A Novel Approach to Polynomial Approximation in Multidimensional Cylindrical Domains via Generalized Kronecker Product Bases

Abstract

The Kronecker product has been commonly seen in various scientific fields to formulate higher-dimensional spaces from lower-dimensional ones. This paper presents a generalization of the Cannon–Kronecker product bases by introducing generalized Kronecker product bases of polynomials within an analytic framework. It investigates the convergence behavior of infinite series formed by these generalized products in various polycylindrical domains, including both open and closed configurations. The paper also delves into essential analytic properties such as order, type, and the ${\mathbb{T}}_{\rho }$-property to analyze the growth and structural characteristics of these bases. Moreover, the theoretical insights are applied to a range of classical special functions, notably Bernoulli, Euler, Gontcharoff, Bessel, and Chebyshev polynomials.

1. Introduction

The Kronecker product of matrices is a powerful and versatile tool in linear algebra, with profound implications across numerous areas of mathematics, engineering, and computational sciences. Originally introduced by the German mathematician Leopold Kronecker in the 19th century, the Kronecker product enables the construction of large structured matrices from smaller ones and facilitates the representation and analysis of complex systems in a compact, algebraically elegant form. The Kronecker product is not merely a theoretical construct but finds extensive applications in real-world problems, system theory and control engineering [1], quantum mechanics [2], image processing and signal analysis [3,4,5], statistics and econometrics [6,7], system identification [8,9], system theory [10,11], matrix calculus [12,13,14,15], including matrix equations [16,17], approximation theory [18] and several other areas [19].

An important area of approximation theory is the study of polynomial bases, a concept initially introduced by Whittaker and Cannon [20,21,22,23,24]. This theory has evolved along three principal directions. The first focuses on the convergence properties of polynomial bases in several complex variables, particularly within hyper-spherical, hyper-elliptical, and polycylindrical domains [25,26,27,28,29,30,31]. The second extends this study to the field of Clifford analysis, emphasizing convergence in hyper-spherical regions [32,33,34,35]. The third approach investigates the approximation of analytic functions in Fréchet spaces using polynomial bases constructed from complex conformable fractional derivatives [36,37,38].

It should be emphasized that the convergence properties, such as effectiveness, order, type, and the $T_{\rho }$-property of various other products of polynomial bases, have been the subject of extensive study. Notably, these include the usual product bases [39,40] Hadamard product bases [41,42,43,44], similar bases [45], and equivalent bases [46]. In practical applications, special functions are frequently expressed in terms of polynomial bases. The convergence properties of polynomial bases related to Bernoulli and Euler functions, Bessel polynomials, Chebyshev polynomials, and the Gontcharoff polynomial basis have been thoroughly investigated (see, for example, [47,48,49,50,51]).

Recent advancements have highlighted the versatility of the Kronecker product in constructing high-dimensional analytical frameworks from lower-dimensional structures. In 2025, a significant study [52] examined the representation of analytic functions over hyper-elliptical regions using infinite series formed from Cannon–Kronecker product polynomial bases. The authors of [52] not only analyzed the growth order, type, and $T_{\rho }$-property of entire functions expressed in this form, but also extended convergence results to Cannon–Kronecker product bases involving special functions such as Bernoulli, Euler, Gontcharoff, Bessel, and Chebyshev polynomials. These findings broaden the theoretical foundation previously established in hyper-spherical domains, offering deeper insights into the structure and behavior of function representations in more generalized geometric settings.

In this paper, we extend the classical concept of Cannon–Kronecker product bases by developing an analytic framework for generalized Kronecker product bases of polynomials (GKPBPs). Our approach involves a rigorous study of the convergence behavior of associated infinite series within various polycylindrical domains. Additionally, we investigate important analytic characteristics of these bases, including their order, type, and the ${\mathbb{T}}_{\rho }$-property, which are essential for understanding their functional and structural properties. Moreover, we study the effectiveness properties of the generalized Kronecker product associated with exponential bases and algebraic bases in polycylindrical domains. Furthermore, we demonstrate the applicability of these generalized bases by establishing connections with several classes of special functions, notably Bessel, Euler, Bernoulli, Chebyshev, and Gontcharoff polynomials.

The structure of this paper is as follows. Section 2 presents key definitions, notations, and previously established results that are utilized throughout the study. In Section 3, we define the generalized Kronecker product of certain bases and show that it forms a valid basis. Section 4 studies the convergence properties of the GKPBPs in closed polycylinders. We provide conditions under which these bases are effective, assuming they are constructed via general polynomial bases that are known to be effective in closed circles for one complex variable. In Section 5, we study the effectiveness of GKPBPs in open polycylinders and closed regions. Section 6 examines the effectiveness of the generalized Kronecker product of exponential bases in closed polycylinders. The effectiveness criteria of the generalized Kronecker product of algebraic bases in closed polycylinders are characterized in Section 7. Section 8 determines the order, type and ${\mathbb{T}}_{\rho }$-property of these GKPBPs. Finally, Section 9 concludes the paper.

2. Notations and Basic Results

In this section, we recall some main results from [25,26,28] which are required throughout this study. We have $\mathbf{z}\in {\mathbb{C}}^{\ell }$, that is $\mathbf{z}=z_1,z_2,\dots z_{\ell }$ are complex variables and ${\mathbf{z}}^{\mathbf{k}}={\prod }_{s=1}^{\ell }{z}_s^{k_s}$, where the modulus of $\mathbf{z}$ is

$${\left|\mathbf{z}\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2+\dots +{\left|z_{\ell }\right|}^2.$$

Let $\mathbf{k}=k_1,k_2,\dots ,k_{\ell },$ are non-negative integers, (in short $k_s,s\in I=\{1,2,\dots ,\ell \}$), ${\mathbf{k}}_{\mathrm{\Sigma }}=k_1+k_2+\dots k_{\ell }$ and $\left[\alpha \mathbf{r}\right]=\left[{\alpha }_{1}r,{\alpha }_{2}r,\dots ,{\alpha }_{\ell }r\right].$

The study of approximating complex functions in terms of bases of polynomials is one of the significant topics in analysis where functions can be represented as infinite series of simpler polynomial terms, facilitating theoretical and practical insights.

Definition 1.

Let

$$\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}=\left\{{\mathcal{G}}_0\left[\mathbf{z}\right],{\mathcal{G}}_1\left[\mathbf{z}\right],\dots ,{\mathcal{G}}_{\mathbf{n}}\left[\mathbf{z}\right],\dots \right\}$$

be a set of polynomials in the variables $z_s$ for $s\in I$. We say that $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is a base (or basis) if every polynomial in the variables $z_s,s\in I$ can be uniquely expressed as a finite linear combination of the elements of the set $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$. Equivalently, the set $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is a base if and only if there exists a unique row-finite matrix $\mathcal{G}=\left\{{\mathcal{G}}_{\mathbf{k},\mathbf{h}}\right\}$, called the coefficient matrix, and its inverse ${\mathcal{G}}^{-1}=\left\{{\mathcal{G}}_{\mathbf{k},\mathbf{h}}^{-1}\right\}$, called the operator matrix, such that

$$\mathcal{G}{\mathcal{G}}^{-1}={\mathcal{G}}^{-1}\mathcal{G}=\mathcal{I},$$

(1)

where $\mathcal{I}$ is the identity matrix. Accordingly, for the base $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]& =\sum _{\mathbf{h}}{\mathcal{G}}_{\mathbf{k},\mathbf{h}}{\mathbf{z}}^{\mathbf{h}},\hfill \\ \hfill {\mathbf{z}}^{\mathbf{k}}& =\sum _{\mathbf{h}}{\mathcal{G}}_{\mathbf{k},\mathbf{h}}^{-1}{\mathcal{G}}_{\mathbf{h}}\left[\mathbf{z}\right].\hfill \end{array}\end{array}$$

(2)

Let $f\left(\mathbf{z}\right)={\displaystyle \sum _{\mathbf{k}}}{a}_{\mathbf{k}}{\mathbf{z}}^{\mathbf{k}}$ be any regular function at the origin $\mathbf{0}$. Using (2), we substitute for ${\mathbf{z}}^{\mathbf{k}}$ and rearrange the terms to obtain the following:

$$f\left(\mathbf{z}\right)=\sum _{\mathbf{k}}{\mathrm{\Pi }}_{\mathbf{k}}{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right],{\mathrm{\Pi }}_{\mathbf{k}}=\sum _{\mathbf{h}}{\mathcal{G}}_{\mathbf{h},\mathbf{k}}^{-1}{a}_{\mathbf{h}},$$

which represents the associated basic series of the function $f\left(\mathbf{z}\right)$.

In the space ${\mathbb{C}}^{\ell }$ of the complex variables $z_s,s\in I$, an open polycylinder of radii $r_s>0,s\in I$ denoted by ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ and its closure by ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. Moreover, $D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)$ denotes an unspecified domain containing the closed polycylinder. In terms of the introduced notations, the regions ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]},$ ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]},$ ${\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}$ and $D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)$ satisfy the following:

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}& ={\mathrm{\Gamma }}_{r_1,r_2,\dots ,r_{\ell }}=\left\{\mathbf{z}\in {\mathbb{C}}^{\ell }:\left|z_s\right|<r_s,s\in I\right\},\hfill \\ \hfill {\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}& ={\overline{\mathrm{\Gamma }}}_{r_1,r_2,\dots ,r_{\ell }}=\left\{\mathbf{z}\in {\mathbb{C}}^{\ell }:\left|z_s\right|\le r_s,s\in I\right\},\hfill \\ \hfill {\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}& ={\overline{\mathrm{\Gamma }}}_{r,r,\dots ,r},\hfill \\ \hfill D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)& =D\left({\overline{\mathrm{\Gamma }}}_{r_1^{+},r_2^{+},\dots ,r_{\ell }^{+}}\right)=\left\{\mathbf{z}\in {\mathbb{C}}^{\ell }:\left|z_s\right|\le r_s^{+},s\in I\right\}.\hfill \end{array}$$

Definition 2.

Let ${\displaystyle \sum _{\mathbf{k}}}{\mathrm{\Pi }}_{\mathbf{k}}{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]$ be the associated basic series. We say that this series represents the function $f\left(\mathbf{z}\right)$ in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ or $D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)$, if the series converges uniformly to $f\left(\mathbf{z}\right)$ in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ or in some polycylinder containing ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, respectively.

Definition 3

([26,28]). The base $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is said to be effective in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ or $D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)$ if the associated basic series represents every regular function in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ or in some polycylinder containing ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, respectively.

Definition 4

([30]). Let $N_{\mathbf{k}}=N_{k_1,k_2,\dots ,k_{\ell }}$ denote the number of non-zero coefficients in the representation (2). A base $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ of polynomials is called a Cannon base, if $N_{\mathbf{k}}$ satisfies that

$$\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim}{\left\{N_{\mathbf{k}}\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}=1.$$

If $\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim}{\left\{N_{\mathbf{k}}\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}=a,$ where $a>1$, then the base $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is called a general base of polynomials (GBP).

To examine the convergence properties of such bases, we use the following notations as indicated below:

$$\begin{array}{cc}\hfill M\left({\mathcal{G}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)& =\underset{{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{sup}\left|{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right|,\hfill \\ \hfill \psi ({\mathcal{G}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})& =\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\underset{i,j,{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|\sum _{\mathbf{h}=i}^j{\mathcal{G}}_{\mathbf{k},\mathbf{h}}^{-1}{\mathcal{G}}_{\mathbf{h}}\left(\mathbf{z}\right)\right|,\hfill \\ \hfill \chi \left(\mathcal{G},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)& =\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left\{\psi \left({\mathcal{G}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}.\hfill \end{array}$$

Regarding the effectiveness of the GBP $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ in polycylindrical regions, the following results have been established [25,41].

Theorem 1.

Let $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ be a GBP. Then

(i) 

$\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ iff $\chi \left(\mathcal{G},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)={\displaystyle \prod _{s=1}^{\ell }}{r}_s.$

(ii) 

$\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ iff $\chi \left(\mathcal{G},{\overline{\mathrm{\Gamma }}}_{\left[\rho \right]}\right)<\alpha \left(\left[\mathbf{r}\right],\left[\rho \right]\right)$ where

$$\alpha \left(\left[\mathbf{r}\right],\left[\rho \right]\right)=max\left\{r_1\prod _{s=2}^{\ell }{\rho }_s,r_{\nu }\prod _{\begin{array}{c}s=1\\ s\ne \nu \end{array}}^{\ell }{\rho }_s,r_{\ell }\prod _{s=1}^{\ell -1}{\rho }_s\right\}.$$

(iii) 

$\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in $D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)$ iff

$$\chi \left(\mathcal{G},D\left({\overline{\mathrm{\Gamma }}}_{\left.[{\mathbf{r}}^{+}\right]}\right)\right)={\displaystyle \prod _{s=1}^{\ell }}{r}_s.$$

3. The Generalized Kronecker Product Bases of Polynomials

The Kronecker product of matrices of arbitrary size was introduced over any ring (see [53,54,55]). It maps the matrices $X\in {\mathbb{C}}^{i\times j}$ and $Y\in {\mathbb{C}}^{s\times t}$ into the matrix $Z=X\otimes Y\in {\mathbb{C}}^{is\times jt}$. Unlike the standard matrix product, which requires that $j=s$ unless one operand is a scalar, the Kronecker product is defined for matrices of any size. The following properties are true for the Kronecker product.

(1)

$X\otimes Y\otimes Z=(X\otimes Y)\otimes Z=X\otimes (Y\otimes Z);$

(2)

$(X\otimes Y)(Z\otimes W)=XZ\otimes YW$ if $XZ$ and $YW$ exist;

(3)

If X and Y are non-singular, then ${(X\otimes Y)}^{-1}=X^{-1}\otimes Y^{-1}$;

(4)

${\mathcal{I}}_m\otimes {\mathcal{I}}_n={\mathcal{I}}_{mn}.$

If $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\};s\in I$ are GBPs of one complex variables $z_s,s\in I$, the product set:

$${\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]=\underset{s=1}{\overset{\ell }{⨂}}{\mathcal{G}}_{s,k_s}\left(z_s\right)$$

(3)

defines the GKPBPs of the several complex variables $z_s,s\in I$ where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]& =\sum _{\mathbf{h}}{\mathcal{K}}_{\mathbf{k},\mathbf{h}}{\mathbf{z}}^{\mathbf{h}},\hfill \\ \hfill {\mathbf{z}}^{\mathbf{k}}& =\sum _{\mathbf{h}}{\mathcal{K}}_{\mathbf{k},\mathbf{h}}^{-1}{\mathcal{K}}_{\mathbf{h}}\left[\mathbf{z}\right].\hfill \end{array}\end{array}$$

Note that $\mathcal{K}=\left({\mathcal{K}}_{\mathbf{k},\mathbf{h}}\right)$ denotes the coefficient matrix and ${\mathcal{K}}^{-1}=\left({\mathcal{K}}_{\mathbf{k},\mathbf{h}}^{-1}\right)$ denotes the operator matrix of the GKPBPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ for which

$$\begin{array}{cc}\hfill \mathcal{K}& ={\mathcal{G}}_1\otimes {\mathcal{G}}_2\dots \otimes {\mathcal{G}}_{\ell },\hfill \\ \hfill {\mathcal{K}}^{-1}& ={\mathcal{G}}_1^{-1}\otimes {\mathcal{G}}_2^{-1}\dots \otimes {\mathcal{G}}_{\ell }^{-1},\hfill \end{array}$$

Here, ${\mathcal{G}}_s$ and ${\mathcal{G}}_s^{-1},s\in I$, denote the coefficient matrix and the corresponding operator matrix of the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, respectively.

Now, we prove that the general Kronecker product set $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is indeed a base of polynomials. Observe that

$$\begin{array}{cc}\hfill \mathcal{K}{\mathcal{K}}^{-1}& =\left({\mathcal{G}}_1\otimes {\mathcal{G}}_2\dots \otimes {\mathcal{G}}_{\ell }\right)\left({\mathcal{G}}_1^{-1}\otimes {\mathcal{G}}_2^{-1}\dots \otimes {\mathcal{G}}_{\ell }^{-1}\right)\hfill \\ \hfill & ={\mathcal{G}}_1{\mathcal{G}}_1^{-1}\otimes {\mathcal{G}}_2{\mathcal{G}}_2^{-1}.\otimes {\mathcal{G}}_{\ell }{{\mathcal{G}}_{\ell }}^{-1}\hfill \\ \hfill & =\mathcal{I}\otimes \mathcal{I}\otimes \dots \otimes \mathcal{I}=\mathcal{I}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathcal{K}}^{-1}\mathcal{K}& =\left({\mathcal{G}}_1^{-1}\otimes {\mathcal{G}}_2^{-1}\dots \otimes {\mathcal{G}}_k^{-1}\right)\left({\mathcal{G}}_1\otimes {\mathcal{G}}_2\dots \otimes {\mathcal{G}}_k\right)\hfill \\ \hfill & ={\mathcal{G}}_1^{-1}{\mathcal{G}}_1\otimes {\mathcal{G}}_2^{-1}{\mathcal{G}}_2.\otimes {\mathcal{G}}_k^{-1}{\mathcal{G}}_k\hfill \\ \hfill & =\mathcal{I}\otimes \mathcal{I}\otimes \dots \otimes \mathcal{I}=\mathcal{I}.\hfill \end{array}$$

According to (1), the general Kronecker product set is a base.

In this study, we will provide a comprehensive investigation of the convergence properties of the constructed GKPBPs through the following scenarios:

• Will the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ be effective in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ if its constituent bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ are effective in the closed discs $|z_s|\le r_s$ where $s\in I$?
• Will the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ be effective in ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ if its constituent bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ are effective in the open discs $|z_s|<r_s$ where $s\in I$?
• Will the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ be effective in $D\left({\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{+}\right]}\right)$ if its constituent bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ are effective in $|z_s|<r_s^{+}$ where $s\in I$?
• How are the orders, types and the ${\mathbb{T}}_{\rho }$-property of the constituent bases related to the order, type and the ${\mathbb{T}}_{\rho }$-property of the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$?
• Are our results applicable to specific special functions like Gontcharoff, Chebyshev, Bessel, Bernoulli, and Euler?

4. Effectiveness of GKPBPs in Closed Polycylinder

Let $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ be GBP of a single complex variable. Then the monomials $z_s^{k_s},\in I$ admit the unique finite representations

$$z_s^{k_s}=\sum _{j_s}{\mathcal{G}}_{s,k_s,j_s}^{-1}{\mathcal{G}}_{j_s}\left(z_s\right),s\in I,$$

where ${\mathcal{G}}_s^{-1}=\left({\mathcal{G}}_{s,k_s,j_s}^{-1}\right)$ and ${\mathcal{G}}_s=\left({\mathcal{G}}_{s,k_s,j_s}\right)$ are the matrices of operators and coefficients of the base $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, $s\in I$, respectively. In this section, we examine the effectiveness criterion of the GKPBPs in closed polycylinders ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]},r_s>0,s\in I.$

Theorem 2.

Let $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ be the GKPBPs of the several complex variables $z_s$, $s\in I$ whose constituents are the GBPs $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$. Then $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]},r_s>0,s\in I,$ if and only if $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, $s\in I$ are effective in $\left|z_s\right|\le r_s,s\in I$, respectively.

Proof. 

Suppose that $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is the GKPBP of the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$. Then as in [52], any monomial ${\mathbf{z}}^{\mathbf{k}}={\displaystyle \prod _{s=1}^{\ell }}{z}_s^{k_s},k_s\ge 0$, $s\in I$, admits a unique finite representation of the form

$${\mathbf{z}}^k=\sum _{\mathbf{j}}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]& =\underset{s=1}{\overset{\ell }{⨂}}{\mathcal{G}}_{s,k_s}\left(z_s\right),\hfill \\ \hfill {\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}& =\prod _{s=1}^{\ell }{\mathcal{G}}_{s,k_s,j_s}^{-1}.\hfill \end{array}\end{array}$$

(4)

Suppose that $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, s $\in I$ are effective in $\left|z_s\right|\le r_s,s\in I$. Then

$$\chi \left({\mathcal{G}}_s,r_s\right)=\underset{k_s\to \infty }{lim\; sup}{\left\{\psi \left({\mathcal{G}}_{s,k_s},r_s\right)\right\}}^{{\displaystyle \frac{1}{k_s}}}=r_s,s\in I$$

(5)

where

$$\psi \left({\mathcal{G}}_{s,k_s},r_s\right)=\underset{{\alpha }_s,{\beta }_s,\left|z_s\right|=r_s}{max}\left|\sum _{j_s={\alpha }_s}^{{\beta }_s}{\mathcal{G}}_{s,k_s,j_s}^{-1}{\mathcal{G}}_{s,j_s}\left(z_s\right)\right|.$$

(6)

Observe that $\chi \left({\mathcal{G}}_s,r_s\right)$ and $\psi \left({\mathcal{G}}_{s,k_s},r_s\right),s\in I$ are the Cannon functions and Cannon sums of the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$. Thus, according to (5), there exists a constant $K>1$ and a positive finite integer $M_s$ such that

$$\psi \left({\mathcal{G}}_{s,k_s},r_s\right)<K{r}_s^{k_s},k_s>M_s,s\in I.$$

(7)

Let $f\left[\mathbf{z}\right]={\displaystyle \sum _{\mathbf{k}}}{a}_{\mathbf{k}}{\mathbf{z}}^{\mathbf{k}}$ be any function that is regular in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. Then

$$\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left\{\prod _{s=1}^{\ell }{R}_s^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_k\right|\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}={\displaystyle \prod _{s=1}^{\ell }}{R}_s^{-1}$$

(8)

where $R_s$, $s\in I$ are positive numbers such that $R_s>r_s$, $s\in I$. Write

$$u_{\mathbf{k}}\left[\mathbf{r}\right]=\underset{{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|a_{\mathbf{k}}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]\right|.$$

Owing to (4), (6)–(8), we obtain

$$\begin{array}{cc}\hfill u_{\mathbf{k}}\left[\mathbf{r}\right]& =\underset{\left|z_s\right|=r_s}{max}\left|a_{\mathbf{k}}\prod _{s=1}^{\ell }{\mathcal{G}}_{s,k_s,h_s}^{-1}{\mathcal{G}}_{s,h_s}\left(z_s\right)\right|\hfill \\ \hfill & \le \left\{\prod _{s=1}^{\ell }{R}_s^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_{\mathbf{k}}\right|\right\}\left\{\prod _{s=1}^{\ell }{R}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\psi \left({\mathcal{G}}_{s,k_s},r_s\right)\right\}\hfill \\ \hfill & <K\left\{\prod _{s=1}^{\ell }{R}_s^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_{\mathbf{k}}\right|\right\}\left\{\prod _{s=1}^{\ell }{R}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}{r}_s^{k_s}\right\}\hfill \\ \hfill & =K\left\{\prod _{s=1}^{\ell }{R}_s^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_{\mathbf{k}}\right|\right\}{\left\{r_1{R}_2{R}_3\dots R_{\ell }\right\}}^{k_1}{\left\{R_1{r}_2{R}_3\dots R_{\ell }\right\}}^{k_2}\dots {\left\{R_1{R}_2\dots R_{\ell -1}{r}_{\ell }\right\}}^{k_{\ell }}\hfill \\ \hfill & \le K\left\{\prod _{s=1}^{\ell }{R}_s^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_{\mathbf{k}}\right|\right\}max\left\{{\left(r_1{R}_2{R}_3\dots R_{\ell }\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}},{\left(R_1{r}_2{R}_3\dots R_{\ell }\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}},\dots ,{\left(R_1{R}_2\dots R_{\ell -1}{r}_{\ell }\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}}\right\}\hfill \end{array}$$

for ${\mathbf{k}}_{\mathrm{\Sigma }}>{max}_s{M}_s$. Hence using (8), it follows that

$$\begin{array}{cc}\hfill \underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left\{u_{\mathbf{k}}\left[\mathbf{r}\right]\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}& \le \left\{\prod _{s=1}^{\ell }{R}_s^{-1}\right\}max\left\{r_1{R}_2{R}_3\dots R_{\ell },R_1{r}_2{R}_3\dots R_{\ell },\dots ,R_1{R}_2\dots R_{\ell -1}{r}_{\ell }\right\}\hfill \\ \hfill & <1.\hfill \end{array}$$

Since the series ${\displaystyle \sum _{\mathbf{k}}}{u}_{\mathbf{k}}\left[\mathbf{r}\right]$ majorities in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ the series ${\displaystyle \sum _{\mathbf{k}}}{a}_{\mathbf{k}}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]$. Then the latter is absolutely and uniformly convergent in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. Moreover, it can be easily seen that the series ${\displaystyle \sum _{\mathbf{k}}}{a}_{\mathbf{k}}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}$ is bounded and convergent for all $\mathbf{j}$.

Now, choose the positive integers $N_i,i=1,2,3,4$ such that ${\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}=0$ for all $\mathbf{j}>N_2$; this is possible since the matrix ${\mathcal{G}}^{-1}$ is row-finite, and for given ${\epsilon }_1,{\epsilon }_2$ we have

$$\underset{{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|\sum _{\mathbf{k}\ge N_3}{a}_{\mathbf{k}}{z}^{\mathbf{k}}\right|<{\displaystyle \frac{{\epsilon }_1}{2}},\underset{{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|\sum _{\mathbf{k}\ge N_4}{a}_{\mathbf{k}}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]\right|<{\displaystyle \frac{{\epsilon }_2}{2}}.$$

This follows directly from the convergence of the two series in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. Thus,

$${\mathbf{z}}^{\mathbf{k}}=\sum _{\mathbf{j}=0}^{N_2}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right].$$

Consider the following difference:

$$\begin{array}{cc}\hfill \underset{{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|f\left[\mathbf{z}\right]-\sum _{\mathbf{j}=0}^{N_2}\sum _{\mathbf{k}=0}^{\infty }{a}_{\mathbf{k}}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]\right|& =\underset{{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|\sum _{\mathbf{k}\ge N_1}{a}_{\mathbf{k}}{z}^{\mathbf{k}}-\sum _{\mathbf{k}\ge N_1}{a}_{\mathbf{k}}\sum _{\mathbf{j}=0}^{N_2}{\mathcal{K}}_{\mathbf{k},\mathbf{j}}^{-1}{\mathcal{K}}_{\mathbf{j}}\left(\mathbf{z}\right)\right|\hfill \\ \hfill & <{\displaystyle \frac{{\epsilon }_1}{2}}+{\displaystyle \frac{{\epsilon }_2}{2}}\left(1+N_2\right)<\epsilon ,\hfill \end{array}$$

(9)

for $\epsilon \ge max\left\{{ϵ}_1,{ϵ}_2\left(1+N_2\right)\right\}$.

Thus, the basic series represents in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ the function $f\left[\mathbf{z}\right]$, which is regular there. Since the function $f\left[\mathbf{z}\right]$ is arbitrarily chosen, we infer that the GKPBP $\left\{{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]\right\}$ is effective in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$.

To prove the other direction of the theorem, suppose that the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in the polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. Then it follows that

$$\begin{array}{cc}\hfill {\displaystyle \prod _{s=1}^{\ell }}{r}_s& =\chi (\mathcal{K},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})=\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left\{\psi ({\mathcal{K}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}\hfill \\ \hfill & \ge \underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left\{\left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]\prod _{s=1}^{\ell }\underset{j_s,s\in I}{max}\underset{\left|z_s\right|=r_s}{max}\left|{\mathcal{G}}_{s,k,j_s}^{-1},{\mathcal{G}}_{s,j_s}\left(z_s\right)\right|\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}\hfill \\ \hfill & \ge \underset{k_1\to \infty }{lim\; sup}{\left\{\left[\prod _{s=2}^{\ell }{r}_s^{k_1}\right]\underset{j_1}{max}\underset{\left|z_1\right|=r_1}{max}\left|{\mathcal{G}}_{1,k_1,j_1}^{-1}{\mathcal{G}}_{1,j_1}\left(z_1\right)\right|\left[\prod _{s=2}^{\ell }\left|{\mathcal{G}}_{s,0,0}^{-1}{\mathcal{G}}_{s,0}\left(0\right)\right|\right]\right\}}^{{\displaystyle \frac{1}{k_1}}}\hfill \\ \hfill & =\underset{k_1\to \infty }{lim\; sup}{\left\{\left[\prod _{s=2}^{\ell }{r}_s^{k_1}\right]\varphi \left({\mathcal{G}}_{1,k_1},r_1\right)\left[\prod _{s=2}^{\ell }\left|{\mathcal{G}}_{s,0,0}^{-1}{\mathcal{G}}_{s,0}\left(0\right)\right|\right]\right\}}^{{\displaystyle \frac{1}{k_1}}}\hfill \\ \hfill & =\prod _{s=2}^{\ell }{r}_s\varphi \left({\mathcal{G}}_1,r_1\right).\hfill \end{array}$$

(10)

Let ${\displaystyle \varphi \left({\mathcal{G}}_{1,k_1},r_1\right)=\underset{j_1}{max}\underset{\left|z_1\right|=r_1}{max}\left|{\mathcal{G}}_{1,k_1,j_1}^{-1}{\mathcal{G}}_{1,j_1}\left(z_1\right)\right|.}$ Accordingly, it follows that

$$\begin{array}{cc}\hfill \varphi \left({\mathcal{G}}_1,r_1\right)& =\underset{k_1\to \infty }{lim\; sup}{\left\{\varphi \left({\mathcal{G}}_{1,k_1},r_1\right)\right\}}^{{\displaystyle \frac{1}{k_1}}}\hfill \\ \hfill & =\underset{k_1\to \infty }{lim\; sup}{\left\{\underset{j_1}{max}\underset{\left|z_1\right|=r_1}{max}\left|{\mathcal{G}}_{1,k_1,j_1}^{-1},{\mathcal{G}}_{1,j_1}\left(z_1\right)\right|\right\}}^{{\displaystyle \frac{1}{k_1}}}\hfill \\ \hfill & \le r_1\hfill \end{array}$$

(11)

Write

$$u\left({\mathcal{G}}_{1,k_1},r_1\right)=\left|a_{k_1}^{\left(1\right)}\right|\varphi \left({\mathcal{G}}_{1,k_1},r_1\right)$$

(12)

and suppose that $f_1\left(z_1\right)={\displaystyle \sum _{k_1}}{a}_{k_1}^{\left(1\right)}{z}_1^{k_1}$ is regular in $\left|z_1\right|\le r_1$, then its radius of regularity $R_1$ will be greater than $r_1$. Hence, by combining (11) and (12), we obtain

$$\underset{k_1\to \infty }{lim\; sup}{\left\{u\left({\mathcal{G}}_{1,k_1},r_1\right)\right\}}^{{\displaystyle \frac{1}{k_1}}}\le {\displaystyle \frac{r_1}{R_1}}<1,0<r_1<R_1.$$

(13)

Therefore, by applying the Weierstrass M-test for uniform convergence, it follows that the series ${\displaystyle \sum _{k_1=0}^{\infty }}u\left({\mathcal{G}}_{1,k_1},r_1\right)$ and ${\displaystyle \sum _{k_1}}{a}_{k_1}^{\left(1\right)}{\mathcal{G}}_{1,k_1,j_1}^{-1}{\mathcal{G}}_{1,j_1}\left(z_1\right)$ are absolutely and uniformly convergent in the disk $\left|z_1\right|\le r_1$. Following similar steps as in (9), the basic series ${\displaystyle \sum _{k_1}}{a}_{k_1}^{\left(1\right)}{\displaystyle \sum _{j_1}}{\mathcal{G}}_{1,k_1,j_1}^{-1}{\mathcal{G}}_{1,j_1}\left(z_1\right)$ associated with the base $\left\{{\mathcal{G}}_{1,k_1}\left(z_1\right)\right\}$ converges uniformly to $f_1\left(z_1\right)$ in the disk $\left|z_1\right|\le r_1$. Consequently, the base $\left\{{\mathcal{G}}_{1,k_1}\left(z_1\right)\right\}$ is effective in the disk $\left|z_1\right|\le r_1$.

Moreover, we can proceed in a manner similar to the above to show that each of the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s=2,3,\dots ,\ell$ is effective in the disks $\left|z_s\right|\le r_s$, $s=2,3,\dots ,\ell$ when the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in ${\overline{\mathrm{\Gamma }}}_{\mathbf{r}]}$. Therefore, we completely establish the result. □

In the following examples, we illustrate the efficiency and applicability of Theorem 2 where the constituents of GKPBPs involve certain bases of polynomials.

Example 1.

The base of polynomials

$${\mathcal{G}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{cc}1,\hfill & k_{s}iseven,\hfill \\ 1+z_s^{k_s}+{\displaystyle \frac{z_s^{2k_s}}{4^{k_s}}},\hfill & k_{s}isodd\forall s\in I,\hfill \end{array}\right.$$

is effective in $|z_s|\le r_s,1\le r_s\le 4,s\in I$. Applying Theorem 2, we conclude that the Kronecker product bases

$$\left\{{\mathcal{K}}_{\mathbf{K}}^{\left(\mathcal{G}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$$

are also effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, $1\le r_s\le 4,s\in I$.

Example 2.

The base of polynomials

$${\mathcal{G}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{cc}{z}_s^{k_s},\hfill & k_{s}iseven,\hfill \\ z_s^{k_s}+z_s^{2k_s},\hfill & k_{s}isodd\forall s\in I,\hfill \end{array}\right.$$

is effective in $|z_s|\le r_s,r_s\le 1,s\in I$. Applying Theorem 2, we conclude that the Kronecker product bases

$$\left\{{\mathcal{K}}_{\mathbf{K}}^{\left(\mathcal{G}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$$

are also effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, $r_s\le 1,s\in I$.

Example 3.

The base of polynomials

$${\mathcal{G}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{cc}1,\hfill & k_s=0,\hfill \\ z_s^{k_s}-a_s{z}_s^{k_s-1},\hfill & k_s\ge 1,\forall s\in I,\hfill \end{array}\right.$$

where $a_s>0,$ is effective in $|z_s|\le r_s,r_s\ge a_s,s\in I$. Applying Theorem 2, we conclude that the Kronecker product bases

$$\left\{{\mathcal{K}}_{\mathbf{K}}^{\left(\mathcal{G}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$$

are also effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, $r_s\ge a_s,s\in I$.

The following example examines the case in which the constituents of GKPBPs involve certain well-known special polynomials.

Example 4.

(1)> 

As deduced [48,49], the base of the proper Bessel polynomials

$${\mathcal{F}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{cc}1,\hfill & k_s=0,\hfill \\ {\displaystyle \sum _{h_s=0}^{k_s}}{\displaystyle \frac{\left(k_s+h_s\right)!}{h_s!\left(k_s-h_s\right)!}}{\left({\displaystyle \frac{z_s}{2}}\right)}^{h_s},\hfill & k_s\ge 1,\forall s\in I,\hfill \end{array}\right.$$

and also the base of the general Bessel polynomials, where $a_s$ and $b_s$ are given numbers,

$${\mathcal{H}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{cc}1,\hfill & k_s=0,\hfill \\ 1+{\displaystyle \sum _{h_s=1}^{k_s}}{\displaystyle \frac{k_s!{\displaystyle \prod _{j_s=1}^{h_s}}\left(k_s+j_s+a_s-2\right)}{h_s!\left(k_s-h_s\right)!}}{\left({\displaystyle \frac{z_s}{b_s}}\right)}^{h_s},\hfill & k_s\ge 1,\forall s\in I,\hfill \end{array}\right.$$

are effective in $|z_s|\le r_s$ for all $r_s>0$. We can conclude, directly applying Theorem 2, that the Kronecker product bases constructed by these bases

$$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{F}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{F}}_{s,k_s}\left(z_s\right)\right\}$$

and

$$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{H}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{H}}_{s,k_s}\left(z_s\right)\right\}$$

are effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ for all $r_s>0,s\in I$.

(2) 

In [50], the Chebyshev polynomials

$${\mathcal{T}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{cc}1,\hfill & k_s=0\hfill \\ {\displaystyle \sum _{h_s=0}^{\left[{\displaystyle \frac{k_s}{2}}\right]}}{\displaystyle \frac{\left(k_s!\right.}{2h_s!\left(k_s-2h_s\right)!}}{z}_s^{k_s-2h_s}{\left(z_s^2-1\right)}^{h_s},\hfill & k_s\ge 1,\forall s\in I\hfill \end{array}\right.$$

are proved to be effective in $|z_s|<1,s\in I$. By applying Theorem 2, the Kronecker product bases

$$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{T}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{T}}_{s,k_s}\left(z_s\right)\right\}$$

are also effective in the closed equi-polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{1}\right]}$.

(3)> 

In [47], the Gontcharoff base of polynomials $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, associated with a given set $\left({\alpha }_s{t}_s^{k_s}\right)$ of points, was given in the form

$$\left\{\begin{array}{c}{\mathcal{G}}_{s,0}\left(z_s\right)=1,\hfill \\ {\mathcal{G}}_{s,k_s}(z_s;{\alpha }_s,{\alpha }_s{t}_s,{\alpha }_s{t}_s^2,\dots ,{\alpha }_s{t}_s^{k_s-1})=\underset{{\alpha }_s}{\overset{z_s}{\int }}d{l}_1^s\underset{{\alpha }_s{t}_s}{\overset{l_1^s}{\int }}d{l}_2^s\underset{{\alpha }_s{t}_s^2}{\overset{l_2^s}{\int }}d{l}_3^s\dots \underset{{\alpha }_s{t}_s^{k_s-1}}{\overset{l_{k_s-1}^s}{\int }}d{l}_{k_s}^{{⠀}^s},k_s\ge 1\hfill \end{array}\right.$$

where ${\alpha }_s$ and $t_s$ are given complex numbers. For the case when $|t_s|<1$, the authors [47] proved that the Gontcharoff polynomials $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ are effective in $|z_s|\le r_s$ for all $r_s\ge \left|{\alpha }_s\right|.$ Applying Theorem 2, we conclude that the Kronecker product bases

$$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{G}\right)}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$$

are effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ for all $r_s\ge \left|{\alpha }_s\right|.$

5. Effectiveness of GKPBPs in Open Polycylinders and Closed Regions

This section characterizes the effectiveness of GKPBPs in open polycylinders and closed regions.

Theorem 3.

The GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in the open polycylinder ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ if and only if the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ are effective in $\left|z_s\right|<r_s,s\in I$, respectively.

Proof. 

Suppose that the GBPs $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ are effective in $\left|z_s\right|<r_s,s\in I$, respectively. Then from the properties of Cannon functions, it follows that

$$\chi \left({\mathcal{G}}_s,{\rho }_s\right)<r_s\forall {\rho }_s<r_s,s\in I.$$

Thus, there exist positive numbers $b_s<1$ such that ${\rho }_s<b_s{r}_s$ and a constant $K>1$ such that

$$\psi \left({\mathcal{G}}_{s,k_s},{\rho }_s\right)<K{\left(b_s{r}_s\right)}^{k_s},k_s\ge 0.$$

(14)

Hence,

$${\displaystyle \prod _{s=1}^{\ell }}\psi \left({\mathcal{G}}_{s,k_s},{\rho }_s\right)<K{b}^{{\mathbf{k}}_{\mathrm{\Sigma }}}{\displaystyle \prod _{s=1}^{\ell }}{\left(r_s\right)}_s^{k_s},k_s\ge 0,s\in I,$$

(15)

where $b={max}_s\{b_s:s\in I\}$. Now, suppose that $f\left[\mathbf{z}\right]={\displaystyle \sum _{\mathbf{k}}}{a}_{\mathbf{k}}{\mathbf{z}}^{\mathbf{k}}$ is any function regular in the polycylinder ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$. Then, it follows that

$$\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left[\prod _{s=1}^{\ell }{\left(r_s\right)}^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_k\right|\right]}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}\le {\displaystyle \prod _{s=1}^{\ell }}{r}_s^{-1}.$$

(16)

Using (15) and (16), we get

$$\begin{array}{cc}\hfill \underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left[\underset{{\overline{\mathrm{\Gamma }}}_{\left[\rho \right]}}{max}\left|a_k{\mathcal{K}}_{k,j}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]\right|\right]}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}& \le \underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left[\left|a_k\right|\prod _{s=1}^{\ell }\psi \left({\mathcal{G}}_{s,k_s},{\rho }_s\right)\right]}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}\hfill \\ \hfill & =\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\left[\left\{\prod _{s=1}^{\ell }{\left(r_s\right)}^{-\left({\mathbf{k}}_{\mathrm{\Sigma }}-k_s\right)}\left|a_k\right|\right\}\left\{\prod _{s=1}^{\ell }{\left(r_s\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\psi \left({\mathcal{G}}_{s,k_s},{\rho }_s\right)\right\}\right]}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}\hfill \\ \hfill & <1.\hfill \end{array}$$

Therefore, the series ${\displaystyle \sum _{\mathbf{k}}}{a}_k{\mathcal{K}}_{k,j}^{-1}{\mathcal{K}}_{\mathbf{j}}\left[\mathbf{z}\right]$ is absolutely and uniformly convergent in ${\overline{\mathrm{\Gamma }}}_{\left[\rho \right]}$, which means that the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$.

To prove the only if statement of the theorem, suppose that the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]},R_s,s\in I$ are any numbers such that $0<R_s<r_s,s\in I$ and $r_i^{\left(s\right)},s\in I,⠀i=0,1,2,\dots$ are chosen such that

$$r_{i+1}^{\left(s\right)}={\displaystyle \frac{1}{2}}\left(r_s+r_i^{\left(s\right)}\right),i=0,1,2,\dots ,s\in I,$$

where

$$0<r_0^{\left(s\right)}<r_s,{\displaystyle \frac{r_0^{\left(s\right)}}{r_0^{\left(j\right)}}}={\displaystyle \frac{r_s}{r_j}},s,j\in I.$$

Thus, it follows that

$${\displaystyle \frac{r_1^{\left(s\right)}}{r_1^{\left(j\right)}}}={\displaystyle \frac{r_s}{r_j}},s,j\in I,i=0,1,2,\dots$$

Therefore, $R_s<r_1^{\left(s\right)}<r_s,s\in I,i=0,1,2,\dots ,$ and

$$r_1{r}_1^{\left(2\right)}{r}_1^{\left(3\right)}\dots r_1^{(\ell )}=r_1^{\left(1\right)}{r}_1^{\left(2\right)}\dots r_1^{(\nu -1)}{r}_v{r}_1^{(\nu +1)}\dots r_1^{(\ell )}=r_1^{\left(1\right)}{r}_1^{\left(2\right)}\dots r_1^{(\ell -1)}{r}_{\ell },$$

where $v=2,3,\dots ,(\ell -1)$. Using the effectiveness of the GKPBPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$, it follows that

$$\begin{array}{cc}\hfill \chi (\mathcal{K},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{R}\right]})& <\chi \left(\mathcal{K},r_1^{\left(1\right)},r_1^{\left(2\right)},\dots ,r_1^{(\ell )}\right)\hfill \\ \hfill & =r_1^{\left(1\right)}{r}_1^{\left(2\right)}\dots r_1^{(\ell -1)}\alpha \hfill \\ \hfill & <r_1^{\left(1\right)}{r}_1^{\left(2\right)}\dots r_1^{(\ell -1)}{r}_J^{(\ell )},0<r_1^{(\ell )}<\alpha <r_j^{(\ell )}<r_{\ell }.\hfill \end{array}$$

(17)

Thus, it follows, as in (10), by using (17), that

$$\prod _{s=2}^{\ell }{r}_1^{\left(s\right)}\chi \left({\mathcal{G}}_1,r_1^{\left(1\right)}\right)\le \chi \left(\mathcal{K},r_1^{\left(1\right)},r_1^{\left(2\right)},\dots ,r_1^{(\ell )}\right)<r_1^{\left(1\right)}{r}_1^{\left(2\right)}\dots r_1^{(\ell -1)}{r}_j^{(\ell )},$$

which leads to

$$\chi \left({\mathcal{G}}_1,r_1^{\left(1\right)}\right)<r_1^{\left(1\right)}{\displaystyle \frac{r_j^{(\ell )}}{r_i^{(\ell )}}}=r_1^{\left(1\right)}{\displaystyle \frac{r_j^{\left(1\right)}}{r_i^{\left(1\right)}}}=r_j^{\left(1\right)}.$$

Since $R_1<r_i^{\left(1\right)}<r_j^{\left(1\right)}<r_1$, we obtain

$$\chi \left({\mathcal{G}}_1,R_1\right)<\chi \left({\mathcal{G}}_1,r_i^{\left(1\right)}\right)<r_j^{\left(1\right)}<r_1$$

(18)

for all $R_1<r_1$.

Thus, if $f_1\left(z_1\right)={\displaystyle \sum _{k_1}}{a}_{k_1}^{\left(1\right)}{z}_1{⠀}^k$ is any function regular in $\left|z_1\right|<r_1$, then we can easily see as in (13) by using (18) that

$$\underset{k_1\to \infty }{lim\; sup}{\left\{u\left({\mathcal{G}}_{1,k_1},R_1\right)\right\}}^{1/k_1}<1,$$

(19)

which means that the series ${\displaystyle \sum _{k_1}}u\left({\mathcal{G}}_{1,k_1},R_1\right)<\infty$ and the series ${\displaystyle \sum _{k_1}}{a}_1^{\left(1\right)}{\mathcal{G}}_{1,k_1,i_1}^{-1}{\mathcal{G}}_{1,i_1}\left(z_1\right)$ converge absolutely and uniformly in $\left|z_1\right|\le R_1<r_1$. Therefore, the base $\left\{{\mathcal{G}}_{1,k_1}\left(z_1\right)\right\}$ is effective in $\left|z_1\right|<r_1$. Now, using the same procedure we can easily prove that the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s=2,3,\dots ,\ell$ will be effective in $\left|z_s\right|<r_s,s=2,3\dots ,\ell$, respectively, when the GKPBP $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$. Hence, Theorem 3 is established. □

Proceeding in a similar way as before, we can prove the following result.

Theorem 4.

The GKPBPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ will be effective in the closed region $D\left(\left[\overline{\mathrm{\Gamma }}\left[{\mathbf{r}}^{+}\right]\right]\right)$ if and only if the GBP $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ is effective in $\left|z_s\right|\le r_s^{+},s\in I,$ respectively.

6. Effectiveness of the General Kronecker Product of Exponential Bases of Polynomials in Closed Polycylinders

Let $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ be a GBP in the several complex variables $z_s$, where $s\in I$, with associated coefficient matrices and operators denoted by $\mathcal{G}$ and ${\mathcal{G}}^{-1}$, respectively. Consider the associated set $\left\{{⠀}_{\mathcal{G}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ whose matrices of coefficients and operator are $e^{\mathcal{G}}$ and $e^{{\mathcal{G}}^{-1}}$. Thus, $e^{\mathcal{G}}{e}^{{\mathcal{G}}^{-1}}=e^0=I$ and the set $\left\{{⠀}_{\mathcal{G}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is a base. This set is defined here as the exponential base of polynomials.

When $\left\{{⠀}_{{\mathcal{G}}_s}{E}_{k_s}\left(z_s\right)\right\},s\in I$ are the exponential bases associated with the general bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, then the product

$$\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{⠀}_{{\mathcal{G}}_s}{E}_{k_s}\left(z_s\right)\right\}$$

defines the general Kronecker product of exponential bases of polynomials (GKPEBP).

Now, we consider the following restriction:

$$\left|{\mathcal{G}}_{s,k_s,h_s}\right|\le M{a}_s^{k_s-h_s}{b}_s^{h_s},0<b_s<1,s\in I$$

(20)

and $a_s$ are positive numbers.

The elements of the power matrices of coefficients ${\mathcal{G}}_s^{\left(j_s\right)}$ of the GBP $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ are

$${\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}=\sum _{{\nu }_s=0}^{\infty }{\mathcal{G}}_{s,k_s,{\nu }_s}{\mathcal{G}}_{s,{\nu }_s,h_s}^{(j_s-1)};j_s\ge 1,s\in I.$$

(21)

Theorem 5.

The GKPEBPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ of the several complex variables $z_s,s\in I$ will be effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ for all $a_s\le r_s<{\displaystyle \frac{a_s}{b_s}},s\in I$, when the general constituent bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ accord to the restriction (20) and may not be effective in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ for all $r_{\mathrm{s}}<a_{\mathrm{s}}$ or $r_{\mathrm{s}}\ge {\displaystyle \frac{a_s}{b_s}},s\in I$.

Proof. 

In the assumption that ${\mathcal{G}}_s^{\left(0\right)}=\mathcal{I}$, $\mathcal{I}$ is the unit matrix. Applying (20) in (21), we get the following inequality:

$$\left|{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}\right|\le M^{j_s}{\left(1-b_s\right)}^{-(j_s-1)}{a}_s^{k_s-h_s}{b}_s^{h_s}.$$

(22)

Using (22) implies

$$\begin{array}{cc}\hfill M\left({⠀}_{{\mathcal{G}}_s}{E}_{k_s},r_s\right)& \le \sum _{h_s=0}^{\infty }\sum _{j_s=0}^{\infty }{\displaystyle \frac{\left|{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}\right|}{j_s!}}{r}_s^{h_s}\hfill \\ \hfill & =\sum _{h_s=0}^{\infty }\left|{\mathcal{G}}_{s,k_s,h_s}^{\left(0\right)}\right|r_s^{h_s}+\sum _{h_s=0}^{\infty }\sum _{j_s=1}^{\infty }{\displaystyle \frac{\left|{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}\right|}{j_s!}}{r}_s^{h_s}\hfill \\ \hfill & \le \left|{\mathcal{G}}_{s,k_s,k_s}^{\left(0\right)}\right|r_s^{k_s}+\sum _{h_s=0}^{\infty }\sum _{j_s=1}^{\infty }{\displaystyle \frac{M^{j_s}}{j_s!}}{\left(1-b_s\right)}^{-(j_s-1)}{a}_s^{k_s-h_s}{\left(b_s{r}_s\right)}^{h_s}\hfill \\ \hfill & =\left[1+\left(e^{M{\left(1-b_s\right)}^{-1}}-1\right)\left(1-b_s\right)\left\{\sum _{h_s=0}^{k_s}{\left({\displaystyle \frac{a_s}{r_s}}\right)}^{k_s-h_s}{b}_s^{h_s}+\sum _{h_s>k_s}^{\infty }{\left({\displaystyle \frac{a_s}{r_s}}\right)}^{k_s}{\left({\displaystyle \frac{b_s{r}_s}{a_s}}\right)}^{h_s}\right\}\right]r_s^{k_s}\hfill \\ \hfill & \le \left[1+\left(e^{M{\left(1-b_s\right)}^{-1}}-1\right)\left(1-b_s\right)\left\{{\left(1-b_s\right)}^{-1}+{\left(1-{\displaystyle \frac{b_s{r}_s}{a_s}}\right)}^{-1}\right\}\right]r_s^{k_s}\hfill \\ \hfill & =K_s{r}_s^{k_s}\le K{r}_s^{k_s}\hfill \end{array}$$

(23)

for all $a_s\le r_s<{\displaystyle \frac{a_s}{b_s}},s\in I,K={max}_{s\in I}{K}_s.$ Using (20) and (23) in the Cannon sum, $\psi \left({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)$, of the GKPEBPs, $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$, yields

$$\begin{array}{cc}\hfill \psi \left({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)& ={\displaystyle \prod _{s=1}^{\ell }}{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\underset{\alpha ,\beta ,{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|\sum _{\mathbf{h}=\alpha }^{\beta }{⠀}_{\mathcal{K}}{E}_{\mathbf{k},\mathbf{h}}^{-1}{⠀}_{\mathcal{K}}{E}_{\mathbf{h}}\left[\mathbf{z}\right]\right|\hfill \\ \hfill & \le {\displaystyle \prod _{s=1}^{\ell }}{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\sum _{\mathbf{h}=0}^{\infty }\left|{⠀}_{\mathcal{K}}{E}_{\mathbf{k},\mathbf{h}}^{-1}\right|M\left({⠀}_{\mathcal{K}}{E}_{\mathbf{h}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)\hfill \\ \hfill & \le \left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]\left[\prod _{s=1}^{\ell }\sum _{h_s=0}^{\infty }\sum _{j_s}^{\infty }{\displaystyle \frac{\left|{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}\right|}{j_s!}}\right]M\left({⠀}_{{\mathcal{G}}_s}{E}_{h_s},r_s\right)\hfill \\ \hfill & \le K\left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]\left[\prod _{s=1}^{\ell }\sum _{h_s=0}^{\infty }\sum _{j_s=0}^{\infty }{\displaystyle \frac{|{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}|}{j_s!}}{r}_s^{h_s}\right]\hfill \\ \hfill & <K\left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}}\right]\left[\prod _{s=1}^{\ell }{K}_s\right]\hfill \\ \hfill & <K^{\ell +1}{\displaystyle \prod _{s=1}^{\ell }}{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}}\hfill \end{array}$$

(24)

for all $a_s\le r_s<{\displaystyle \frac{a_s}{b_s}},s\in I.$ Thus, the Cannon function for the GKPEBPs $\left\{{⠀}_{\mathcal{K}}{\mathrm{E}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is given by

$$\chi ({⠀}_{\mathcal{K}}{\mathrm{E}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})=\underset{\mathbf{k}\to \infty }{lim\; sup}{\left\{\psi \left({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)\right\}}^{{\displaystyle \frac{1}{{\mathbf{k}}_{\mathrm{\Sigma }}}}}\le {\displaystyle {\displaystyle \prod _{s=1}^k}}{r}_s,a_s\le r_s\le {\displaystyle \frac{a_s}{b_s}}$$

but $\chi ({⠀}_{\mathcal{K}}{\mathrm{E}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})\ge {\displaystyle {\displaystyle \prod _{s=1}^k}}{r}_s.$ Therefore, $\chi ({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})={\displaystyle \prod _{s=1}^k}{r}_s$, $a_s\le r_s\le {\displaystyle \frac{a_s}{b_s}}$ and the GKPEBP $\left\{{⠀}_{\mathcal{K}}{\mathrm{E}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is effective in the polycylinder ${\mathrm{\Gamma }}_{\left[\mathbf{r}\right]}$ for all $a_s\le r_s\le {\displaystyle \frac{a_s}{b_s}}$.

When $r_s<a_s$ or $r_s\ge {\displaystyle \frac{a_s}{b_s}}$ for some $s\in I$, the GKPEBPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ may fail to be effective in the polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. We illustrate this with the following example.

Example 5. Consider the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, $s\in I$, constructed as follows:

$${\mathcal{G}}_{s,k_s}\left(z_s\right)=\left\{\begin{array}{c}1,k_s=0\hfill \\ {\displaystyle \sum _{h_s=0}^s}M{a}^{k_s-h_s}{b}^{h_s}{z}_s^{h_s},k_s\ne 0,s\in I\hfill \end{array}\right.$$

(25)

Now, the Cannon sum of the GKPEBPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ for the equi-polyclyinder ${\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}$ will be such that

$$\begin{array}{cc}\hfill \psi \left({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}\right)& =r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\underset{\alpha ,\beta ,{\overline{\mathrm{\Gamma }}}_{\left[{\mathbf{r}}^{*}\right]}}{max}\left|\sum _{\mathbf{h}=\alpha }^{\beta }{⠀}_{\mathcal{K}}{E}_{\mathbf{k},\mathbf{h}}^{-1}{⠀}_{\mathcal{K}}{E}_{\mathbf{h}}\left[\mathbf{z}\right]\right|\hfill \\ \hfill & \ge r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\left|{⠀}_{\mathcal{K}}{E}_{\mathbf{k},0}^{-1}\right|M\left({⠀}_{\mathcal{K}}{E}_0,{\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}\right)\hfill \\ \hfill & =r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\prod _{s=1}^{\ell }\left|{⠀}_{{\mathcal{G}}_s}{E}_{k_s,0}^{-1}\right|M\left({⠀}_{{\mathcal{G}}_s}{E}_0,r\right)\hfill \\ \hfill & \ge r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\prod _{s=1}^{\ell }\left|\sum _{j_s=1}^{\infty }{(-1)}^{j_s}{\displaystyle \frac{{\mathcal{G}}_{s,k_s}^{\left(j_s\right)}}{j_s!}}\right|\left|{\left\{e^{{\mathcal{G}}_s}\right\}}_{0,0}\right|\hfill \\ \hfill & ={\left(a{r}^{\ell -1}\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}}\prod _{s=1}^{\ell }\left(e^{-\ell {(1-b)}^{-1}}-1\right)\left(e^{M{(1-b)}^{-1}}-1\right){\left(1-b\right)}^2\hfill \end{array}$$

Thus, $\chi \left({⠀}_{\mathcal{K}}E,{\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}\right)\ge a{r}^{\ell -1}>r^k$ for $r<a$, and the GKPEBPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is not effective in ${\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}$ for $r<a$.

Moreover, applying (25) in the Cannon sum of the GKPEBPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$, it follows that

$$\begin{array}{cc}\hfill \psi \left({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}\right)& \ge r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\left|{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}^{-1},\mathbf{k}\right|M\left({⠀}_{\mathcal{K}}{E}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}}\right)\hfill \\ \hfill & =r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\prod _{s=1}^{\ell }\left|{⠀}_{{\mathcal{G}}_s}{E}_{k_s,k_s}^{-1}\right|M\left({⠀}_{{\mathcal{G}}_s}{E}_{k_s},r\right)\hfill \\ \hfill & \ge r^{{\mathbf{k}}_{\mathrm{\Sigma }}(\ell -1)}\prod _{s=1}^{\ell }\left|\sum _{j_s=1}^{\infty }{(-1)}^{j_s}{\displaystyle \frac{{\mathcal{G}}_{s,k_s,k_s}^{\left(j_s\right)}}{j_s!}}\right|\left|{\left\{e^{{\mathcal{G}}_s}\right\}}_{k_s,k_s{\mathbf{k}}_{\mathrm{\Sigma }}}\right|r^{k_s{\mathbf{k}}_{\mathrm{\Sigma }}}\hfill \\ \hfill & \ge K{\left(b{r}^{(k-1)}\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}}{\left(br\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}^2}{a}^{{\mathbf{k}}_{\mathrm{\Sigma }}e-{\mathbf{k}}_{\mathrm{\Sigma }}^2}\hfill \\ \hfill & \ge K{\left[b{\left({\displaystyle \frac{\mathrm{Na}}{\mathrm{b}}}\right)}^{k-1}\right]}^{{\mathbf{k}}_{\mathrm{\Sigma }}}{\left[b{\displaystyle \frac{\mathrm{Na}}{\mathrm{b}}}\right]}^{{\mathbf{k}}_{\mathrm{\Sigma }}^2}{a}^{{\mathbf{k}}_{\mathrm{\Sigma }}-{\mathbf{k}}_{\mathrm{\Sigma }}^2}\hfill \\ \hfill & =K{\left({\displaystyle \frac{a^k}{b^{k-2}}}\right)}^{{\mathbf{k}}_{\mathrm{\Sigma }}}{N}^{{\mathbf{k}}_{\mathrm{\Sigma }}-{\mathbf{k}}_{\mathrm{\Sigma }}^2},r={\displaystyle \frac{Na}{b}},N>1\hfill \end{array}$$

Thus,

$$\chi \left({⠀}_{\mathcal{K}}E,{\left[{\displaystyle \frac{Na}{b}}\right]}^{*}\right)\ge {\displaystyle \frac{a^k}{b^{k-2}}}\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim}{N}^{1+{\mathbf{k}}_{\mathrm{\Sigma }}}=\infty ,$$

which implies that the GKPEBPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathrm{z}\right]\right\}$ is not effective in ${\overline{\mathrm{\Gamma }}}_{{\left[\mathbf{r}\right]}^{*}},r>{\displaystyle \frac{a}{b}}.$ Hence, the result is fully established. □

7. Effectiveness of the General Kronecker Product of Algebraic Bases of Polynomials in Closed Polycylinders

The base of polynomials $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is called algebraic of degree j when its matrix of coefficients $\mathcal{G}$ satisfies the usual identity of least degree

$${\alpha }_0{\mathcal{G}}^j+{\alpha }_1{\mathcal{G}}^{j-1}+\dots +{\alpha }_{j}I=0$$

Thus, we have a relation of the form

$${{\mathcal{G}}^{-1}}_{\mathbf{k},\mathbf{h}}={\delta }_{\mathbf{k},\mathbf{h}}{\gamma }_0+\sum _{s=1}^{j-1}{\gamma }_s{\mathcal{G}}_{\mathbf{k},\mathbf{h}}^{\left(s\right)}$$

where ${\mathcal{G}}_{\mathbf{k},\mathbf{h}}^{\left(s\right)}$ denote the elements of the power matrix ${\mathcal{G}}^{\left(s\right)}$ and ${\gamma }_s,s=0,1,2,\dots ,j-1,$ are constant numbers.

When $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ are the algebraic general bases, then the product

$$\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}=\underset{s=1}{\overset{\ell }{⨂}}\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$$

defines the general Kronecker product of algebraic bases of polynomials (GKPABP).

Theorem 6.

The GKPABPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$, whose constituents are all algebraic general bases of polynomials will be effective in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ for all $a_s<r_s<{\displaystyle \frac{a_s}{b_s}},s\in I$ when the elements of the matrices of coefficients $\left({\mathcal{G}}_{s,k_s,h_s}\right),s\in I$ of the general constituent bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ satisfy (20) and may not be effective in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ for all $r_s<a_s$ or $r_s\ge {\displaystyle \frac{a_s}{b_s}},s\in I.$

Proof. 

Suppose that the GBPs $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ are algebraic and obey the condition (20), then there exists a representation of the form:

$${\mathcal{G}}_{s,k_s,h_s}^{-1}=\sum _{j_s=0}^{N_s}{C}_{j_s}{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}$$

(26)

where $C_j,j_s=0,1,2,\dots ,N_s$ are constants and ${\mathcal{G}}_s^{-1}=\left({\mathcal{G}}_{s,k_s,h_s}^{-1}\right)$ is the matrix of operators of the base $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$. Now, since $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ satisfy the condition (20), then we have

$$\begin{array}{cc}\hfill M\left({\mathcal{G}}_{s,k_s},r_s\right)& =\underset{\left|z_s\right|=r_s}{max}\left|\sum _{h_s=0}^{\infty }{\mathcal{G}}_{s,k_s,h_s}{z}_s^{h_s}\right|\hfill \\ \hfill & \le M\sum _{b=0}^{\infty }{a}_s^{k_s-h_s}{\left(b_s{r}_s\right)}^{h_s}\hfill \\ \hfill & <M\left[\left({\left(1-b_s\right)}^{-1}+{\left(1-{\displaystyle \frac{b_s{r}_s}{a_s}}\right)}^{-1}\right)\right]r_s^m\hfill \\ \hfill & <K{r}^{k_s}\hfill \end{array}$$

(27)

for all $a_s<r_s<{\displaystyle \frac{a_s}{b_s}}.$ A combination among (22), (24), (26) and (27) in the Cannon sum of the GKPABPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ yields

$$\begin{array}{cc}\hfill \psi \left({\mathcal{K}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)& ={\displaystyle \prod _{s=1}^{\ell }}{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\underset{\alpha ,\beta ,{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}}{max}\left|\sum _{\mathbf{h}=\alpha }^{\beta }{\mathcal{K}}_{\mathbf{k},\mathbf{h}}^{-1}{\mathcal{K}}_{\mathbf{h}}\left[\mathbf{z}\right]\right|\hfill \\ \hfill & \le {\displaystyle \prod _{s=1}^{\ell }}{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\sum _{h=0}^{\infty }\left|{\mathcal{K}}_{\mathbf{k},\mathbf{h}}^{-1}\right|M\left({\mathcal{K}}_{\mathbf{h}},\left[r\right]\right)\hfill \\ \hfill & \le \left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]\left[\prod _{s=1}^{\ell }\sum _{h_s=0}^{\infty }\left|{\mathcal{G}}_{s,k_s,h_s}^{-1}\right|M\left({\mathcal{G}}_{s,h_s,r_s}\right)\right]\hfill \\ \hfill & \le \left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]\left[\prod _{s=1}^{\ell }\sum _{h_s=0}^{\infty }\sum _{j_s=0}^{N_s}\left|C_{j_s}{\mathcal{G}}_{s,k_s,h_s}^{\left(j_s\right)}\right|M\left({\mathcal{G}}_{s,h_s,r_s}\right)\right]\hfill \\ \hfill & \le K\left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]\left[\prod _{s=1}^{\ell }\sum _{h_s=0}^{\infty }\sum _{j_s=0}^{N_s}\left|C_{j_s}\right|M^{j_s}{(1-b_s)}^{-(j_s-1)}{a}_s^{k_s-h_s}{\left(b_s{r}_s\right)}^{h_s}\right]\hfill \\ \hfill & =K\left[\prod _{s=1}^{\ell }{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}-k_s}\right]{\displaystyle \prod _{s=1}^{\ell }}C(N_s+1)M^{N_s}{(1-b_s)}^{-(N_s-1)}\sum _{h_s=0}^{\infty }{a}_s^{k_s-h_s}{b}_s^{h_s}{r}_s^{h_s}\hfill \\ \hfill & \le K{\displaystyle \prod _{s=1}^{\ell }}{r}_s^{{\mathbf{k}}_{\mathrm{\Sigma }}},\hfill \end{array}$$

where $C={max}_{0\le j_s\le N_s,s\in I}\left|C_{j_s}\right|.$ Thus, the Cannon function $\chi (\mathcal{K},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})$ of the set $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ will accord to the following inequality

$$\chi (\mathcal{K},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})\le {\displaystyle {\displaystyle \prod _{s=1}^k}}{x}_s$$

for all $a_s\le r_s<{\displaystyle \frac{a_s}{b_s}},s\in I$. Therefore, the theorem is deduced.

If $r_s<a_s$ or $r_s\ge {\displaystyle \frac{a_s}{b_s}}$ for some $s\in I$, the GKPABPs $\left\{{⠀}_{\mathcal{K}}{E}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ may not be effective in the polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$. To illustrate this, we refer to Example 6.1 and proceed using analogous steps. □

8. Mode of Increase and the ${\mathbb{T}}_{\mathbf{\rho }}$-Property of GKPBPs

We determine the order and type of the GKPBPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ in relation with the constituent GBPs $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$. Suppose that the GBPs $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ are of respective increase orders $\left(f_s\right)$ and types $\left(g_s\right)$. Moreover, let the $\left\{{\mathcal{G}}_{1,k_1}\left(z_1\right)\right\}$ have the greatest rate of increase. In other words, we consider either $f_1>f_2,f_3,\dots ,f_k$ or $f_1=f_2=f_3=\dots =f_k$ and $g_1>g_2>g_3>\dots >g_k$ or $g_1=g_2=g_3=\dots =g_k$. We now evaluate the order F and type G of the GKPBPs in terms of the increase in the constituents.

Definition 5.

The order F of the base of polynomials ${\mathcal{G}}_{\mathbf{K}}\left[\mathbf{z}\right]$ in the closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\alpha \mathbf{r}\right]}$ is given by

$$F=\underset{r\to \infty }{lim}\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\displaystyle \frac{log\psi \left({\mathcal{G}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\alpha \mathbf{r}\right]}\right)}{{\mathbf{k}}_{\mathrm{\Sigma }}log{\mathbf{k}}_{\mathrm{\Sigma }}}}.$$

If $0<F<\infty$, then the type G of the base of polynomial is given by

$$G=\underset{r\to \infty }{lim}{\displaystyle \frac{e}{F}}\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\displaystyle \frac{{\left\{\psi \left({\mathcal{G}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\alpha \mathbf{r}\right]}\right)\right\}}^{\frac{1}{\mathrm{\Omega }{\mathbf{k}}_{\mathrm{\Sigma }}}}}{{\mathbf{k}}_{\mathrm{\Sigma }}}}.$$

Remark 1.

Note that when the GBP ${\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]$ is of finite order F and finite type G, then it represents every entire function of order less than ${\displaystyle \frac{1}{F}}$ and type less than ${\displaystyle \frac{1}{G}}$ in any finite polycylinder.

Theorem 7.

Let $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$, $s\in I$ be GBPs of respective increase orders $f_s$ and types $g_s$, where $\left\{{\mathcal{G}}_{1,k_1}\left(z_1\right)\right\}$ has the greater rate of increase. Then the order F of the GKPBPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ equals $f_1$ and the type G is determined as follows:

(i) 

If $f_1\ge {\displaystyle \frac{1}{2}}$, then $G=g_1$.

(ii) 

If $f_1<{\displaystyle \frac{1}{2}}$, then $g_1\le G\le g_1{2}^{{\displaystyle \frac{1}{2f_1}}-1}.$

Proof. 

The proof can be carried out very similarly to the case of hyper-elliptical regions (see [52]); therefore, it is omitted. □

Now, we define the ${\mathbb{T}}_{\rho }$-property of the GBPs $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ as follows:

Definition 6.

If the GBP $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ represents all entire functions of order less than ρ in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$, then it is said to have property ${\mathbb{T}}_{\rho }$ in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$.

Let

$$F\left(\mathcal{G},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)=\underset{{\mathbf{k}}_{\mathrm{\Sigma }}\to \infty }{lim\; sup}{\displaystyle \frac{log\psi \left({\mathcal{G}}_{\mathbf{k}},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}\right)}{{\mathbf{k}}_{\mathrm{\Sigma }}log{\mathbf{k}}_{\mathrm{\Sigma }}}}.$$

The following theorem gives the property ${\mathbb{T}}_{\rho }$ of the base $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$.

Theorem 8.

A GBP $\left\{{\mathcal{G}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is said to have the property ${\mathbb{T}}_{\rho }$ for all entire functions of order less than ρ in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ if and only if $F(\mathcal{G},{\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]})\le {\displaystyle \frac{1}{\rho }}.$

Proof. 

As the proof closely parallels that for complete Reinhardt domains (polycylindrical regions) (see [27]), it is omitted. □

Next, we construct the ${\mathbb{T}}_{\rho }$-property of the GKPBPs in $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ in closed polycylinder ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ in terms of the ${\mathbb{T}}_{\rho }$-property of their constituents in $\left|z_s\right|\le r_s,s\in I$.

Theorem 9.

Let $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\},s\in I$ be the GBPs and suppose that $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ is their GKPBPs. Then the base $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ has $T_{\rho }$-property in ${\overline{\mathrm{\Gamma }}}_{\left[\mathbf{r}\right]}$ iff the bases $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ have the property ${\mathbb{T}}_{{\rho }_s}$ in $\left|z_s\right|\le r_s,$ where $s\in I$ and $\rho =min\{{\rho }_s,s\in J\}.$

Proof. 

The proof proceeds in much the same way as in the case of hyper-elliptical regions (see [52]) and is therefore omitted. □

To illustrate Theorems 7 and 8, we provide examples involving certain special functions.

Example 6.

The order, the type and ${\mathbb{T}}_{\rho }$-property were determined for certain special polynomials such as Bernoulli polynomials $\left\{{\mathcal{B}}_{s,k_s}\left(z\right)\right\}$ [51], the Euler polynomials $\left\{{\mathcal{E}}_{s,k_s}\left(z\right)\right\}$ [51] and Gontcharoff polynomials $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$ (when $|t_s|=1,$) [56].

Table 1. Order, type and ${\mathbb{T}}_{\rho }$-property of certain bases.

Polynomial’s Name	Order	Type	${\mathbb{T}}_{\mathit{\rho }}$-Property
Bernoulli polynomials $\left\{{\mathcal{B}}_{s,k_s}\left(z\right)\right\}$	1	${\displaystyle \frac{1}{2\pi }}$	${\mathbb{T}}_1$
$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{B}\right)}\left[\mathbf{z}\right]\right\}={\displaystyle \underset{s=1}{\overset{\ell }{⨂}}}\left\{{\mathcal{B}}_{s,k_s}\left(z_s\right)\right\}$	1	${\displaystyle \frac{1}{2\pi }}$	${\mathbb{T}}_1$
Euler polynomials $\left\{{\mathcal{E}}_{s,m_s}\left(z\right)\right\}$	1	${\displaystyle \frac{1}{\pi }}$	${\mathbb{T}}_1$
$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{E}\right)}\left[\mathbf{z}\right]\right\}={\displaystyle \underset{s=1}{\overset{\ell }{⨂}}}\left\{{\mathcal{E}}_{s,k_s}\left(z_s\right)\right\}$	1	${\displaystyle \frac{1}{\pi }}$	${\mathbb{T}}_1$
Gontcharoff polynomials $\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$	1	${\displaystyle \frac{\left|a\right|}{\mu }}$,	${\mathbb{T}}_1$
$\left\{{\mathcal{K}}_{\mathbf{k}}^{\left(\mathcal{G}\right)}\left[\mathbf{z}\right]\right\}={\displaystyle \underset{s=1}{\overset{\ell }{⨂}}}\left\{{\mathcal{G}}_{s,k_s}\left(z_s\right)\right\}$	1	${\displaystyle \frac{\left|a\right|}{\mu }}$,	${\mathbb{T}}_1$

below lists these details, and according to Theorems 7 and 8, we conclude the corresponding properties of the general Kronecker product bases constructed via these special polynomials.

Note that μ in the last two rows of Table 1 denotes the modulus of a zero of the function $f\left(z_s\right)={\displaystyle \sum _{k_s=0}^{\infty }}{t}_s^{{\displaystyle \frac{k_s(k_s-1)}{2}}}{\displaystyle \frac{z_s^{k_s}}{k_s!}}$ of least modulus.

Remark 2.

Similar results for the GKPBPs $\left\{{\mathcal{K}}_{\mathbf{k}}\left[\mathbf{z}\right]\right\}$ in hyper-elliptical regions can be obtained when the constituent bases are taken to be GBPs. This is considered a generalization of a recent study published in [52], extending the Cannon–Kronecker product bases to the generalized Kronecker product bases in hyper-elliptical regions.

9. Closing Remarks and Conclusions

In this study, I developed and analyzed generalized Kronecker product bases of polynomials within polycylindrical regions, offering substantial extensions of the classical Cannon–Kronecker framework. Our investigation into the convergence of infinite series generated by these generalized bases across a variety of polycylindrical domains, both open and closed, has revealed rich analytic structures governed by properties such as order, type, and the ${\mathbb{T}}_{\rho }$-property. These findings contribute to a deeper understanding of the growth behavior and analytic character of polynomial bases in multidimensional settings. Furthermore, the broad utility and expressive capability of the generalized Kronecker product approach were demonstrated by applying this generalized theory to a wide class of classical special functions including Bernoulli, Euler, Gontcharoff, Bessel and Chebyshev polynomials. The results suggest promising avenues for further exploration, particularly in the approximation theory of multivariate analytic functions, as well as potential applications in mathematical physics and numerical analysis. This work lays a foundational platform for future studies aimed at refining and extending the use of Kronecker-based constructions in higher-dimensional function spaces.