1 Introduction

It is well known that the theory of bases in function spaces plays an important role in mathematics and its applications, e.g., in approximation theory, partial differential equations, geometry and mathematical physics.

The theory of polynomial bases (or the theory of basic set of polynomials) in one complex variable has been initiated principally by Whittaker [42].

Later on this theory has been developed by Cannon [17, 20]. The terminology of the basic set (or the base) means here the Hamel base of the complex linear space of all polynomials in one complex variable with complex coefficients. Under some consideration, many well-known polynomials such as Laguerre, Legendre, Hermite, Bernoulli, Euler and Bessel polynomials form simple bases of polynomials. In this respect we may refer for instance to [6]. The central problem of this theory of basic set of complex polynomials may be described as follows: “Given a linearly independent set of polynomials, under which conditions can each function belonging to a certain class of holomorphic functions be expanded into these polynomials?” These conditions are called “conditions of effectiveness” a terminology that proceeds from Cannon-Whittaker [42] where Cannon has obtained in [17,18,19,20] necessary and sufficient conditions for bases to be effective in classes of holomorphic functions with finite radius of regularity and of entire functions. A significant contribution to the subject was made by Newns [36] who developed the theory from the topological point of view, to some generalized spaces. In this respect J. Cnops and Abul-Ez has contributed in [21] by some interesting results in nuclear Fréchét spaces.

As Clifford analysis is the revolution of the one of holomorphic functions of the complex variable in higher dimensions, Abul-Ez and et al gave in [1,2,3,4,5,6,7,8,9,10], the extension of the theory of bases of polynomials in one complex variable to the setting of Clifford analysis. This is 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 axially monogenic functions is considered, for which a Cannon theorem on the effectiveness in different regions is introduced [1, 2, 9, 40]. It should be noted that, although straight forward generalization are possible, the proof the main central theorem of the theory (Cannon theorem) concerning effectiveness (see [1, 2]) in a true \(n-\)dimensional domain is very complected and not yet be touched. Therefore, Abul-Ez and Constales in [1, 2]. confine themselves to consider representation of special monogenic functions in axially symmetric domains, for what we call here axially monogenic functions.

From this starting point many results on the polynomial bases in complex case of one variable were refined and generalized to Clifford setting (see [1,2,3,4,5,6,7,8,9,10] and also, [12,13,14, 28, 29] ). Relating to this work in Clifford analysis, we quote the work in, e.g., [11, 15, 16, 22, 24,25,26, 30, 32].

Now, given a base of axially monogenic polynomials we can associate with it by one way or another, a new base of axially monogenic polynomials. It is an interesting question to give a clear answer of the effectiveness property of these associated bases in terms of the effectiveness property of generating base or bases. With this in hand some authors exhibit some of these questions for what they call inverse bases, product bases, similar bases, \(\mu -\) th root bases (see, for examples [3, 8, 28]).

The main goal of our work is to give deeply investigation of the problem of representation of axially monogenic function (or special monogenic function) according to different domains in terms of the inverse bases of polynomials. So we will discuss the question of effectiveness for integral Clifford valued functions, and then the question of effectiveness of inverse bases in closed balls, open balls and at the origin.

In fact it will be seen here that our attempts will extend, in an easy way, the results given in [3, 28], to more general types of bases, as well as our results here are considered to be a natural generalization of the results in complex analysis setting given by Mikhaill [33, 34], Halim [27], Eweida [23], Nassif [37] and Tantawi [41]. The paper is constructed as follows. In Sect. 2, we briefly introduce the most important concepts of Clifford analysis and the basic tools in the theory of bases of special monogenic polynomials. In Sect. 3, we introduce the effectiveness study in closed balls, open balls and at the origin. In Sect. 4 we establish a result to estimate the upper and lower order of the inverse base in terms of order of original base with illustrative examples.

2 Preliminaries

Clifford analysis offers a function theory which is a higher-dimensional analogue of the theory of holomorphic functions of one complex variable. For more details concerning this function theory and its applications we refer the reader to [31, 39].

The real Clifford algebra over \( \mathbb {R} ^{m}\) is defined as

$$\begin{aligned} \mathcal {A}_{m}=\left\{ \sum _{A\mathcal {\subseteq \{}1,\ldots ,m\}}a_{A}e_{A}:a_{A}\in \mathbb {R} \right\} \end{aligned}$$

where \(e_{A}=e_{\alpha _{1}}\ldots e_{\alpha _{h}\text { }}\) with \(A=\{\alpha _{1},\ldots ,\alpha _{h}\},\alpha _{1}<\ldots <\alpha _{h}.\) Moreover, \(e_{\phi }=e_{o}=1,\)\(e_{\{k\}}=e_{k},\)\(k=1,\ldots ,m\) and the product in \(\mathcal { A}_{m}\) is determined by the relations \(e_{k}^{2}=-1,\)\(k=1,\ldots ,m\) and \(e_{k}e_{j}+e_{j}e_{k}=0,\)\(k\ne j,\)\(k,j=1,\ldots ,m\). An involution on \( \mathcal {A}_{m}\) is defined by , where \(\bar{e}_{\mathcal {A} }=\bar{e}_{\alpha _{h}}\ldots \bar{e}_{\alpha _{1}},\) and \(\bar{e}_{\alpha _{j}}=-e_{\alpha _{j}}.\)

Since \(\mathcal {A}_{m}\) is isomorphic to \( \mathbb {R} ^{2^{m}}\) we may provide it with the \( \mathbb {R} ^{2^{m}}\)-norm \(\left| a\right| \) and one sees easily that for any \( a,b\in \mathcal {A}_{m},\left| a\cdot b\right| \le 2^{\frac{m}{2} }\left| a\right| \cdot \left| b\right| .\)

The elements \((x_{o},x)=(x_{o},x_{1},\ldots ,x_{m})\in \mathbb {R} ^{m+1}\) will be identified with the Clifford numbers \(x_{o}+\overrightarrow{ x}=x_{o}+\overset{m}{\sum _{j=1}}e_{j}x_{j}\). Note that if \(x=x_{o}+ \overrightarrow{x}\in \mathbb {R} ^{m+1},\)\(\bar{x}=x_{o}-\overrightarrow{x}\). Let \(\Omega \subseteq \mathbb {R} ^{m+1}\) be open and let \(f\in C_{1}(\Omega ;\mathcal {A}_{m})\). Then f is called left monogenic in \(\Omega \) if \(Df=\overset{m}{\sum _{j=0}}e_{j}\left( \frac{\partial }{\partial x_{j}}\right) f=0\) in \(\Omega \), or right monogenic if \( fD=0.\) Here \(D=\overset{m}{\sum _{j=0}}e_{j}\left( \frac{\partial }{\partial x_{j}} \right) ,\) stands for the generalized Cauchy–Riemann operator.

The right \(\mathcal {A}_{m}-\)module \(\mathcal {A}_{m}[x]\) defined by \( \mathcal {A}_{m}[x]=\hbox { span }_{\mathcal {A}_{m}}\{z_{n}(x):n\in \mathbb {N} \}\) is called the space of special monogenic polynomials, where \(\mathcal {A} _{m}\) is the Clifford algebra and x is the Clifford variable. \(z_{n}(x)\) is defined by (see [1]),

$$\begin{aligned} z_{n}(x)=\sum _{i+j=n}\frac{\left( \frac{m-1}{2}\right) _{i}\left( \frac{m+1}{2}\right) _{j}}{i!j!} \overline{x}x^{j}, \end{aligned}$$

where for \(b\in \mathbb {R} ,\ (b)_{i}\) stands for \(b(b+1)\ldots (b+l-1),\ \overline{x}\) is the conjugate of x, \(x\in \mathbb {R} ^{m+1}, \mathbb {R} ^{m+1}\) is identified with a subset of \(\mathcal {A}_{m}\).

If \(P_{n}(x)\) is homogeneous special monogenic polynomials of degree n in x,  then (see [1]) \(P_{n}(x)=z_{n}(x)\alpha \), \(\alpha \) is some constant in \(\mathcal {A}_{m}\) and

$$\begin{aligned} \sup _{\left| x\right| =R}\left| z_{n}(x)\right| =\left( \begin{array}{c} m+n-1 \\ n \end{array} \right) R^{n}=\frac{(m)_{n}}{n!}R^{n}, \end{aligned}$$

where \(((m)_{n}/n!)=(m+n-1)!/n!(m-1)!\)

Remark 1

A homogeneous special monogenic polynomials \(P_{n}(x)\) of degree n are necessarily of the form \(P_{n}(x)=z_{n}(x)c_{n}.\) Hereby \(c_{n}\in \mathcal { A}_{m}\) and \(z_{n}(x)\) is given by generating function \(\frac{1-\bar{x}}{ \left| 1-x\right| ^{m+1}}\) (see [1]).

Definition 1

Let \(\Omega \) be a connected open subset of \(\mathbb {R} ^{m+1}\) containing 0, then a monogenic function f in \(\Omega \) is said to be special monogenic in \(\Omega \) if and only if its Taylor series near zero (which is known to exist) has the form

$$\begin{aligned} f(x)=\overset{\infty }{\sum _{n=0}}z_{n}(x)c_{n},~\ \ c_{n}\in \mathcal {A} _{m}. \end{aligned}$$

A function f is said to be special monogenic on \(\bar{B}(R)\) if it is special monogenic on some connected open neighborhood \(\Omega _{f}\) of \(\bar{B}(R).\) The set of all functions which are special monogenic on \(\bar{B }(R)\) is denoted by SM(R). Clearly SM(R) is a submodule of the right \( \mathcal {A}_{m}-\)module \(M(\bar{B}(R))\) of functions which are monogenic in a neighborhood of \(\bar{B}(R)\).

The fundamental references for special monogenic functions are (cf. [15, 31, 39]).

In [1, 2] the theory of basic sets (bases) of special monogenic polynomials has been developed as follows.

Definition 2

A set \(\beta =\left\{ P_{k}\left( x\right) :k\in \mathbb {N} \right\} \) of special monogenic polynomials is called basic if and only if it is a base for the space \(\mathcal {A}_{m}\left[ x\right] \) of special monogenic polynomials, i.e.,

(i) Every \(z_{n}\left( x\right) \) can be expressed as a right \(\mathcal {A} _{m}\)-linear combination of elements from \(\beta \).

$$\begin{aligned} z_{n}\left( x\right) =\sum \limits _{i=0}^{\infty }P_{i}\left( x\right) \pi _{ni},\ \ \ \ \ \ \ \pi _{ni}\in \mathcal {A}_{m} \end{aligned}$$
(1)

where only a finite number of terms differ from zero and \(P_{i}\left( x\right) \) is given by

$$\begin{aligned} P_{i}\left( x\right) =\sum \limits _{j=0}^{\infty }z_{j}\left( x\right) p_{ij}\in \mathcal {A}_{m}\mathcal {~\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}$$

(ii) Let \(l\in \mathbb {N} \) and let \(a_{o},a_{1},\ldots ,a_{l}\in \mathcal {A}_{m}\) be such that

$$\begin{aligned} \sum \limits _{k=0}^{\ell }P_{k}\left( x\right) a_{k}=0\Rightarrow a_{k}=0,~\ \ \ k=0,1,\ldots ,\ell \text {.} \end{aligned}$$

Let \(N_{n}\) denotes the number of nonzero elements \(\pi _{ni}\) in (1). If \(\lim \underset{n\rightarrow \infty }{\sup }N_{n}^{(\frac{1}{n} )}\rightarrow 1,\) the base \(\beta \) is called a Cannon base, and if this condition not satisfied the basic set will be called general.

It is shown in [4] that \(\{P_{n}(x)\}\) is a basic set or a base, iff its Clifford matrix of coefficients \(P=(p_{ni})\) has a unique row-finite inverse \(\Pi =(\pi _{nk})\) which we call the matrix of operators.

If f(x) is a special monogenic function as defined above, then there is formally an associated basic series given by

$$\begin{aligned} f(x)=\overset{\infty }{\sum _{k=0}}P_{n}(x)\left( \overset{\infty }{\sum _{n=0} }\pi _{nk}c_{n}\right) . \end{aligned}$$

A base (or a basic set) \(\{P_{n}(x)\}\) is said to be effective if for every special monogenic function f, defined in a closed neighborhood of zero \( \overline{B}(R)\) of the radius \(R>0\), the above associated basic series converges normally to f in \(\overline{B}(R)\).

By effectiveness in the region \(\overline{B}(R^{+})\) we mean that the basic series associated with every special monogenic function f(x), which is monogenic in and on \(\overline{B}(R)\), represents f(x) in some ball surrounding \(\overline{B}(R)\), the size of the ball depending on the function.

The inverse base \(\left\{ \overline{P}_{n}\left( x\right) \right\} \) of special monogenic polynomials is defined in [3]. As the base whose Clifford matrix of coefficients is \(\Pi \), and consequently its Clifford matrix of operators is P then we have

$$\begin{aligned} \overline{P}_{i}\left( x\right) =\sum \limits _{j=0}^{i}z_{j}\left( x\right) \pi _{ij},~\ \ \ \ \ \ \ \ \ \pi _{ij}\in \mathcal {A}_{m} \end{aligned}$$
(2)

and

$$\begin{aligned} z_{n}\left( x\right) =\sum \limits _{i}\overline{P}_{i}\left( x\right) p_{ni},~\ \ \ \ \ \ p_{ni}\in \mathcal {A}_{m} \end{aligned}$$
(3)

The function (the Cannon function)

$$\begin{aligned} \lambda \left( R\right) =\underset{n\rightarrow \infty }{\lim \sup }\left\{ \omega _{n}\left( R\right) \right\} ^{\frac{1}{n}}, \end{aligned}$$

is defined in [1] associated with the base \(\left\{ P_{n}\left( x\right) \right\} \), with

$$\begin{aligned} \omega _{n}\left( R\right) =\sum \limits _{i=0}^{\infty }\sup \left| P_{i}\left( x\right) \pi _{ni}\right| ~\ \ \ \ (\text {the Cannon sum).} \end{aligned}$$

Notice that \(\omega _{n}\left( R\right) \), and \(\lambda \left( R\right) \) are non-decreasing.

It has been showed in [1] that a simple base \(\left\{ P_{n}\left( x\right) \right\} \) is effective in \(\overline{B}\left( R\right) \) iff

$$\begin{aligned} \lambda \left( R\right) =R. \end{aligned}$$
(4)

The order \(\omega \) of the base was defined in [2] by

$$\begin{aligned} \omega =\lim _{R\rightarrow \infty }\underset{n\rightarrow \infty }{\lim \sup }\frac{\log \omega _{n}\left( R\right) }{n\log n}. \end{aligned}$$
(5)

Also, it has been shown in [1] that the order of the basic set plays an important role in the representation of entire functions by series of special monogenic polynomials.

3 Convergence Properties of the Inverse Base

A simple example illustrates the fact that when a base possesses a certain convergence property, its inverse base does not necessarily possesses the same convergence property. Thus, the base defined by

$$\begin{aligned} P_{n}\left( x\right) =\left[ \begin{array}{c} \left[ z_{n}\left( x\right) -z_{n+1}\left( x\right) \right] a^{n^{2}},\left( a\in \mathbb {N} >1\right) ,\qquad n\text { odd} \\ z_{n}(x),~ \qquad \qquad \;\;\qquad \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n~\text {even} \end{array} \right. \end{aligned}$$

is effective in every ball \(\overline{B}\left( R\right) ,~R>0;\) but the inverse base given by

$$\begin{aligned} \bar{P}_{n}(x)=\left[ \begin{array}{l} z_{_{n}}\left( x\right) a^{-n^{2}}+z_{n+1}(x),\qquad \quad \;\;n\text { odd} \\ z_{n}(x),~\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n\text { even} \end{array} \right. \end{aligned}$$

is effective in no ball \(\bar{B}(R)\).

In order that the inverse base should have a certain convergence property, we must consider some restrictions on the base \(\{P_{n}(x)\}\) further than possessing the assigned convergence property.

As in the case of complex analysis we consider a simple monic base \( \{P_{n}(x)\}\) for which the coefficients of \(z_{n}(x)\) in the formula (1) is the unity.

3.1 Effectiveness in Closed, Open Balls and at the Origin

We shall use the terms

$$\begin{aligned} \mu (R)= & {} \underset{n\rightarrow \infty }{\lim \sup }\{A_{n}(R)\}^{\frac{1}{n} }, \end{aligned}$$
(6)
$$\begin{aligned} \nu (R)= & {} \underset{n\rightarrow \infty }{\lim \inf }\{A_{n}(R)\}^{\frac{1}{n} }~\ \end{aligned}$$
(7)

Theorem 1

Let the base \(\{P_{n}(x)\}\) of special monogenic polynomials satisfies the two conditions

$$\begin{aligned} \mu (R^{+})\leqslant & {} aR \end{aligned}$$
(8)
$$\begin{aligned} \nu (\rho )< & {} aR \end{aligned}$$
(9)

then if \(\{P_{n}(x)\}\) is effective in \(\bar{B}(R^{+})\), the inverse base \(\{ \bar{P}_{n}(x)\}\) is effective in \(\bar{B}(aR^{+})\).

Proof

We shall consider \(R_{5\text { }}\) to be any number \(>R\) and then choose the intervening \(R^{^{\prime }}s\), \(R<R_{1}<R_{2}<R_{3}<R_{4}<R_{5}\), to suit their requirements.

Since \(\{P_{n}(x)\}\) is effective in \(\bar{B}(R^{+})\), then \(\kappa (R^{+})=R, \ \kappa (R)\) stands for the Cannon function of the general base.

Hence,

$$\begin{aligned} \kappa (R_{3})< & {} R_{4}~ \\ F_{n}(R_{3})< & {} k_{1}R_{4}^{n}\ \ \ \ \forall n,k_{1}\ \text {is constant} \end{aligned}$$

Also from (7) we have

$$\begin{aligned} \nu (R)=\underset{n\rightarrow \infty }{\lim \inf }[A_{n}(R)]^{\frac{1}{n}};\; \; \end{aligned}$$

and then from (9) since \(\nu (R_{3})>aR_{2}~\ \ \ \ for\)\(R_{3}>R_{2}\) , then \(k_{2}A_{n}(R_{3})>(aR_{2})^{n}\) for all \(n,~k_{2}\) is constant.

Hence,

$$\begin{aligned} \bar{A}_{n}(aR_{1})= & {} \sup _{\left| x\right| =aR_{1}}\left| \bar{ P}_{n}(x)\right| \\= & {} \sup _{\left| x\right| =aR_{1}}\left| \overset{D_{n}}{ \sum \limits _{i=0}}z_{i}(x)\pi _{ni}\right| \;\;\;\;\;\text {where}\; D_{n}= \max _{i}\\\le & {} \sup _{\left| x\right| =aR_{1}} \overset{D_{n}}{ \sum \limits _{i=0}} \left| z_{i}(x)\right| \left| \pi _{ni}\right| ;\;\;\;\;\;\;\;\;\sup _{\left| x\right| =R}\left| z_{n}(x)\right| = \frac{(m)_{n}}{n!}R^{n},\\\le & {} 2^{\frac{m}{2}}k_{2}\frac{(m)_{D_{n}}}{(D_{n})!}\overset{\infty }{ \sum \limits _{i=0}}A_{i}(R_{3})\left| \pi _{ni}\right| \left( \frac{R_{1}}{ R_{2}}\right) ^{i} \\< & {} 2^{\frac{m}{2}}k_{2}\frac{(m)_{D_{n}}}{(D_{n})!}N_{n} \max _{i=0}\;A_{i}(R_{3})\left| \pi _{ni}\right| \left( \frac{R_{1}}{ R_{2}}\right) ^{i} \\< & {} 2^{\frac{m}{2}}k_{3}\frac{(m)_{D_{n}}}{(D_{n})!}N_{n}F_{n}(R_{3}) \\< & {} 2^{\frac{m}{2}}k_{4}\frac{(m)_{D_{n}}}{(D_{n})!}N_{n}R_{4}^{n}. \end{aligned}$$

It follows that

$$\begin{aligned} \bar{F}_{n}(aR_{1})= & {} \underset{i,j}{\max }\underset{\left| x\right| =(aR_{1})}{\sup }\left| \underset{k=i}{\overset{j}{\sum }} \bar{P}_{k}(x)p_{nk}\right| \\\le & {} \underset{i,j}{\max }2^{\frac{m}{2}}\overset{j}{\sum _{k=i}}\underset{ \left| x\right| =(aR_{1})}{\sup }\left| \bar{P} _{k}(x)\right| \left| p_{nk}\right| \\< & {} 2^{\frac{m}{2}}\;\underset{i,j}{\max }\overset{j}{\sum _{k=i}}2^{\frac{m}{2} }k_{4}\frac{(m)_{D_{k}}}{(D_{k})!}N_{k}R_{4}^{k}\left| p_{nk}\right| . \end{aligned}$$

Using Cauchy’s inequality (cf. [2]) then

$$\begin{aligned} \left| p_{nk}\right| \le \sqrt{\frac{n!}{(m)_{k}}}\frac{A_{n(R_{5})} }{R_{5}^{k}}. \end{aligned}$$

One can get

$$\begin{aligned} \overline{F}_{n}(aR_{1})< & {} \;2^{m}k_{4}\;\max _{k=0} \frac{(m)_{D_{k}}}{(D_{k})!}N_{k}R_{4}^{k}\sqrt{\frac{n!}{(m)_{k}}}\frac{ A_{n}(R_{5})}{R_{5}^{k}} \\< & {} 2^{m}k_{4}N_{n}\frac{(m)_{n}}{n!}\sqrt{\frac{n!}{(m)_{n}}}A_{n}(R_{5})\left( \frac{R_{4}}{R5}\right) ^{n},\ \ \ \ \forall n>D_{n} \\< & {} k_{5}A_{n}(R_{5}). \end{aligned}$$

Then we have \( \bar{\kappa }(aR_{1})\le \mu (R_{5})\).

As \(R_{5}\longrightarrow R^{+},\ aR_{1}\longrightarrow aR^{+},\) and making use of (8) we get

$$\begin{aligned} \overline{\kappa }(aR^{+})\le aR. \end{aligned}$$

Hence, \(\overline{\kappa }(aR^{+})=aR,\) and the theorem follows. \(\square \)

Theorem 2

Let the base \(\{P_{n}(x)\}\) of special monogenic polynomials, satisfying the two condition

$$\begin{aligned}&\mu (r)<aR,\ \ \ \ \forall r<R \end{aligned}$$
(10)
$$\begin{aligned}&\nu (R^{-})\ge aR. \end{aligned}$$
(11)

Then if the base \(\{P_{n}(x)\}\) is effective in the open ball B(R), its inverse base \(\{\bar{P}_{n}(x)\}\) is effective in the open ball B(aR).

Proof

Suppose that \(R_{5}\) is any number such that \(R>R_{5}\) and again choose the intervening such that \(R^{^{\prime }}s\)

\(R>R_{1}>R_{2}>R_{3}>R_{4}>R_{5}\) to suit their requirements.

Since \(\{P_{n}(x)\}\) is effective in B(R), then \(\kappa (r)<R\) for all \(r<R\). Thus, \(\kappa (R_{3})<R_{2}\). Hence,

$$\begin{aligned} F_{n}(R_{3})<k_{1}R_{2}^{n}~,\ \ \ \ \ \ \ \forall n,k_{1~}\text {is constant. } \end{aligned}$$

Also from (11) we have

$$\begin{aligned} \nu (R_{3})>aR_{4},~\ \ \ \ \ \text {for}~R_{3}>R_{4}. \end{aligned}$$

So that

$$\begin{aligned} (aR_{4})^{n}<k_{2}A_{n}(R_{3}),\ \ \ \ \ \forall ~n,k_{2}~\text {is constant.} \end{aligned}$$

The steps applied to estimate the supremum of \(\bar{P}_{n}(x)\) in the proof of Theorem (1) can be similarly carried out to derive the following.

$$\begin{aligned} \bar{A}_{n}(aR_{5})= & {} \underset{\left| x\right| =aR_{5}}{\sup } \left| \bar{P}_{n}(x)\right| \\= & {} \underset{\left| x\right| =aR_{5}}{\sup }\left| \underset{i=0}{\overset{D_{n}}{\sum }z_{i}(x)\pi _{ni}}\right| \;\;\;\;\;\;\text {where} \; D_{n}= \max _{i}\\< & {} 2^{\frac{m}{2}}\overset{D_{n}}{\underset{i=0}{\sum }}\frac{(m)_{i}}{i!} (aR_{4})^{i}\left| \pi _{ni}\right| \left( \frac{R_{5}}{R_{4}}\right) ^{i} \\< & {} 2^{\frac{m}{2}}k_{2}\frac{(m)_{Dn}}{(D_{n})!}\underset{i=0}{\overset{ \infty }{\sum }A_{i}}(R_{3})\left| \pi _{ni}\right| \left( \frac{R_{5}}{ R_{4}}\right) ^{i} \\< & {} 2^{\frac{m}{2}}k_{2}\frac{(m)_{D_{n}}}{(D_{n})!}N_{n}\overset{\infty }{ \max _{i=0}}A_{i}(R_{3})\left| \pi _{ni}\right| \\< & {} 2^{\frac{m}{2}}k_{3}\frac{(m)_{D_{n}}}{(D_{n})!}N_{n}F_{n}(R_{3}) \\< & {} 2^{\frac{m}{2}}k_{4}\frac{(m)_{D_{n}}}{(D_{n})!}N_{n}R_{2}^{n}. \end{aligned}$$

Consequently, one can get the cannon sum in the form

$$\begin{aligned} \bar{F}_{n}(aR_{5})= & {} \underset{i,j}{\max }\underset{\left| x\right| =(aR_{5})}{\sup }\left| \overset{j}{\sum _{k=i}}\overline{P} _{k}(x)p_{nk}\right| \\\le & {} 2^{\frac{m}{2}}\underset{i,j}{\max }\underset{k=i}{\overset{j}{\sum }} \bar{A}_{k}(aR_{5})\left| p_{nk}\right| \\< & {} 2^{\frac{m}{2}}\max _{i,j}\overset{j}{\sum _{k=i}}2^{\frac{m}{2}}k_{4}\frac{ (m)_{D_{k}}}{(D_{k})!}N_{k}R_{2}^{k}\left| p_{nk}\right| . \end{aligned}$$

Relying on \(\hbox {Cauchy}^{,}\hbox {s}\) inequality (c.f. [2]) for the special monogenic polynomials \(P_{n}(x)\), we infer that

$$\begin{aligned} \left| p_{nk}\right| \le \sqrt{\frac{n!}{(m)_{k}}}\frac{A_{n}(R_{1}) }{R_{1}^{k}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \bar{F}_{n}(aR_{5})< & {} 2^{m}k_{4}\overset{\infty }{\underset{k=0}{\max }} \frac{(m)_{D_{n}}}{(D_{k})!}N_{k}R_{2}^{k}\sqrt{\frac{n!}{(m)_{k}}}\frac{ A_{n}(R_{1})}{R_{1}^{k}} \\< & {} 2^{m}k_{4}N_{n}\frac{(m)_{n}}{n!}\sqrt{\frac{n!}{(m)_{n}}}A_{n}(R_{1})\left( \frac{R_{2}}{R_{1}}\right) ^{k},\forall n>D_{n},R_{2}<R_{1} \\< & {} k_{5}A_{n}(R_{1}). \end{aligned}$$

Then \(\bar{\kappa }(aR_{5})\le \mu (R_{1})\). Hence, by (10) one can see \(\bar{\kappa }(aR_{5})<aR\).

This is true for all \(aR_{5}<aR,\), and therefore, the theorem is established. \(\square \)

Conclusion We can now summarizes the particular cases given by different authors:

Since effectiveness in \(\bar{B}(R^{+}),\) reduces to effectiveness in \(\bar{B }(R)\), when \(D_{(n)}=O(n),\) we have

(i)Theorem 1 covers Abul-Ez’s result in [3] when the base \(\{P_{n}(x)\}\) of special monogenic polynomials is a simple monic one, which in its turn is an extension of Mursi-Makar’s result [35] to Clifford setting.

(ii)Theorem 1 is considered to be a generalization of Mikhail’s result in complex case [35] to the Clifford setting, where \(\left| p_{nn}\right| =O\left( n^{\hslash }\right) ,~\hslash \) is finite.

(iii) Theorem 1 covers Abul-Ez’s result in [4] which in its turn is an extension of Halim’s result [27] where the base is simple and \(\left| p_{nn}\right| ^{\frac{1}{n}}\longrightarrow a\) as \(n\longrightarrow \infty \).

For effectiveness at the origin , Theorem 1 with \(R=0\) generalizes the following results to the Clifford setting:

(iv)Ewida’s result [23] in which the base is a simple monic one.

(v)Nassif’s result [37] in which the base is a general one satisfying

$$\begin{aligned} 0<\alpha r\le \nu (r)\le \mu (r)\le \beta (r)<\infty . \end{aligned}$$

Theorem 1 also is considered to be an extension to Clifford analysis of:

(vi)Tantawi’s result [41] in which the base is a general one satisfying

$$\begin{aligned} \nu (R)=\mu (R)=aR. \end{aligned}$$

Remark 2

When the effectiveness in \(\bar{B}(R^{+})\), strict effectiveness in \(\bar{B} (R)\), Theorem 2 generalizes Tantawi’s work [41] in that respect.

4 The Order of the Inverse Axially Monogenic Base

The order \(\omega \) of the base \(\{P_{n}(x)\}\) given by the expression in (5) is of essential importance in that a base of order \(\omega \) represents every entire axially monogenic function of order less than \(\frac{1}{\omega }\) (see [1]).

From this point of view we estimate order \(\overline{\omega }\) of the inverse base as it is stated in the following result:

Theorem 3

Let \(\{P_{n}(x)\}\) be a simple base of polynomials of order \(\omega \) and such that

$$\begin{aligned} \frac{\log \left| p_{nn}\right| }{n\log n}\longrightarrow 0~\ \ \ \ \text {as}~\ \ \ n\longrightarrow \infty . \end{aligned}$$
(12)

Then the order \(\varpi \) of the inverse base is such that

$$\begin{aligned} \frac{\omega }{2}\le \varpi \le 2\omega . \end{aligned}$$

The upper and lower bounds are attainable.

To prove the theorem we make use of the following lemma given in ([5], p.124).

Lemma 1

If \(\{P_{n}(x)\}\) is a base of special monogenic polynomials, such that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim \sup }\frac{D_{n}}{n}\longrightarrow a~\ \ (a\ge 1). \end{aligned}$$

Then its order is given by

$$\begin{aligned} \omega =\underset{n\rightarrow \infty }{\lim \sup }\frac{\log \underset{i,j}{ \max }\{\left| p_{ij}\right| \left| p_{ni}\right| \}}{n\log n } \end{aligned}$$

where \(D_{n}\) is the largest degree of any \(P_{i}(x)\) in the expansion (1).

Proof

(Theorem 3) From the above lemma we have for the inverse base

$$\begin{aligned} \varpi= & {} \underset{n\rightarrow \infty }{\lim \sup }\frac{\log \underset{i,j }{\max }\{\left| \overline{p}_{ij}\right| \left| p_{ni}\right| \}}{n\log n}~\ \ \ \ \ \ ,\ p_{ni}=\pi _{ni} \\= & {} \underset{n\rightarrow \infty }{\lim \sup }\frac{\log \left| \overline{p}_{i(n),j(n)}\right| \left| p_{n,i(n)}\right| }{n\log n} \\= & {} \underset{n\rightarrow \infty }{\lim \sup }\frac{\log \left| \overline{p}_{i(n),j(n)}\right| \left| p_{j(n),j(n)}\right| }{ n\log n}+\underset{n\rightarrow \infty }{\lim }\sup \frac{\log \left| \overline{p}_{n,n}\right| \left| p_{n,i(n)}\right| }{n\log n}, \end{aligned}$$

since \(\pi _{nn}=\bar{p}_{nn},\) and \(p_{nn}\) satisfies (12).

Now using the above lemma and noting that \(j(n)\le i(n)\le n\), one can get \(\overline{\omega }\le \omega +\omega =2\omega .\)

But the inverse base \(\{\bar{p}_{n}(x)\}\) is such that

$$\begin{aligned} \frac{\log \left| \pi _{nn}\right| }{n\log n}\rightarrow 0~\ \ \text { as}~\ \ n\rightarrow \infty . \end{aligned}$$

Hence, by what we have just proved \(\omega \le 2\varpi \). \(\square \)

5 Conclusion

(i) Abul-Ez [3] has proved the above result, but with \(p_{nn}=1\), by an entirely different method, so our result here is considered to give a generalization of that in [3].

(ii) The above elegant proof of Theorem 3 is the analogue of the one given by Mikhail [33] using the condition \(p_{nn}=o(n^{\hslash }),\hslash \) finite, in place of (12), so that our result here extends Mikhail’s one are in [33] to the Clifford setting.

(iii) The proof given for Theorem 3 covers also bases satisfying (12).

The fact that the lower and upper bounds of the order \(\varpi \) are attainable, even when the base \(\{P_{n}(x)\}\) is monic is illustrated by the following example (see \(\left[ 14\right] \)). Consider the simple monic base \(\{P_{n}(x)\}\) given by:

$$\begin{aligned} P_{4n}(x)= & {} z_{4n}(x) \\ P_{4n+1}(x)= & {} \mu _{n}z_{4n}(x)+z_{4n+1}(x) \\ P_{4n+2}(x)= & {} \mu _{n}z_{4n}(x)+z_{4n+1}(x)+z_{4n+2}(x) \\ P_{4n+3}(x)= & {} \mu _{n}z_{4n}(x)+z_{4n+1}\left( x\right) +\mu _{n}z_{4n+2}\left( x\right) +z_{4n+3}\left( x\right) \end{aligned}$$

where \(\mu _{n}=\left( 4n+1\right) ^{\frac{\omega }{2\left( 4n+1\right) } \text {. }},\)\(n=0,1,2,\ldots \). Then \(\left\{ P_{n}\left( x\right) \right\} \) is of order \(\omega \), whilst the inverse base \(\left\{ \overline{P}_{n}\left( x\right) \right\} \) given by:

$$\begin{aligned} \overline{P}_{4n}\left( x\right)= & {} z_{4n}\left( x\right) \\ \overline{P}_{4n+1}\left( x\right)= & {} -\mu _{n}z_{4n}\left( x\right) +z_{4n+1}\left( x\right) \\ \overline{P}_{4n+2}\left( x\right)= & {} -z_{4n+1}\left( x\right) +z_{4n+2}\left( x\right) \\ \overline{P}_{4n+3}\left( x\right)= & {} [\mu _{n}-1]z_{4n+1}\left( x\right) -\mu _{n}z_{4n+2}\left( x\right) +z_{4n+3}\left( x\right) \end{aligned}$$

is of order \(\overline{\omega }=\frac{\omega }{2}\).

The case of non-simple bases has been considered, originally in complex setting by Tantawi [41] and Mikhail [33], but their attempts did not lead to any interesting results, and consequently the generalization to the Clifford case is an open question to get such example for more general base, to attain the two bounds as stated in the above theorem.