y is said to be an Am-linear operator if, for all f,g∈X and λ,μ∈Am, (1)Tfλ+gμ=Tfλ+Tgμ. The set of all Am-linear operators from X into Y is denoted by L(X,Y).Definition 4 (proper system of seminorms).
LetX be a unitary Am-module. Then a family P of functions p:X→[0,∞) is said to be a proper system of seminorms on X if the following conditions are fulfilled:[P1]
There exists a constantC0≥1 such that, for all p∈P, λ∈Am and f,g∈X:
(i)
p(f+g)≤p(f)+p(g).(ii)
p(fλ)≤C0λp(f), and p(fλ)=λp(f) if λ∈R.[P2]
For any finite numberp1,p2,…,pk∈P, there exist p∈P and C>0 such that, for all f∈X, (2)supj=1,2,…,k⁡pjf≤Cpf.[P3]
Ifp(f)=0 for all p∈P then f=0.Definition 5 (Fréchet module).
A Fréchet moduleE over Am is a Hausdorff space with a countable proper system of seminorms P=(pi)i≥0 satisfying(i)
i<j⇒pi(f)≤pj(f);  (f∈E),(ii)
a subsetU of E is open if, ∀f∈U, there exist ϵ>0 and N≥0 such that (3)g∈E:pif-g≤ϵ⊂U,∀i≤N,(iii)
E is complete with respect to this topology.
We denote byTE the topology defined by the family P of seminorms on E.Definition 6 (convergent sequences in the topologyTE).
The sequence(fk)k≥0 of elements of E converges in the topology TE to the element g of E, if and only if, for all pi∈P, we have(4)limk→∞pifk-g=0.We may also equivalently say that the sequence (fk)k≥0 converges in E to g with respect to TE.It is a familiar property for the Fréchet moduleE that a seminorm σ on E is TE-continuous, if and only if there is a seminorm pi∈P and a positive finite constant Ci such that(5)σf≤Cipif;∀f∈E.It is also known that a linear operatorL on E is continuous if and only if there is a seminorm pi∈P and a constant Ki such that(6)Lf≤Kipif;∀f∈E.


## 3. Basis and Absolute Basis

Definition 7 (basis for Fréchet moduleE).
LetE be a Fréchet module over Am. A sequence (zn)n≥0 of nonzero elements of E is called a basis for E if, for each element f∈E, there is one and only one sequence (Zn(f))n≥0 of the Clifford algebra Am, such that(7)f=∑n=0∞znZnf.Definition 8 (Cauchy’s inequality).
We shall assume that Cauchy’s inequality holds for the basis(zn)n≥0 in the form that for each pi∈P there is a positive finite constant Mi such that(8)Znfpizn≤Mipiffor all integers n and for all f∈E.Also, the nature of the problems considered here necessitates that wheneveri<j, there is a finite positive constant Mi,j, such that(9)∑n=0∞piznpjzn=Mi,j.Definition 9 (absolute basis for Fréchet moduleE).
The basis(zn)n≥0 is called an absolute basis for E if the series (10)∑n=0∞Znfpiznis convergent in R for all integers i and for all f∈E. Thus, in this case, we can write(11)∑n=0∞Znfpizn=Qif<∞;i≥0,f∈E.We start with the following introductory theorems.Theorem 10.
If(zk)k≥0 is a basis for E and if Cauchy’s inequality (8) is satisfied, then Zn is a continuous linear operator on E, orthonormal to (zk)k≥0.Proof.
It easily follows from the uniqueness of representation (7) that if f,g∈E and λ,μ∈Am, then (12)Znfλ+gμ=Znfλ+Zngμ,so that Zn is a linear operator on E. Also, putting f=zk in (7), it can be verified that(13)Znzk=δnk,and Zn is orthonormal to (zk)k≥0.
We deduce the continuity ofZn from (6) and (8).Theorem 11.
Let(zn)n≥0 be an absolute basis for E and let Qi be given by (11). Then the family (Qi)i≥0 forms a proper system of continuous seminorms. Moreover, for i<j, there exists a constant Ni,j such that(14)Qif≤Ni,jpjffor all f∈E.Proof.
Firstly, we prove that the family(Qi)i≥0 is a proper system of seminorms as follows.
[ P 1 ] We observe, from the linearity of Zn and properties (i) and (ii) of seminorms, that (15)Qif+g=∑n=0∞Znf+gpizn=∑n=0∞Znf+Zngpizn≤∑n=0∞Znf+Zngpizn=Qif+Qig,Qifλ≤C0λQif,C0≥1,whenever f,g∈E and λ∈Am.
[P2] Let Q1,Q2,…,Qk be defined by (11). Since P is a proper system of seminorms, then there exist p∈P and C>0 such that, for all f∈X, (16)sup1≤j≤kQjf≤∑n=0∞Znfsup1≤j≤kpjzn≤C∑n=0∞Znfpzn=CQf.
[ P 3 ] Suppose that Qi(f)=0∀i; then |Zn(f)|pi(zn)=0∀i,n. Since P is a proper system of seminorms and (zk)k≥0 is a basis, for each n∈N there exists j such that pj(zn)≠0. So Zn(f)=0 for all n∈N and hence (7) implies that f=0.
Finally, wheni<j, inequality (14) can be obtained from (8), (9), and (11) as follows:(17)Qif=∑n=0∞Znfpizn=∑n=0∞Znfpjznpiznpjzn≤MjMi,jpjf≔Ni,jpjf.It follows from condition (5) that the seminorm Qi is continuous on E. Therefore the family (Qi)i≥0 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 Theorem13) that when the basic series converges, it will converge to the element with which it is associated.Let(Pk)k≥0 be a sequence of nonzero elements of E, and suppose that (πk(zn))n,k≥0 is a matrix of coefficients in the Clifford algebra Am such that, for each n≥0, we have the unique representation(18)zn=∑k=0∞Pkπkzn.In this case, we shall call the sequence (Pk)k≥0 a basic set on E.Letf be any element of E and substitute (18) in (7) to obtain the formal series(19)f~∑k=0∞PkΠkf,where(20)Πkf=∑n=0∞πkznZnf.When series (20) converges in Am, Πk(f) exists and is called the kth basic coefficient of f relative to the set (Pk)k≥0. When the basic coefficient Πk(f) exists for all k, series (19) is called the basic series associated with f.The following theorem is concerned with the basic coefficients(Πk(f))k≥0.Theorem 12.
IfΠk(f) is defined for all f in E, the map Πk:E→Am is a continuous linear operator on E.Proof.
Let(21)TNf=∑n=0NπkznZnf. It is clear that TN is a continuous linear operator on E as a finite sum of continuous linear functional Zn.
Now, ifΠk(f) is defined for all f in E, the sequence (TN(f))N≥0 converges pointwise to Πk(f) in E. Therefore, by the Banach-Steinhaus theorem for Fréchet space [35], we deduce that Πk is equally a continuous linear operator on E, and the theorem is established.We now write(22)Bnf=∑k=0nPkΠkffor the nth partial sum of basic series (19). The following theorem establishes the required conformity of the limit of Bn(f) with the space E.Theorem 13.
If, for everyf∈E, Bn(f) is defined for all n and if the sequence (Bn(f))n≥0 converges in E to some limit B(f), then B(f)=f for all elements f∈E.Proof.
We prove thatB is a continuous linear operator on E as a limit of finite sum of continuous linear operators as in Theorem 12.
Now, it can be proven, from (13) and (20), that (23)Πkzn=πkzn;n,k≥0. Hence, (18) and (22) together yield(24)Bzn=∑k=o∞Pkπkzn=zn;n≥0.Let f be any element of E and write(25)gk=∑j=okzjZjf.Then, in view of (24) and (25), we have(26)Bgk=gk,and, by continuity of B, we deduce that B(f)=f and Theorem 13 is therefore established.


## 5. Effectiveness of Basic Sets

We have seen that when(Bn(f))n≥0 converges for each f of E, Bn(f) converges to f. This means that basic series (19) associated with the element f converges to f itself for all f∈E. In this case, we say that the set (Pk)k≥0 is effective for E. To find a necessary and sufficient condition for the effectiveness of a basic set (Pk)k≥0 for the space E, we consider, for each seminorm pi∈P, the mapping qi:E→R defined by(27)qif=supμ,νpi∑k=μνPkΠkf,f∈E.Suppose that qi(f) is finite for all f∈E. We first show that qi is a seminorm on E.Letf,g∈E and λ∈Am; it follows from (27) and the linearity of Πk that (28)qif+g≤qif+qig,qifλ≤C0λqif.The first result concerning the effectiveness of basic set(Pk)k≥0 is the following theorem.Theorem 14.
For(Pk)k≥0 to be effective for E, it is necessary and sufficient that, for each pi∈P, the seminorm qi exist and be continuous.Proof.
Necessity. When the basic set (Pk)k≥0 is effective for E, basic series (19) associated with each element f∈E converges to f, and it follows that Bn is a continuous linear operator on E.
Therefore, if we write(29)Cν,μf=Bνf-Bμ-1f;ν≥μ>0,then from (27) we shall have (30)qif=supμ,νpiCν,μf.If μ→∞, then Cν,μ(f)→0. Then for every ϵ>0 there exists a natural number m, such that if μ≥m, then pi(Cν,μ(f))<ϵ. Hence, we have (31)qif≤sup⁡supμ≤m,ν≥μpiCν,μf,ϵ.If ν→∞ with fixed μ, then Cν,μ(f)→f-Bμ-1(f). Hence, for some ϵ>0, there is N such that, for all ν>N and μ≤m, (32)piCν,μf-f-Bμ-1f<ϵ,qif≤sup⁡supν≤N,μ≤mpiCν,μf,supμ≤mpif-Bμ-1f+ϵ,ϵ.This shows that the seminorm qi exists.
Now, to prove the continuity ofqi, let (fn)n≥0 be a sequence in E which converges to an element f∈E. By this hypothesis, if ϵ>0, there exists a natural number M such that if n>M, then pi(fn-f)<ϵ.
Hence, ifK(ϵ)=sup⁡{M,N,m}, it follows that if n≥K(ϵ), then (33)qifn-f≤sup⁡supν≤N,μ≤mpiCν,μfn-f,pifn-f-Bμ-1fn-f+ϵ,ϵ≤sup⁡ϵ,3ϵ,ϵ=3ϵ.Thus limn→∞qi(fn)=qi(f) and qi is a continuous seminorm on E.
Sufficiency. We observe from (22) and (27) that (34)piBnf≤qif;n≥0.Since qi is continuous on E, we deduce that the sequence (Bn)n≥0 is an equicontinuous sequence on E.
Now define the subspaceY of E by (35)Y≔g∈Y,the  sequenceBngn≥0is  a  Cauchy  sequence.It follows from the equicontinuity of (Bn)n≥0 that the set Y is closed. Hence, the set Y⊂E is everywhere dense and is closed, so that Y=Y¯=E. Therefore (Bn(f))n≥0 is a Cauchy sequence on E and since E is complete, the sequence (Bn(f))n≥0 converges for all f∈E and hence it converges to f in E. Thus, the set (Pk)k≥0 is effective for E and Theorem 14 is established.Theorem 15.
Suppose that(zn)n≥0 is an absolute basis for E. Then the basic set (Pk)k≥0 will be effective for E if and only if, for any continuous seminorm pi∈P, there is a continuous seminorm pj∈P and a positive finite number Ki,j such that(36)qizn≤Ki,jpjzn;∀n∈N.Proof.
Necessity. If the basic set (Pk)k≥0 is effective for E, then, by Theorem 14, the application qi is a continuous seminorm on E. Hence, by (5), there is a seminorm pj∈P and a positive number Ki,j such that (37)qif≤Ki,jpjf;∀f∈E.Putting f=zn,  n≥0, we obtain (36).
Sufficiency. Multiplying the basic coefficient Πk(f) of (20) by Pk and using (11), (27), and (36), we obtain (38)pi∑k=μνPkΠkf≤C0∑n=0∞pi∑k=μνPkΠkznZnf≤C0∑n=0∞qiznZnf≤C0Kij∑n=0∞pjznZnf=C0KijQjf,where Qj(f) is defined by (11). According to inequality (14), there is a seminorm pl∈P and a positive number Mjl such that (39)qif=supμ,νpi∑k=μνPkΠkf≤C0KijQjf≤C0KijMjlplf,∀f∈E.It follows from condition (5) that qi is continuous on E. Then, by Theorem 14, we deduce that the set (Pk)k≥0 is effective for E and the proof of Theorem 15 is therefore terminated.


## 6. Alternative Treatment of the Problem

In this treatment, we consider the Fréchet moduleE as a subspace of a Banach module F with a continuous norm σ such that(40)σf≤pif,∀i∈N,∀f∈E,where P=(pi)i≥0 is the family of seminorms defined, as before, in the space E. Thus, the topology induced in E by the topology TF of F determined by the norm σ is coarser than the topology TE defined on E by the family P of seminorms.Let(zk)k≥0 be a basis for E and let (Pk)k≥0 be a sequence of nonzero elements of F. We suppose that (πk(zn))n,k≥0 is a matrix of Am such that, for each n≥0, we have the unique representation(41)zn=∑k=0∞Pkπkzn,and the convergence is in F. In this case, we call the sequence (Pk)k≥0 a basic set on F. Let f be an element of E and substitute (41) in (7) to obtain the formal series(42)f~∑k=0∞PkΠkf,where(43)Πkf=∑n=0∞πkznZnf.When series (43) converges in Am, we call Πk(f) the basic coefficient of f, and when Πk(f) exists for each k, series (42) is called the basic series of f. Recall that the partial sum Bn:E→F is defined (see (22)) by(44)Bnf=∑k=0nPkΠkf.Theorem 12 remains unchanged, while the alternative form of Theorem 13 is the following.Theorem 16.
If, for everyf∈E, Bn(f) is defined for all n and if the sequence (Bn(f))n≥0 converges in F to some element B(f), then B(f)=f for all elements f∈E.Proof.
We prove, as before, that, for each integern, Bn is a linear operator from E to F. We show now that Bn is continuous. In fact, if (fm)m≥0 is a sequence of elements of E converging to an element g of E, it can be deduced from (40) and (44) and Theorem 12 that (45)limm→∞⁡σBnfm-Bng=0.Hence Bn(fm) converges to Bn(g) in F and hence Bn:E→F is a continuous linear operator. Proceeding exactly as in the proof of Theorem 13, we can deduce that B:E→F is a continuous linear operator.
Now, setgk=∑j=okzjZj(f). It is clear that gk→f in E and B(gk)=gk; hence B(f)=f, as required.We see that when(Bn(f))n≥0 converges in F, for every element f∈E, Bn(f) converges in F to f. This means that basic series (42) associated with the element f converges in F to the element f, for all f∈E. In this case, we say that the basic set (Pk)k≥0 is effective for E in F.The necessary and sufficient condition for effectiveness of(Pk)k≥0 for E in F is obtained through the expression(46)qσf=supμ,ν⁡σ∑k=μνPkΠkf;f∈E.It can be proven in exactly the same way as before that qσ is a seminorm on E. The revised version of Theorem 14 is as follows.Theorem 17.
For the basic set(Pk)k≥0 to be effective for E in F, it is necessary and sufficient that qσ exist and be continuous on E.Proof.
Necessity. Since Bn is continuous linear operator from E to F, we apply the same method as in the proof of Theorem 14.
Sufficiency. We see here, from (22) and (46), that(47)σBnf≤qσf;∀f∈E.Since qσ is continuous on E, then the sequence (Bn)n≥0 will be equicontinuous from E to F. The proof is then completed in exactly the same way as in the proof of Theorem 14.Now, if(zn)n≥0 is an absolute basis for E, the effectiveness of the set (Pk)k≥0 for E in F will be estimated through the expression qσ(zn) as it is seen from the following Theorem which is the alternative form of Theorem 15.Theorem 18.
Suppose that(zn)n≥0 is an absolute basis for E. Then the basic set (Pk)k≥0 will be effective for E in F if and only if there is a seminorm pi∈P and a constant Ki such that(48)qσzn≤Kipizn;∀n≥0.Proof.
Necessity. If (Pk)k≥0 is effective for E in F, then, by Theorem 17, qσ is a continuous norm on E. Hence, by (5), there is a seminorm pi∈P and a constant Ki such that (49)qσf≤Kipif;∀f∈E.Putting f=zn, we obtain (48).
Sufficiency. As in the proof of Theorem 15, we deduce from (11), (43), (46), and (48) that (50)σ∑k=μνPkΠkf≤C0∑n=0∞σ∑k=μνPkΠkznZnf≤C0∑n=0∞qσznZnf≤C0Ki∑n=0∞piznZnf=C0KiQif.Applying inequality (14), we obtain (51)qσf≤C0KiMi,jpjf=C0Li,jpjf,∀f∈E.So we deduce that qσ is continuous on E and hence Theorem 17 implies that (Pk)k≥0 is effective for E in F. 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).
LetΩ⊂Rm+1 be an open set; then an Am-valued function f is called left (resp. right) monogenic in Ω if it satisfies Df=0 (resp. fD=0) in Ω. Here, D=∑i=0mei(∂/∂xi), defined in Section 2, is the generalized Cauchy-Riemann operator.Definition 20 (special monogenic polynomial).
A polynomialP(x) is special monogenic if and only if DP(x)=0 (so P(x) is monogenic) and there exists ai,j∈Am for which P(x)=∑i,jfinitex¯ixjai,j.Definition 21 (special monogenic function).
LetΩ be a connected open subset of Rm+1 containing 0 and let f be monogenic in Ω. Then f is called special monogenic in Ω if and only if its Taylor series near zero (which exists) has the form f(x)=∑n=0∞Pn(x) for certain special monogenic polynomials Pn(x).The fundamental references for special monogenic function are [36, 37].Remark 22.
Note that ifPn(x) is a homogeneous special monogenic polynomial of degree n, then (see [21, 29]) Pn(x)=Pn(x)α: α is some constant in Am and(52)Pnx=n!mn∑k+l=nm-1/2kk!m+1/2ll!x¯kxl,x∈Rm+1,where, for b∈R, (b)l=b(b+1)⋯(b+l-1).
It is well known that(Pn(x))n≥0 is an Appell sequence with respect to ∂/∂x0 or (1/2)D¯ (which represent the same operator for monogenic functions): (1/2)D¯Pn(x)=nPn-1(x) and D¯=∑i=0me¯i(∂/∂xi) in Rm+1 (see [38–40]).
The maximum value ofPnx in x=R is given by (see [21])(53)PnxR≔supx=RPnx=Rn.An open ball is usually denoted byB(r), and closed ball is denoted by B¯(r), where (54)Br=x∈Rm+1,x<r,B¯r=x∈Rm+1,x≤r.Also, the class of special monogenic functions in an open ball B(r) is written as H(r) and H¯(r) denotes the class of special monogenic functions in closed ball B¯(r).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 Section5, conditions for effectiveness of basic sets in open balls. For this case, we take the Fréchet module E to be the class H(r), r>0, of special monogenic functions in the open ball B(r).Letρ0 be a certain positive number less than r and construct the sequence (ρn)n≥0 as follows: (55)ρ1=r+ρ02,ρ2=r+ρ12,⋮ρn+1=r+ρn2,⋮So(56)ρ0<ρ1<ρ2<⋯<ρn<⋯<r.The countable family P of seminorms (pi)i≥0 on the Fréchet module H(r) is defined as follows.Forf∈H(r), we set(57)pif=supB¯ρifx=Mf,ρi.Thus, when i<j,  ρi<ρj. So(58)pif≤pjf;∀f∈Hr,and, therefore, condition (i) of Definition 5 is satisfied.The topology onH(r) defined by the family (pi)i≥0 is the topology of normal convergence over the compact sets B¯(ρi),i≥0. It is easy to show that the Am-module H(r) is complete for this topology; that is to say, H(r) is a Fréchet module.We shall take a basis forH(r), the Appell sequence (Pn(x))n≥0. In fact every function f∈H(r) has the unique expansion(59)fx=∑n=0∞Pnxanf.Thus (7) is true. In this case, Cauchy’s inequality (8) takes the form (see [22])(60)anfρin≤supB¯ρifx=fρi.It can be verified also that(Pn(x))n≥0 is an absolute basis for H(r) in the sense that series (11), which is rewritten here as(61)∑n=0∞anfρin,is convergent ∀i≥0.Finally, wheni<j,  ρi<ρj, so(62)∑n=0∞piPnxpjPnx=∑n=0∞ρinρjn<∞and then relation (9) holds.Now, let(Pk)k≥0=(Pk(x))k≥0 be a basic set. Expression (18) is the unique representation(63)Pnx=∑k=0∞Pkxπn,kand if f∈H(r), then by substituting (63) in (59) we obtain the basic series of f:(64)fx~∑k=0∞PkxΠkf,where(65)Πkf=∑n=0∞πn,kanf;k≥0is the basic coefficient of f.In this case, the expressionqi(Pn(x)) is called the Cannon sum for the set (Pk) and is denoted by Fn(ρi):(66)qiPnx=supμ,νsupB¯ρi∑k=μνPkxπn,k=Fnρi.The Cannon function for the same set in B¯(ρ) is(67)Kρ=limsupn→∞Fnρ1/n.It should be observed that (63), (66), and (67) together yield(68)Kρ≥ρand it is easily seen that K(ρ) is a monotonic increasing function of ρ.The fundamental theorem for effectiveness forH(r) is deducible from Theorem 15. It is stated in the following form.Theorem 23.
The necessary and sufficient condition for the basic set(Pk(x))k≥0 to be effective for H(r) is that(69)Kρ<r,∀ρ<r.Proof.
Suppose thatρ is any positive number less than r; then there exists a number ρi such that(70)ρ≤ρi<r.If the set (Pk(x))k≥0 is effective for H(r), then, by Theorem 15, there exist ρj<r and a constant Ki,j such that (71)qiPnx≤Ki,jpjPnx;∀n∈N.Hence, (66) gives (72)Fnρi≤Ki,jρjn.Using (67) and (70), we can deduce that(73)Kρ≤Kρi≤ρj<rand condition (69) follows. Thus (69) is necessary.
On the other hand, suppose that condition (69) is satisfied and let ρi be any element of sequence (56). So we have (74)Kρi<r.Since the sequence (ρi)i≥0 converges to r as i tends to infinity, then there exists an integer j>i such that(75)Kρi<ρj<r.Then, by definition (67) of K(ρ), there exists Ci,j such that (76)Fnρi≤Ci,jρjn.Applying (53), (57), and (66), it follows that(77)qiPnx≤Ci,jpjPnx;∀n∈N.Hence, by Theorem 15, the set (Pk)k≥0 is effective for H(r), and the theorem is satisfied.


### 7.2. Effectiveness in Closed Balls

LetR be any fixed positive number and take the number r to be any finite number greater than R. The Am-module F of Section 6 will be taken as the class H¯(R) of special monogenic functions in the closed ball B¯(R), with the norm σ defined by(78)σf=supB¯Rfx=Mf;R,∀f∈H¯R.Thus, the topology TH¯(R) determined by the norm σ is the topology of normal convergence on B¯(R). It is well known that H¯(R) is complete for this topology; that is to say, H¯(R) is a Banach module. The subspace E of H¯(R) will be taken as the Fréchet module H(r) (see Section 7.1) which will be equipped with the family of seminorms (pi)k≥0 defined by(79)pif=Mf,ρi;i∈N,f∈Hr,where (80)ρ0=R,ρ1=r+ρ02,ρ2=r+ρ12,⋮ρn+1=r+ρn2,⋮It is clear that (81)σf≤pif;i∈N,f∈Hr.So condition (40) is satisfied.The basis forH(r) is taken, as before, to be the Appell sequence (Pn(x))n≥0 which accords to Cauchy’s inequality and it is also an absolute basis. Moreover, condition (9) is satisfied.Now, let(Pk(x))k≥0 be a basic set of H¯(R) and suppose that Pn(x) admits the representation (82)Pnx=∑k=0∞Pkxπn,k,where the convergence is in H¯(R), so that (83)maxB¯RPnx-∑k=0mPkxπn,k⟶0,asm⟶∞.Write the basic coefficient and the basic series as in (64) and (65). We are concerned with the convergence in H¯(R); that is to say, basic series (64) converges to f in H¯(R) if (84)supB¯Rfx-∑k=0mPkxΠkf⟶0,asm⟶∞.The basic set (Pk)k≥0 will be effective for H(r) in H¯(R) if the basic series of each special monogenic function in B(r) converges to f normally in B¯(R). In this case, we say that the basic set (Pk)k≥0 is effective for H(r) in B¯(R), R<r.The theorem about such effectiveness is deducible from Theorem18.As in (66), we can see, by using (78), that(85)qσPnx=FnR.We shall establish the following Theorem.Theorem 24.
The necessary and sufficient condition for the basic set(Pk(x))k≥0 to be effective for H(r) in B¯(R), r>R>0, is that(86)KR<r.Proof.
Suppose that(Pk)k≥0 is effective for H(r) in B¯(R). Hence, according to Theorem 18, there exist a norm pi∈P and a constant Ki such that (87)qσPnx≤KipiPnx,n≥0.Hence, (67), (79), and (85) together yield (88)KR≤ρi<rand condition (86) is necessary.
On the other hand, suppose that condition (86) is satisfied. Since ρn→r as n→∞, there is a number ρi such that K(R)<ρi<r. Then, by definition (67) of K(ρ), there exists Ki such that (89)FnR≤Kiρin,n≥0.
Hence, from (53), (57), and, (85) it follows that (90)qσPnx≤KipiPnx,n≥0.By Theorem 18, it follows that the basic set (Pk(x))k≥0 is effective for H(r) in B¯(R) as required.Takingr sufficiently near to R, Theorem 24 leads to the following corollary.Corollary 25.
The necessary and sufficient condition for the basic set(Pk(x))k≥0 to be effective in B¯(R) is that(91)KR=R.


### 7.3. Cannon Sets

When(Pk)k≥0 is a basic set of special monogenic polynomials, representation (63) is finite. Thus, if Nn is the number of nonzero coefficients in (63), then Nn is finite. If this number accords further to the restriction that(92)Nn1/n⟶1,asn⟶∞,the corresponding basic set (Pk(x))k≥0 of polynomials is called Cannon set of special monogenic polynomials (see [21]). Write(93)ωnρ=∑kPkxπn,kρ≔∑ksupB¯ρPkxπn,k;ρ>0,(94)Psnxπn,snρ=maxkPkxπn,kρ.Then, in view of (66), we shall have (95)Psnxπn,snρ≤Fnρ≤ωnρ≤NnPsnxπn,snρ.Hence, if we put(96)λρ=limsupn→∞ωnρ1/n,then, for Cannon sets, relation (67) implies that(97)λρ=Kρ. Therefore, for Cannon sets, the Cannon sum and the Cannon function ωn(ρ) and λ(ρ) are given by (93) and (96), respectively.For Cannon sets, the Cannon functionλ(ρ) will replace K(ρ) 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(Pk(x))k≥0 to be effective for H(r) is that (98)λρ<r,∀ρ<r.Corollary 27.
The necessary and sufficient condition for the Cannon sets of special monogenic polynomials(Pk(x))k≥0 to be effective for H(r) in B¯(R), r>R>0, is that(99)λR<r.Corollary 28.
The necessary and sufficient condition for the Cannon sets of special monogenic polynomials(Pk(x))k≥0 to be effective for in B¯(R) is that(100)λR=R.


## 7.1. Effectiveness in Open Balls

We propose to derive in the present section, from the results of Section5, conditions for effectiveness of basic sets in open balls. For this case, we take the Fréchet module E to be the class H(r), r>0, of special monogenic functions in the open ball B(r).Letρ0 be a certain positive number less than r and construct the sequence (ρn)n≥0 as follows: (55)ρ1=r+ρ02,ρ2=r+ρ12,⋮ρn+1=r+ρn2,⋮So(56)ρ0<ρ1<ρ2<⋯<ρn<⋯<r.The countable family P of seminorms (pi)i≥0 on the Fréchet module H(r) is defined as follows.Forf∈H(r), we set(57)pif=supB¯ρifx=Mf,ρi.Thus, when i<j,  ρi<ρj. So(58)pif≤pjf;∀f∈Hr,and, therefore, condition (i) of Definition 5 is satisfied.The topology onH(r) defined by the family (pi)i≥0 is the topology of normal convergence over the compact sets B¯(ρi),i≥0. It is easy to show that the Am-module H(r) is complete for this topology; that is to say, H(r) is a Fréchet module.We shall take a basis forH(r), the Appell sequence (Pn(x))n≥0. In fact every function f∈H(r) has the unique expansion(59)fx=∑n=0∞Pnxanf.Thus (7) is true. In this case, Cauchy’s inequality (8) takes the form (see [22])(60)anfρin≤supB¯ρifx=fρi.It can be verified also that(Pn(x))n≥0 is an absolute basis for H(r) in the sense that series (11), which is rewritten here as(61)∑n=0∞anfρin,is convergent ∀i≥0.Finally, wheni<j,  ρi<ρj, so(62)∑n=0∞piPnxpjPnx=∑n=0∞ρinρjn<∞and then relation (9) holds.Now, let(Pk)k≥0=(Pk(x))k≥0 be a basic set. Expression (18) is the unique representation(63)Pnx=∑k=0∞Pkxπn,kand if f∈H(r), then by substituting (63) in (59) we obtain the basic series of f:(64)fx~∑k=0∞PkxΠkf,where(65)Πkf=∑n=0∞πn,kanf;k≥0is the basic coefficient of f.In this case, the expressionqi(Pn(x)) is called the Cannon sum for the set (Pk) and is denoted by Fn(ρi):(66)qiPnx=supμ,νsupB¯ρi∑k=μνPkxπn,k=Fnρi.The Cannon function for the same set in B¯(ρ) is(67)Kρ=limsupn→∞Fnρ1/n.It should be observed that (63), (66), and (67) together yield(68)Kρ≥ρand it is easily seen that K(ρ) is a monotonic increasing function of ρ.The fundamental theorem for effectiveness forH(r) is deducible from Theorem 15. It is stated in the following form.Theorem 23.
The necessary and sufficient condition for the basic set(Pk(x))k≥0 to be effective for H(r) is that(69)Kρ<r,∀ρ<r.Proof.
Suppose thatρ is any positive number less than r; then there exists a number ρi such that(70)ρ≤ρi<r.If the set (Pk(x))k≥0 is effective for H(r), then, by Theorem 15, there exist ρj<r and a constant Ki,j such that (71)qiPnx≤Ki,jpjPnx;∀n∈N.Hence, (66) gives (72)Fnρi≤Ki,jρjn.Using (67) and (70), we can deduce that(73)Kρ≤Kρi≤ρj<rand condition (69) follows. Thus (69) is necessary.
On the other hand, suppose that condition (69) is satisfied and let ρi be any element of sequence (56). So we have (74)Kρi<r.Since the sequence (ρi)i≥0 converges to r as i tends to infinity, then there exists an integer j>i such that(75)Kρi<ρj<r.Then, by definition (67) of K(ρ), there exists Ci,j such that (76)Fnρi≤Ci,jρjn.Applying (53), (57), and (66), it follows that(77)qiPnx≤Ci,jpjPnx;∀n∈N.Hence, by Theorem 15, the set (Pk)k≥0 is effective for H(r), and the theorem is satisfied.


## 7.2. Effectiveness in Closed Balls

LetR be any fixed positive number and take the number r to be any finite number greater than R. The Am-module F of Section 6 will be taken as the class H¯(R) of special monogenic functions in the closed ball B¯(R), with the norm σ defined by(78)σf=supB¯Rfx=Mf;R,∀f∈H¯R.Thus, the topology TH¯(R) determined by the norm σ is the topology of normal convergence on B¯(R). It is well known that H¯(R) is complete for this topology; that is to say, H¯(R) is a Banach module. The subspace E of H¯(R) will be taken as the Fréchet module H(r) (see Section 7.1) which will be equipped with the family of seminorms (pi)k≥0 defined by(79)pif=Mf,ρi;i∈N,f∈Hr,where (80)ρ0=R,ρ1=r+ρ02,ρ2=r+ρ12,⋮ρn+1=r+ρn2,⋮It is clear that (81)σf≤pif;i∈N,f∈Hr.So condition (40) is satisfied.The basis forH(r) is taken, as before, to be the Appell sequence (Pn(x))n≥0 which accords to Cauchy’s inequality and it is also an absolute basis. Moreover, condition (9) is satisfied.Now, let(Pk(x))k≥0 be a basic set of H¯(R) and suppose that Pn(x) admits the representation (82)Pnx=∑k=0∞Pkxπn,k,where the convergence is in H¯(R), so that (83)maxB¯RPnx-∑k=0mPkxπn,k⟶0,asm⟶∞.Write the basic coefficient and the basic series as in (64) and (65). We are concerned with the convergence in H¯(R); that is to say, basic series (64) converges to f in H¯(R) if (84)supB¯Rfx-∑k=0mPkxΠkf⟶0,asm⟶∞.The basic set (Pk)k≥0 will be effective for H(r) in H¯(R) if the basic series of each special monogenic function in B(r) converges to f normally in B¯(R). In this case, we say that the basic set (Pk)k≥0 is effective for H(r) in B¯(R), R<r.The theorem about such effectiveness is deducible from Theorem18.As in (66), we can see, by using (78), that(85)qσPnx=FnR.We shall establish the following Theorem.Theorem 24.
The necessary and sufficient condition for the basic set(Pk(x))k≥0 to be effective for H(r) in B¯(R), r>R>0, is that(86)KR<r.Proof.
Suppose that(Pk)k≥0 is effective for H(r) in B¯(R). Hence, according to Theorem 18, there exist a norm pi∈P and a constant Ki such that (87)qσPnx≤KipiPnx,n≥0.Hence, (67), (79), and (85) together yield (88)KR≤ρi<rand condition (86) is necessary.
On the other hand, suppose that condition (86) is satisfied. Since ρn→r as n→∞, there is a number ρi such that K(R)<ρi<r. Then, by definition (67) of K(ρ), there exists Ki such that (89)FnR≤Kiρin,n≥0.
Hence, from (53), (57), and, (85) it follows that (90)qσPnx≤KipiPnx,n≥0.By Theorem 18, it follows that the basic set (Pk(x))k≥0 is effective for H(r) in B¯(R) as required.Takingr sufficiently near to R, Theorem 24 leads to the following corollary.Corollary 25.
The necessary and sufficient condition for the basic set(Pk(x))k≥0 to be effective in B¯(R) is that(91)KR=R.


## 7.3. Cannon Sets

When(Pk)k≥0 is a basic set of special monogenic polynomials, representation (63) is finite. Thus, if Nn is the number of nonzero coefficients in (63), then Nn is finite. If this number accords further to the restriction that(92)Nn1/n⟶1,asn⟶∞,the corresponding basic set (Pk(x))k≥0 of polynomials is called Cannon set of special monogenic polynomials (see [21]). Write(93)ωnρ=∑kPkxπn,kρ≔∑ksupB¯ρPkxπn,k;ρ>0,(94)Psnxπn,snρ=maxkPkxπn,kρ.Then, in view of (66), we shall have (95)Psnxπn,snρ≤Fnρ≤ωnρ≤NnPsnxπn,snρ.Hence, if we put(96)λρ=limsupn→∞ωnρ1/n,then, for Cannon sets, relation (67) implies that(97)λρ=Kρ. Therefore, for Cannon sets, the Cannon sum and the Cannon function ωn(ρ) and λ(ρ) are given by (93) and (96), respectively.For Cannon sets, the Cannon functionλ(ρ) will replace K(ρ) 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(Pk(x))k≥0 to be effective for H(r) is that (98)λρ<r,∀ρ<r.Corollary 27.
The necessary and sufficient condition for the Cannon sets of special monogenic polynomials(Pk(x))k≥0 to be effective for H(r) in B¯(R), r>R>0, is that(99)λR<r.Corollary 28.
The necessary and sufficient condition for the Cannon sets of special monogenic polynomials(Pk(x))k≥0 to be effective for in B¯(R) is that(100)λR=R.



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