EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6793
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Exponential Bases of Cliffordian Polynomials in Fréchet
Modules

Mohra Zayed1,∗

1 Mathematics Department, College of Science, King Khalid University, Abha 61413,
Saudi Arabia

Abstract. This paper presents a generalization of the exponential base of special monogenic poly-
nomials within the framework of Fréchet modules (F-modules). The study focuses on examining
the convergence properties, specifically the effectiveness, of both the exponential simple base of
special monogenic polynomials (ESBSMPs) and the exponential Cannon base of special monogenic
polynomials (ECBSMPs) in Fréchet modules. These properties are investigated on hyper-closed
and open balls, in open regions surrounding hyper-closed balls, for all entire special monogenic
functions, as well as at the origin. Furthermore, an explicit upper bound for the order of the
exponential simple base is established and shown to be attainable. Finally, we extend the discus-
sion to equivalent and similar bases, verifying that the derived results remain valid under such
transformations, which confirms the robustness and general applicability of the findings.

2020 Mathematics Subject Classifications: 30G35, 30D15, 41A10

Key Words and Phrases: Clifford analysis, special monogenic polynomials, Fréchet modules,
bases of polynomials, growth of bases, effectiveness

1. Introduction

The development of bases theories in functional spaces has gained increasing impor-
tance in diverse mathematical areas including approximation theory, partial differential
equations, and mathematical physics. Approximation theory has a crucial role provid-
ing tools for analyzing and solving problems arising in applied sciences and engineering.
Recent developments demonstrate the diversity of its applications. The authors of [1]
addressed approximate numerical solutions of time-fractional partial differential equations
in three dimensions, achieving high accuracy even on irregular domains. In [2], the au-
thors proposed a derivative-free iterative method with optimal fourth-order.convergence
for finding multiple roots.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6793

Email addresses: mzayed@kku.edu.sa (M. Zayed)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 2 of 19

Clifford analysis is considered as an elegant higher dimensional analogy to complex
analysis. It allow an extension of the theory of holomorphic functions to higher dimen-
sional setting by using Clifford algebra valued functions which are the solutions the gener-
alized Cauchy–Riemann system (Dirac operator ). These functions are known as Clifford
holomorphic functions (monogenic functions).

In the context of single complex variable, the topic of basic sets was introduced by
Whittaker [3, 4] and Cannon [5, 6]. In [7, 8], the authors proposed an extended adap-
tation of Whittaker-Cannon theory of bases of polynomial in the complex setting to the
framework of Clifford algebra. They provided the effectiveness criteria in the convergence
domain which means the action of approximating special monogenic functions (SMFs) via
bases of special monogenic polynomials (SMPs) with or without some restrictions. Their
study was concerned with the approximation of subclass of MFs which are generated by
a particular special polynomials in axially symmetric domains. Subsequent papers pro-
posed by several authors which expose the significant of this approach. The construction
and effectiveness of certain derived bases of SMPs were studied in [9–11]. The conver-
gence properties for the Bernoulli polynomials, Euler polynomials, Bessel polynomials,
and Chebyshev in Clifford analysis [12–15].

Constructing bases of SMPs by the utilization the fundamentals of the functional anal-
ysis was proposed in [16] where the authors characterized the convergence of certain classes
of bases for various F-modules. The effectiveness and the growth of the equivalent base
constructed with SMPs in F-modules were studied in [17]. Recently, the authors in [18]
deduced the extended Ruscheweyh differential operator and examined the representation
of its derived bases of SMPs in different convergence regions. Recently in [19], the authors
investigated the representation of a SMFs in terms of infinite series of Cliffordian Hasse
derivative bases in hyper regions was introduced.

The representation of regular functions of several complex variables by means of expo-
nential base of polynomials in hyperelliptical regions were discussed in [20]. In the Clifford
context, the exponential function exp(x), x a Clifford variable was introduced in [21] as
an extension of the classical complex function ez. In [21], the author discussed the conver-
gence properties of the exponential base of SMPs with the bases associated with the base
of polynomials

{Qn(x)} = {
∑
k

qk(x)Qn,k},

where the Qn,k are real Clifford coefficients. Precisely, the restriction Qn,n = 1 for n ∈ N
was imposed on the diagonal of the matrix of entries of the matrix Q = (Qn,k). In the
current study, we relinquish the aforementioned condition and consider a more generalized
form for these diagonal elements to be Qn,n = αn for all n ∈ N, where {αn} is a bounded
sequence of the positive numbers. We assign certain conditions to the coefficients of the
power associated infinite matrices of an original base. Consequently, the representation
of a SMF is provided in terms of ESBSMPs. We show that exponential base satisfies
the higher-dimensional effectiveness criteria for the F-module WB̄(R). Furthermore, we
find that the order of the ESBSMPs is bounded above by an attainable upper bound.
Moreover, we investigate the convergence properties of the ECBSMPs for the F-modules



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 3 of 19

WB̄(R), WB(R), WB+(R), W∞, W0+ which will be defined in Section 2.

2. Notations and basic results

The associated 2m-dimensional real algebra Am constructed from the space Rm with an
orthogonal base {e1, e2, . . . , em} is identified such that Rm ⊂ Am. Suppose that {ej}mj=1 is
an orthonormal base of Rm. Then the non-commutative multiplication in Rm is subjected
to ekeℓ + eℓek = −2δkℓ, where k, ℓ = 1, . . . ,m and δkℓ stands for the Kronecker symbol
(for details on the main concepts of Am, see [22]). The set {eS : S ⊂ {0, 1, . . . ,m}} where
S = {s1, s2, . . . sh} , 0 ≤ s1 < s2 < · · · < sh ≤ m, eS = es1es2 . . . esh , eϕ, e0 = 1, creates a
base of R0,m. Embedding Rm+1 in R0,m. it is identified that x = (x0, x1, . . . , xm) ∈ Rm+1

corresponds to x =
∑m

i=0 xiei. Conjugation in R0,m is defined as the anti-involution for

which ek = −ek (1 ≤ k ≤ m). The norm of an element u ∈ Am is defined by
(∑

A |uA|2
) 1

2

and we have always use the formula |uv| ≤ 2
m
2 |u| |v| where the products uu or vv produce

real numbers.
Clifford analysis provide a function theory that is a higher dimensional analog of func-

tions theory of single complex variable (see e.g. [22, 23]). In this context, a monogenic
functions is a null solution of the Dirac operator D =

∑m
i=0 ei

∂
∂xi
, in Rm+1 (see [24]).

Consider a regular function φ having values in Am defined in some open subset M
of Rm+1. If Dφ = 0 or φD = 0, then φ is called left-monogenic or right-monogenic
respectively.

Definition 1. Let Q(x) be MP. Then Q(x) is SMP if there exist uk,ℓ ∈ Am, and we have

Q(x) =

finite∑
k,ℓ

xkxℓuk,ℓ.

Definition 2. Let φ be a MF in M where M ⊂ Rm+1 is an open and connected. which
contain 0. The function φ is SMF in M iff it has Taylor expansion near zero and can be

written in the form φ(x) =
∞∑
ℓ=0

Qℓ(x)uℓ for some SMPs Qℓ(x) where uℓ ∈ Am.

The right Am module has the the following form:

Am[x] = spanAm
{qℓ(x) : ℓ ∈ N},

where qℓ(x) was first constructed in [7] as follows:

qℓ(x) =
ℓ!

(m)ℓ

∑
s+t=ℓ

(m−1
2 )s(

m+1
2 )t

s!t!
xsxt, (1)

where (a)h = a(a+ 1) . . . (a+ h− 1) is the Pochhamer symbol for a ∈ R.



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 4 of 19

Remark 1. If Qℓ(x) is a SMP which is homogeneous and of degree ℓ and Qℓ(x) = qℓ(x)u,
for u ∈ Am ( [7]), then

∥qℓ∥R = sup
B(R)

|qℓ(x)| = Rℓ.

Definition 3. A space F is called an F-module over Am if F is a Hausdorff space asso-
ciated with a countable proper sets of seminorms N = {∥.∥k}k≥0 such that:

(i) For φ ∈ F , and k < l, we have ∥φ∥k ≤ ∥φ∥l.

(ii) A subset W ⊂ F is open if for all φ ∈ W , there exists ϵ > 0, L ≥ 0 such that
{ψ ∈ F : ∥φ− ψ∥k) ≤ ϵ} ⊂W, for all k ≤ L.

(iii) F is complete regarding the topology determined by the family N.

Definition 4. Let {un} be sequence in an F-module F . Then, the sequence {un} converges
to φ in F iff limn→∞ ∥un − φ∥k = 0 for all ∥.∥k ∈ N.

Let φ(x) where x ∈ Rm+1 be a SMF. Each notation in Table 1 below expresses a class
of SMFs in the indicated regions including: hyper open ball B(R), hyper closed ball B(R),
any hyper open ball enclosing hyper closed ball B+(R), for all entire special monogenic
functions and at the origin. Note that all these spaces are actually F-modules associated
with the corresponding semi-norms system.

Table 1: Types of F-Modules

F-Module Notation Associated Semi-Norms

Space containing SMFs in B(R): WB(R) ∥φ∥r = supB̄(r) |φ(x)|, ∀r < R

Space containing SMFs B(R): WB̄(R) ∥φ∥R = supB̄(R) |φ(x)|
Space containing SMFs in B+(R) WB+(R) ∥φ∥r = supB̄(r) |φ(x)|, ∀R < r}

Space of entire SMFs W∞ ∥φ∥n = supB(n) |φ(x)|, n <∞
Space of SMFs at 0 W0+ ∥φ∥ϵ = supB̄(ϵ) |φ(x)|, ϵ > 0

Definition 5. Let {Qn(x)} be a sequence of an F-module F . Then {Qn(x)} forms a base
if qn(x) is given in the form:

qn(x) =

∞∑
k=0

Qk(x) Q̃n,k, Q̃n,k ∈ Am, (2)

where Q̃ = (Q̃n,k) is the Clifford matrix of operators of the base {Qn(x)} which has the
form:

Qn(x) =
∞∑
k=0

qk(x) Qn,k, Qn,k ∈ Am. (3)



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 5 of 19

The matrix Q = (Qn,k) is the Clifford matrix of coefficient of {Qn(x)}. As deduced in
[7], the set {Qn(x)} form a base iff

QQ̃ = Q̃Q = I, (4)

where I is the identity matrix.

Suppose that φ(x) =

∞∑
n=0

qn(x) un(g) is any SMF of F . By using the formula of qn(x)

as in (2), the basic series

g(x) ∼
∞∑
n=0

Qn(x) Πn(g), (5)

follows, where

Πn(g) =
∞∑
k=0

Q̃k,n ak(g) . (6)

Definition 6. If the associated basic series (5) is normally convergent to every φ(x) in
an F-module F , then the base {Qn(x)} is called effective for F .

More details on the convergence properties of bases {Qn(x)} in the sense of F-modules
can be found in [16, 17].

∥Qn∥R = sup
B(R)

|Qn(x)|, (7)

where

∥Qk Q̃n,k∥R = sup
B(R)

|Qk(x) Q̃n,k|.

Examining the effectiveness of bases throughout this study is mainly conducted using
the value of the Cannon function

Ω(Q, R) = lim sup
n→∞

{Ω(Qn, R)}
1
n , (8)

where
Ω(Qn, R) =

∑
k

∥Qk Q̃n,k∥R, (9)

is called the Cannon sum as we shall see in the following theorem which determines the
effectiveness criteria (see [16, 17]).

Theorem 1. (i) A base {Qn(x)} is effective for WB̄(R), iff Ω(Q, R) = R;

(ii) A base {Qn(x)} is effective for WB+(R), iff Ω(Q, R+) = R;



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 6 of 19

(iii) A base {Qn(x)} is effective for WB(R) iff Ω(Q, r) < R ∀ r < R;

(iv) A base {Qn(x)} is effective for W∞ iff Ω(Q, R) <∞ ∀ R <∞;

(v) A base {Qn(x)} is effective for W0+ iff Ω(Q, 0+) = 0.

For the base defined in (3), we have the Clifford version Cauchy inequality [16]:

|Qn,k| ≤
∥Qn∥R
Rk

. (10)

Definition 7. If {Qn(x)} be a base of SMPs, then the series in (2) is finite. Let N(n)

denote the number of terms in (2) for which Qn,k ̸= 0 and lim supn→∞{N(n)}
1
n = 1.

Then, the base {Qn(x)} is said to be a Cannon base of SMPs (CBSMPs).

Definition 8. Any base Qn(x) is simple it has degree n. In a particular case when
Qn,n = 1 for all n ∈ N, then Qn(x) i a simple monic base

Now, we characterize the growth of a base as introduced in [7, 8] where the order in
Clifford setting was given by:

ρ = lim
R→∞

lim sup
n→∞

log Ω(Qn, R)

n log n
. (11)

Remark 2. A significant relation between the order of base and the entire functions be
represented in the sense that if the base {Qn(x)} is of order ρ, then it will represent every
entire SMF which has order ρ′ < 1

ρ in any finite ball.

3. Effectiveness and order of the exponential simple base of special
monogenic polynomials

In this section, we investigate the convergence properties of the ESBSMPs for the
F-module WB̄(R) as follows.

3.1. Effectiveness of the exponential simple base of special monogenic
polynomials for the F-module W B̄(R)

Let {Qn(x)} be a simple base of SMPs (SBSMPs) of a Clifford variable x ∈ Rm+1,
whose Clifford matrix of coefficients is Q. Then the set {QEn(x)}, whose Clifford matrices
of coefficients and operators are E = eQ and Ẽ = e−Q is a base. This base was defined in
[21] as the ESBSMP. Suppose that the elements {Qn,k} of the Clifford matrix of coefficients
of the SBSMPs {Qn(x)} satisfy{

|Qn,k| ≤M bn−k

n+1 αk, 0 ≤ k < n, M ≥ 1

Qn,n = αn,
(12)

where {αn} is a bounded sequence of positive numbers, k and b are finite positive numbers.
Before we proceed with our results, we state and prove the following important lemma.



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 7 of 19

Lemma 1. If Q(j)
n,k ∈ Am are the elements of the power Clifford matrix Q(j), j ≥ 1 and

satisfying (12), then ∣∣∣Q(j)
n,x

∣∣∣ ≤ 2−
m
2

(
2

m
2 MC

)j
bn−k, 0 ≤ k < n, (13)

where C is a real number such that αk ≤ C, 0 ≤ k < n.

Proof. The elements Q(j)
n,k of the power Clifford matrix Qj , j ≥ 1 are given as

Q(j)
n,k =

n∑
t=k

Q(j−1)
t,k Qn,t (14)

Assume that j = 2 in (14) and by using (12), we obtain that

∣∣∣Q(2)
n,k

∣∣∣ = ∣∣∣∣∣
n∑

t=k

Qt,kQn,t

∣∣∣∣∣
≤ 2

m
2

n∑
t=k

|Qt,k| |Qn,t|

≤ 2
m
2

n∑
t=k

M
bt−k

t+ 1
αk ·M

bn−t

n+ 1
αt

=
2

m
2 (MC)2 · bn−k

n+ 1

n∑
t=k

1

t+ 1

≤ 2
m
2 (MC)2bn−k

= 2−
m
2

(
2

m
2 MC

)2
bn−k,

where 0 ≤ k < n. We assume that∣∣∣Q(j)
n,k

∣∣∣ ⩽ 2−
m
2

(
2

m
2 Mαk

)j
bn−k (15)

Applying (12), (14) and (15) implies that

∣∣∣Q(j+1)
n,k

∣∣∣ ≤ 2
m
2

n∑
t=k

∣∣∣Q(j)
t,k

∣∣∣ |Qn,t|

⩽ 2
m
2

n∑
t=k

2−
m
2

(
2

m
2 MC

)j
bt−k · Mbn−t

n+ 1
αt

⩽ 2−
m
2

(
2

m
2 MC

)j+1
bn−k.

which gives the desire result by mathematical induction.



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 8 of 19

Theorem 2. Suppose that {Qn(x)} is a SBSMPs satisfying (12). Then the ESBSMPs
{QEn(x)} is effective for WB̄(R), where R ≥ b, however, it may not be effective for WB̄(R)

when R < b.

Proof. Since QE = eQ is the Clifford matrix of coefficients of the ESBSMPs {QEn(x)},
then it follows that

QEn(x) =

n∑
k=0

qk(x) QEn,k

=

n∑
k=0

qk(x)

 ∞∑
j=0

Q(j)
n,k

j!

 (16)

Applying (13) in (16), it follows that

∥QEn∥R = sup
B̄(R)

|QEn(x)|

≤ 2
m
2

n∑
k=0

Rk

 ∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!


⩽

n∑
k=0

Rkbn−k
∞∑
j=0

(2
m
2 MC)j

j!

⩽ Rne2
m
2 MC

n∑
k=0

(
b

R

)n−k

≤ e2
m
2 MC(n+ 1)Rn, for all R ≥ b. (17)

Moreover, since the Clifford matrix of operators of the ESBSMPs is {QEn(x)} is QĒ =
e−Q, hence by using (13) and (17) in the Cannon sum of {QEn(x)} yields:

Ω
(
QEn, B̄(R)

)
=

n∑
k=0

∥∥QEkQĒn,k

∥∥
R

=

n∑
k=0

sup
B̄(R)

∣∣QEk(x)QĒn,k

∣∣
≤ 2

m
2

n∑
k=0

∥QEk∥R
∣∣QĒn,k

∣∣
= 2

m
2

n∑
k=0

∥QEk∥R

∣∣∣∣∣∣
∞∑
j=0

(−1)j
Q(j)

n,k

j!

∣∣∣∣∣∣
≤ 2

m
2

n∑
k=0

∥QEk∥R
∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 9 of 19

≤ e2
m
2 MC

n∑
k=0

(k + 1)Rk
∞∑
j=0

bn−k (2
m
2 MC)j

j!

≤ e2
(m2 +1)MC

n∑
k=0

(k + 1)Rkbn−k

= e2
(m2 +1)MCRn

n∑
k=0

(k + 1)

(
b

R

)n−k

≤ e2
(m2 +1)MC(n+ 1)2Rn for all R ≥ b.

It follows that

Ω
(
QE, B̄(R)

)
= lim sup

n→∞

{
Ω
(
QEn, B̄(R)

)} 1
n

≤ R for all R ≥ b.

As Ω
(
QE, B̄(R)

)
≥ R, then Ω

(
QE, B̄(R)

)
= R. and therefore the ESBSMPs {QEn(x)}

will be effective for WB̄(R) for all R ≥ b.
When R < b the ESBSHPs {QEn(x)} may not be effective for WB̄(R) where R < b.

To show this fact, we illustrate the following example

Example 1. Consider the SBSMPs {Qn(x)} for which

Qn,k =

{
αn, k = n,
bn−k

n+1 αk, 0 ≤ k < n,
(18)

where αi =
1

i+1 for i = k, n.

As in (13) we can apply (18) to prove by mathematical induction that

Q(j)
n,0 ≥ bnαj

n. (19)

Now, the Cannon sum for ESBSMPs {QEn(x)} is given as

Ωn

(
QEn, B̄(R)

)
=

n∑
k=0

∥∥∥QEk QẼn,k

∥∥∥
R

≥
∥∥∥QEnQẼn,n

∥∥∥
R

= eαn ∥QEn∥R
≥ eαn |QEn,0|

= eαn

∞∑
j=0

Q(j)
n,0

j!

≥ eαnbn
∞∑
j=0

αj
n

j!
= e2αnbn ≥ bn. (20)



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 10 of 19

Thus Ω
(
QE, B̄(R)

)
≥ b > R for R < b, and the ESBSMPs is not effective for WB̄(R)

for R < b as required.

Remark 3. In Theorem 2, if {Qn(x)} is taken to be a simple monic base (Qn,n = 1 for
n ∈ N), then the results in [21] becomes a particular case of our Theorem 2.

3.2. The Order of the exponential simple base of special monogenic poly-
nomials

Let {Qn(x)} be a SBSMP for which{
|Qn,k| ≤M nλ(n−k)

n+1 αk 0 ⩽ k < n

Qn,n = αn, n ∈ N
(21)

where λ is positive constant and 1 ≤ M < ∞. To justify the main result of this section,
we first introduce the following lemma.

Lemma 2. If Q(j)
n,k ∈ Am are the elements of the power Clifford matrix Q(j), j ≥ 1

satisfying (21), then ∣∣∣Q(j)
n,k

∣∣∣ ⩽ 2−
m
2

(
2

m
2 MC

)j
nλ(n−k), 0 ⩽ k < n. (22)

Proof. As proceed in the proof of Lemma (1), the inequality (22). can be verified.

Theorem 3. Let {Qn(x)} be a SBSMP satisfies (21). Then the ESBSMPs {QEn(x)} is
of order Γ ≤ λ. Moreover, the value λ is attainable.

Proof. Applying (22) in (16), we get

∥QEn∥R ≤ 2
m
2

n∑
k=0

Rk

 ∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!


≤

n∑
k=0

Rk

 ∞∑
j=0

(
2

m
2 MC

)j

j!
nλ(n−k)


≤ e2

m
2 MCnλn

n∑
k=0

(
R

nλ

)k

≤ e2
m
2 MCnλn(n+ 1) = K1(m) nλn(n+ 1), for nλ > R. (23)

Where K1(m) = e2
m
2 MC .

Introducing (22) and (23) in the Cannon sum of the ESBSMP {QEn(x)} we have



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 11 of 19

Ω
(
QEn, B̄(R)

)
≤ 2

m
2

n∑
k=0

∥QEk∥R
∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!

≤ 2
m
2

n∑
k=0

k1(m)kλk(k + 1)
∞∑
j=0

2−m/2


(
2

m
2 MC

)j

j!
nλ(n−k)


≤ K2

1 (m)(n+ 1)2nλn (24)

When n approaches infinity, and using the definition of the order, it follows that the
order Γ of ESBSMPs {QEn(x)} is at most λ.

The following example demonstrates the fact that the bound λ is attainable.

Example 2. Consider the SBSMPS {Qn(x)} given by

Qn,k =

{
M nλ(n−k)

n+1 αk, 0 ⩽ k < n

αn, k = n
(25)

where αk = 1
k+1 and αn = 1

n+1 .

As in (19) we can use (25) to prove by mathematical induction that

Q(j)
n,0 ⩾ nλn (Mαn)

j .

Now, it can be verified, as in (20) that

Ω
(
QEn, B̄(R)

)
⩾ eαn

∑
j=0

Q(j)
n,0

j!

≥ eαnnλn
∞∑
j=0

(Mαn)
j

j!

≥ e
1+M
1+n nλn.

Thus, Γ ≥ λ, but Γ ≤ λ which means that Γ = λ and the bound is λ is attainable.
This completes the proof.

Remark 4. In Theorem 3, if {Qn(x)} is a simple monic base (Qn,n = 1 for n ∈ N), then
the result in [21] becomes a particular case of our Theorem 3.



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 12 of 19

4. Effectiveness of the the exponential Cannon base of special
monogenic polynomials

In the current section, we study the convergence properties of the ECBSMPs for the
F-modules WB̄(R), WB(R), WB+(R), W0+ and W∞.

Theorem 4. The ECBSMPs {QEn(x)} is effective for WB̄(R) for all b ⩽ R < b
a if the

Cannon base {Qn(x)} satisfy

|Qn,k| ≤M bn−kak, 0 < a < 1 (26)

and may not be effective for WB̄(R) for all R < b or R ⩾ b
a .

Proof. Suppose that {Qn(x)} is a Cannon base, the relation (14) has the form

Q(j)
n,k =

∞∑
t=0

Q(j−1)
t,k Qn,t (27)

Thus, as in (13), applying (26), we obtain the following inequality∣∣∣Q(j)
n,k

∣∣∣ ⩽M j(1− a)−(j−1)bn−kak. (28)

Using (28), we get

∥QEn∥R ≤ 2
m
2

∑
k

Rk

 ∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!


≤ 2

m
2 eM(1−a)−1

(1− a)
∑
k

Rkbn−kak

= 2
m
2 eM(1−a)−1

(1− a)

∑
k≤n

(
b

R

)n−k

ak +
∑
k>n

(
b

R

)n(Ra
b

)k
]
Rn

= K2R
n, for all b ≤ R <

b

a
. (29)

where

K2 = 2
m
2 eM(1−a)−1

(1− a)

∑
k≤n

(
b

R

)n−k

ak +
∑
k>n

(
b

R

)n(Ra
b

)k
 .

Hence, making use of of (28) and (29) in Ωn

(
QEn, B̄(R)

)
of the exponential base

{QEn(x)} associated with the Cannon base {Qn(x)} yields.

Ωn

(
QEn, B̄(R)

)
≤ 2

m
2

∑
k

∥QEk∥R
∣∣QĒn,k

∣∣



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 13 of 19

≤ 2
m
2

∑
k

∥QEk∥R
∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!

≤ K2 e
M(1−a)−1

(1− a)
∑
k

Rkbn−kak

≤ K2
2 R

n

for all b ≤ R < b
a . Thus, the Cannon function for {QEn(x)} satisfies Ω

(
QE, B̄(R)

)
≤ R,

but Ω
(
QE, B̄(R)

)
≥ R. Therefore, Ω

(
QE, B̄(R)

)
= R, for b ≤ R < b

a , and the ECBSMPs

{QEn(x)} is effective for WB̄(R) for b ≤ R < b
a .

When R < b or R ⩾ b
a , the ECBSMPs {QEn(x)} may not be effective in for WB̄(R).

To show this fact consider the CBSMPs {Qn(x)} for which

Qn,k =

{
1, k = 0

Mbn−kak, k ̸= 0.

Similar steps as in (20) can be obtained Ω
(
QE, B̄(R)

)
> R for R < b, and the

ECBSMPs {QEn(x)} is not effective for WB̄(R). Also, Ω
(
QE, B̄(R)

)
= ∞ for R ≥ b

a and

the ECBSMPs {Qn(x)} is not effective for WB̄(R) for R ≥ b
a .

In examining the effectiveness of the ECBSMPs {QEn(x)} for the F-module WB+(R),
we assume that the base Qn(x) satisfies the prescribed condition.

µ(Q, R+) ≤ R, (30)

where

µ(Q, R+) = lim sup
n→∞

{∥Qn∥R+}
1
n .

and

∥Qn∥R+ = sup
B+(R)

|Qn(x)| .

In the sequel, we present a proof for the following theorem

Theorem 5. If the CBSMPs {Qn(x)} satisfying (30), then the ECBSMPs {QEn(x)}
associated with {Qn(x)} is effectiveness for WB+(R).

Proof. Suppose that ρ1, ρ2 and r are chosen such that R < ρ1 < ρ2 < r. Then one can
construct a sequence (Rj) of positive numbers satisfying R < ρ1 < ρ2 < R1 < R2 < · · · <
Rj < · · · < r and

∥Qn∥Ri
< KRn

i+1;n ≥ 0 (31)



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 14 of 19

where K is a positive finite constant. For the power base
{
Q(j)

n (x)
}
corresponding to the

original base {Qn(x)}, it follows that

Q(j)
n (x) =

∑
k

Q(j−1)
k (x)Qn,k, (32)

where {Qn,k} represent the elements of the matrix Q, and by substituting s = 2 into (32),
then applying Cauchy’s inequality together with relation (31), one obtains

∥∥∥Q(2)
n

∥∥∥
R1

= sup
B̄(R1)

∣∣∣Q(2)
n (x)

∣∣∣
≤ 2

m
2 ∥Qn∥R3

∑
k

∥Qk∥R1

Rk
3

< K2
m
2

∑
k

(
R2

R3

)k

∥Qn∥R3

= K2
m
2 S (R2, R3) ∥Qn∥R3

.

Throughout the following discussion, we assume that

∥∥∥Q(j)
n

∥∥∥
R1

<
(
K 2

m
2

)j−1
j−1∏
N=1

S (R2N , R2N+1) ∥Qn∥R2j−1 (33)

From (31), (32), and (33), together with an application of Cauchy’s inequality, it
follows that

∥∥∥Q(j+1)
n

∥∥∥
R1

≤ 2
m
2

∥∥Qn∥R2j+1

∥∥∑
k

∥∥∥Q(j)
k

∥∥∥
R1

Rk
2j+1

< 2
m
2

(
K 2

m
2

)j−1
j−1∏
N=1

S (R2N , R2N+1) ∥Qn∥R2j+1

∑
k

∥Qk∥R2j−1

Rk
2j−1

<
(
K 2

m
2

)j
j∏

N=1

S (R2N , R2N+1) ∥Qn∥R2j+1

Hence, the validity of assumption (33) is established by induction. Defining k1 =
max1≤N≤j−1K 2

m
2 S (R2N , R2N+1), relation (33) takes the form∥∥∥Q(j)

n

∥∥∥
R1

< Kj−1
1 ∥Qn∥R2j−1

(34)

Observing that QE = eQ serves as the coefficient matrix associated with the exponen-
tial basis {QEn(x)}, we deduce that



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 15 of 19

QEn(x) =
∑
k

qk(x)QEn,k

=
∑
k

qk(x)

 ∞∑
j=0

Q(j)
n,k

u!

 (35)

From (31), (34), together with Cauchy’s inequality applied in (35), we obtain

∥QEn∥ρ1 = sup
B̄(ρ1)

|QEn(x)|

≤ 2
m
2

∑
k

ρk1

∣∣∣∣∣∣
∞∑
j=0

Q(j)
n,k

j!

∣∣∣∣∣∣
< 2

m
2

∑
k

∞∑
j=0

∥Qn∥R1

j!

(
ρ1
R1

)k

< 2
m
2

∑
k

∞∑
j=0

Kj−1
1 ∥Qn∥R2j−1

j!

(
ρ1
R1

)k

< 2
m
2 rn

∑
k


∞∑
j=0

Kj
1

j!

(ρ1
R

)k

= 2
m
2 eK1S (ρ1, R1) r

n.

It can be concluded that

∥QE∥ρ1 = lim sup
n→∞

{
∥QEn∥ρ1

} 1
n ≤ r

Taking the limit as r → R+, we obtain

∥QE∥R+ ≤ R.

Then, for any number ρ1 > R, one can choose ρ2 > ρ1 > R such that ∥QE∥ρ1 < ρ2.
Consequently, it follows that ∥QEn∥ρ1 < K2ρ

n
2 , ∀ n ≥ 0, where K2 ≥ 1.

From the definition of the exponential base {QEn(x)}, its Cannon sum is given by

Ω
(
QEn, B̄ (ρ1)

)
=

∑
k

∥∥
QEk QĒn,k

∥∥
ρ1

≤ 2
m
2

∑
k

∥QEk∥ρ1
∞∑
j=0

∣∣∣Q(j)
n,k

∣∣∣
j!



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 16 of 19

≤ 2
m
2

∑
k

∥QEk∥ρ1
∞∑
j=0

∥∥∥Q(j)
n

∥∥∥
R1

j!Rk
1

< K2 2
m
2

∑
k

∞∑
j=0

∥Qn∥R2j−1

hj−1
1

j!

(
ρ2
R1

)k

< K2 2
m
2 · ek1S (ρ2, R1) r

n.

Taking the limit as n tend to infinity, we obtain

Ω
(
QE, B̄ (ρ1)

)
= lim sup

n→∞

{
Ω
(
QEn, B̄ (ρ1)

)}
≤ r

which yields Ω (QE,R
+) ≤ R as r → R+. However, since Ω (QE , R

+) ≥ R, it follows that
Ω (QE,R

+) = R. R. Consequently, the exponential base {QEn(x)} is effective for WB+(R).

Remark 5. When R→ 0 in condition (30) we obtain the effectiveness of the ECBSMPs,
{QEn(x)} for the space W0+.

In order to examine the effectiveness of the ECBSMPs {QEn(x)} for the F-module
WB(R), we assume that the base {Qn(x)} fulfills the condition

µ(Q, r) < R, ∀r < R, (36)

with

µ(Q, r) = lim sup
n→∞

{∥Qn∥r}
1
n .

Theorem 6. If the CBSMPs {Qn(x)} satisfying (36), then the ECBSMPs {QEn(x)}
associated with {Qn(x)} is effectiveness for WB(R).

Proof. The proof of the following theorem is omitted, as it can be established by
arguments analogous to those employed in Theorem 4, together with the aid of Theorem 1
(see [16]).

Remark 6. If R→ ∞ then condition (36) will be replaced by the condition

µ(Q, r) <∞, ∀r <∞, (37)

and we have the effectiveness for the ECBSMPs {QEn(x)} for W∞.



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 17 of 19

Now, let {Qs,n(x)}, s = 1, 2, 3 be three bases of SMPs where x ∈ Rm+1. the equivalent
base {Tn(x)} defined by

{Tn(x)} = {Q̃3,n(x)}{Q2,n(x)}{Q1,n(x)} (38)

where {Q̃3,n(x)} is the inverse base of {Q3,n(x)}.
In a recent paper [17] the authors prove that the equivalent base {Tn(x)} satisfy the

conditions

µ(T, r) < R, ∀r < R, (39)

and

µ(T,R+) ≤ R, (40)

when the bases {Qs,n(x)}, s = 1, 2, 3 is algebraic according to [10] and satisfying the
following conditions, respectively

µ(Qs, r) < R, ∀r < R, (41)

and

µ(Qs, R
+) ≤ R, (42)

Applying Theorems 5 and 6, we conclude the following result.

Corollary 1. Let {Qs,n(x)}, s = 1, 2, 3 be three algebraic bases of SMPs satisfying the
conditions (41) and (42), then the ECBSMPs {TEn(x)} associated with the equivalent
base {Tn(x)} are effectiveness for the spaces WB(R), WB+(R), W0+ and W∞.

Looking back to the equivalent base {Tn(x)} and by taking {Q3,n(x)} = {Q1,n(x)},
we get the similar base {Sn(x)} as studied in [25]. Therefore all the results of Corollary 1
will be satisfied but for the similar base {Sn(x)} instead of equivalent base {Tn(x)}.

5. Conclusions

The current study discusses the representation of the exponential base of special mono-
genic polynomials (EBSMPs) in Fréchet modules. The coefficients of the power associated
infinite matrices of an original base have some restriction for which certain classes of SMFs
can be represented by ESBSMPs in hyber closed ball. The upper bound of the order of
the ESBSMPs is is determined. Moreover, the effectiveness of the exponential base as-
sociated with Cannon sets is studied in different hyper regions . In the previous studies
[9–11, 20, 25–27] in complex and Clifford analysis the convergence properties (effective-
ness, order, type, and the Tρ-property) of some associated bases of polynomials have been
examined, such as, Hadamard product base, product base, inverse base, square root base,



M. Zayed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6793 18 of 19

similar base and similar transposed base. Theses constituents are simple monic bases. It is
interesting to explore more extended associated bases from those obtained in the existing
by considering the constituents to be general bases. As far as we know from the literature
that one of the untouched problems in Clifford context, is to represent any entire mono-
genic functions by a set of polynomials, in non-spherical regions, such as Faber regions.
Having such tools would pave the way to explore these problems and may be fruitful to
enrich the approximation theory in Clifford setting.

Acknowledgements

The author thanks the anonymous reviewers and the handling editor for their con-
structive comments, which significantly improved the paper.

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