# Criteria in Nuclear Fréchet Spaces and Silva Spaces with Refinement of the Cannon-Whittaker Theory

Authors: Mahmoud Abul-Ez; Mohra Zayed
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Along with the theory of bases in function spaces, the existence of a basis is not always guaranteed. The class of power series spaces contains many classical function spaces, and it is of interest to look for a criterion for this class to ensure the existence of bases which can be expressed in an easier form than in the classical case given by Cannon or even by Newns. In this article, a functional analytical method is provided to determine a criterion for basis transforms in nuclear Fréchet spaces ((NF)-spaces), which is indeed a refinement and a generalization of those given in this concern through the theory of Whittaker on polynomial bases. The provided results are supported by illustrative examples. Then, we give the necessary and sufficient conditions for the existence of bases in Silva spaces. Moreover, a nuclearity criterion is given for Silva spaces with bases. Subsequently, we show that the presented results refine and generalize the fundamental theory of Cannon-Whittaker on the effectiveness property in the sense of infinite matrices.


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## Body

## 1. Introduction

The existence of bases is one of the fundamental problems in the classical theory of analytic functions. A functional analytic approach to the theory of bases in function spaces emerges naturally when studying classes of functions which play a vital role in applied mathematics and mathematical physics. This paper is entirely devoted to the study of bases in the spaces of holomorphic functions in one or two complex variables. Let us consider two important problems which arise in the study of function spaces as follows:(1)
Does the space under consideration possess a basis?(2)
If this is the case, how can any other basis of this space be characterized?Assume these problems are answered in a positive way. Then, whenE denotes the space and xnn∈ℕ stands for a basis in E, each element x∈E admits a (unique) decomposition of the form ∑n=1∞anxxn whereby for each n∈ℕ, an is a linear functional on E. In practice (e.g., in approximation theory), the choice of a suitable basis is very important.The present work essentially deals with these two fundamental problems in the case where the considered function spaces admit a set of polynomials as a basis (or in the terminology of Cannon-Whittaker, a basic set of polynomials) [1–5]. Basic examples of such function spaces are given by the space of holomorphic functions in an open disc or the space of analytic functions on a closed disc. Of course, as the theory of holomorphic functions in the plane allows generalizations to higher dimensions, analogous problems may be considered in the corresponding function spaces, see [2, 6–14].Cannon and Whittaker [5, 15–17] studied the existence of basic sets of polynomials of one complex variable (as bases) in the classical spaces OBr, and later, many authors considered such problem in the space OB¯r×B¯r, see for example [2, 14, 18, 19]. The main tool in their investigations is a Cannon criterion, determining whether the considered set of polynomials forms a basis for the space (or in the Whittaker terminology, this set of polynomials is effective). So that by the effectiveness of the basic set of polynomials Pnz, z∈ℂ, in the closed ball B¯r, it means that the set forms a base for the class E of holomorphic functions regular in B¯r with norm given by Mf,r, f∈E.In our approach here, we introduce the topic from a different point of views, as the change of basis problem presented in the abstract setting of nuclear Fréchet spaces ((NF)-spaces) and nuclear Silva spaces ((NS)-spaces) having in mind essentially the basic exampleOBr. Criteria are obtained which tell us under which conditions an infinite matrix P is a basis transform in both a (NF)-space and a NS-space with basis. These criteria are then applied to the case of power series spaces, thus yielding a refinement of Cannon’s criterion in the case OB¯r or even those given in [4, 13, 18, 20]. It is worth mentioning that this problem has been treated by many authors from different angles for which we may mention [4, 5, 21, 22]. Furthermore, significant advances of the subject in higher dimensional spaces have been investigated as in the space of several complex variables ℂn [2], monogenic function spaces [8–12, 23, 24], and the matrix function spaces [22]. Much close to the effectiveness problem is the study of spaces of entire functions having finite growth, wherein [25] the author showed that such spaces are also (NF)-spaces.The main purpose of this paper is to find criteria under which a sequence of vectorsxnn∈ℕ in a (NF)-space E with basis is itself a basis for E. This leads to the notion of basis-preserving homeomorphisms between (NF)-spaces with bases. The analog study on (NS)-spaces is also provided. Our results are then applied to the case of power series spaces to get a link with the theory of Whittaker on polynomial bases [5] and the theory of Newns who treated the problem of effectiveness by using the topological method approach (see [4]). To begin our investigation, we first give a survey of all necessary definitions and basic results from the general theory of locally convex topological vector spaces and the theory of nuclear spaces. For a more general account, we refer to [26–29]. Then, we point out that the results obtained in this paper might form the starting point of further investigations concerning basis transforms in more general locally convex spaces and higher dimension context.


## 2. Preliminaries

In the sequel,P=Pij, Q=Qij, and T=Tij be infinite complex matrices and the infinite identity matrix will be denoted by I. All vector spaces will be Fréchet spaces over ℂ, although all results are without changes valid for vector spaces over ℝ. Since we are interested in basis transforms, it will be convenient to look upon a space together with a basis, and notation of the form V,b will be used, where V is the vector space and b=bnn=1+∞ is a topological basis for V. If V and W are isomorphic, then there exists an isomorphism φ, called a basis-preserving isomorphism, mapping the basis b of V to a basis c of W (i.e., φbn=cn for all n); V,b and V,c are called similar. This relation is written V,b≃V,c. Obviously, it is not necessarily true that if b and f are two bases for V then V,b≃V,f. If it is true however we speak of conjugate bases. If x=xn is a sequence of vectors in V and for each k, the series
(1)yk=∑n∈ℕPknxn,k∈ℕconverges, we simply write y=Px.


### 2.1. Köthe Sequence Spaces

Letω denote the space of all infinite sequence of complex numbers and let A=ank be an infinite matrix of real nonnegative numbers such that ank≤ank+1 for all n,k and for each n, these exist k such that ank≠0. Here, k and n stand for the row and the column indices, respectively. With such a matrix A, we associate the following subset of ω:
(2)KA=ξ=ξn∈ω:ξk=∑n=1∞ankξn<+∞,k∈ℕ,with the topology given by the seminorms ∙k. Note that P=∙:k∈ℕ is a sequence of norms on KA, which is moreover a system of norms on KA. Putting for each n∈ℕ, en=δknk∈ℕ, we cite the following important result proved by Pietsch in [29].Theorem 1 (see [29]).
KA is an (F)-space and e=enn∈ℕ is a basis for KA.Remark 2.
It is well known that ifV,x is a (NF)-space and the topology is given by seminorms pk, then V,x≃KA,e, where A is given by the relation ank=pkxn. This makes it possible to use alternatively the seminorms pk, φ.k, or φ.k where φ is the basis-preserving isomorphism.


### 2.2. Nuclear Fréchet Spaces with Basis

We shall write shortly (F)-space for Fréchet space and (NF)-space for nuclear Fréchet space, as is common in the literature (see [27, 30–32]). Now, let E,P be an (F)-space where it is henceforth assumed that the countable system of seminorms P=pk:k∈ℕ is in fact a sequence of norms. We therefore write pk=∙k.The following result given in [28, 33] is useful in the sequel.Theorem 3 (Banach’s homeomorphism theorem).
LetE,P and F,Q be (F)-spaces and let T:E→F be a bijective and bounded linear operator. Then, T is a homeomorphism and thus T−1 is also bounded.The definition of the (NF)-spaces was introduced in [26] as follows.Definition 4 ((NF)-space with basis).
LetE,P be an (F)-space. Then, it is called a (NF)-space if for each k∈ℕ, there exist ℓ∈ℕ and a sequences ynn∈ℕ in E and Lnn∈ℕ in E′ such that for each x∈E,
(i)
x=.k∑n∈ℕLnxyn(ii)
∑n=1∞Lnℓ′ynk<+∞
whereLnℓ′=supx∈E,xℓ≤1Lnx and if Lnℓ′=∞, then ynk=0.Now, we recall some functional theorems concerning (NF)-spaces with basis.Theorem 5 (Dynin-Mitiagin [34, 35]).
LetE,x be a (NF)-space with basis. Then, for each k∈ℕ, there exist ℓ∈ℕ and Kk>0 such that, if for y∈E, y=∑n=1∞αnyxn, where αny is the coefficient function, then
(3)∑n=1∞αnyxnk<Kkyℓ.Corollary 6 (see [34, 35]).
Each base in a (NF)-space with basis is absolute.Theorem 7 (see [31]).
LetE be a (NF)-space with a complete biorthogonal system xn,x′nn∈ℕ. Then, x=xnn∈ℕ is a basis for E if and only if for each k∈ℕ, there exist ℓ∈ℕ, and Kk>0 such that for each y∈E,
(4)supn∈ℕx′nyxnk≤Kkyℓ.Theorem 8 (Haslinger’s criterion for a (NF)-space with basis [30]).
LetE,x be a (NF)-space with basis and let yn,y′nn∈ℕ be a complete biorthogonal system in E. Then, ynn∈ℕ is a basis in E if and only if for each k∈ℕ, there exist ℓ∈ℕ and Kk>0 such that for each j∈ℕ,(5)supn∈ℕy′nxjxnk≤Kkxjℓ.


### 2.3. Nuclear Köthe Sequence Spaces

As we have seen that a Köthe sequence spaceKA is an (F)-space with basis e, we have also according to [29].Theorem 9. (see [29]).
LetKA be a Köthe sequence space. Then, KA is nuclear (and hence a (NF)-space with basis) if and only if for each k∈ℕ, there exists ℓ∈ℕ such that
(6)∑n∈ℕankanℓ<+∞.Now, suppose thatKA is a Köthe sequence space and put
(7)SA=ξ∈ω:supn∈ℕξnank<+∞,ξk=supn∈ℕξnank,k∈ℕ,ξ∈SA.Then, clearly,Q=∙k:k∈ℕ is a set of norms on SA. We call SA the supremum space associated with A. Then, we haveTheorem 10 (see [29]).
SA,P is an (F)-space.Remark 11.
(i)
Notice that, althoughSA is always an (F)-space, it need not have a basis. Indeed, it is sufficient to consider A=ank with ank=1 for all n,k∈ℕ(ii)
Notice also that for this matrixA, the condition (6) needed for the nuclearity of KA is not satisfied. This might suggest that the nuclearity of KA could be related to the existence of a basis for SA. This is indeed the case as it was shown in [26, 30–32, 36], which can be stated in the following theoremTheorem 12.
(i)
KA is nuclear if and only if SA is nuclear(ii)
KA and SA are homeomorphic(iii)
The standard basise=enn∈ℕ of KA is also a basis for SA, and thus, SA is a (NF)-space with basisThis theorem implies immediately that ifKA is nuclear, then KA,e≃SA,e.


### 2.4. Important Remarks

For (F)-spaces, and even for (NF)-spaces, the existence of a basis is not always guaranteed, as was shown in [34, 35]. The Haslinger’s criterion mentioned previously in Theorem 8 is essential for our study, namely, to establish whether or not an infinite matrix P determines a basis transform in either (NF)-space or (NS)-space with basis. That is what we strive to achieve in the sequel. This will give a refinement of the criteria of effectiveness problem in the sense of Cannon-Whittaker theory on basis of polynomials.Beforehand, we give a fruitful study concerning a criterion for an (F)-space to be nuclear showing by supporting examples that the provided criterion is attainable.


## 2.1. Köthe Sequence Spaces

Letω denote the space of all infinite sequence of complex numbers and let A=ank be an infinite matrix of real nonnegative numbers such that ank≤ank+1 for all n,k and for each n, these exist k such that ank≠0. Here, k and n stand for the row and the column indices, respectively. With such a matrix A, we associate the following subset of ω:
(2)KA=ξ=ξn∈ω:ξk=∑n=1∞ankξn<+∞,k∈ℕ,with the topology given by the seminorms ∙k. Note that P=∙:k∈ℕ is a sequence of norms on KA, which is moreover a system of norms on KA. Putting for each n∈ℕ, en=δknk∈ℕ, we cite the following important result proved by Pietsch in [29].Theorem 1 (see [29]).
KA is an (F)-space and e=enn∈ℕ is a basis for KA.Remark 2.
It is well known that ifV,x is a (NF)-space and the topology is given by seminorms pk, then V,x≃KA,e, where A is given by the relation ank=pkxn. This makes it possible to use alternatively the seminorms pk, φ.k, or φ.k where φ is the basis-preserving isomorphism.


## 2.2. Nuclear Fréchet Spaces with Basis

We shall write shortly (F)-space for Fréchet space and (NF)-space for nuclear Fréchet space, as is common in the literature (see [27, 30–32]). Now, let E,P be an (F)-space where it is henceforth assumed that the countable system of seminorms P=pk:k∈ℕ is in fact a sequence of norms. We therefore write pk=∙k.The following result given in [28, 33] is useful in the sequel.Theorem 3 (Banach’s homeomorphism theorem).
LetE,P and F,Q be (F)-spaces and let T:E→F be a bijective and bounded linear operator. Then, T is a homeomorphism and thus T−1 is also bounded.The definition of the (NF)-spaces was introduced in [26] as follows.Definition 4 ((NF)-space with basis).
LetE,P be an (F)-space. Then, it is called a (NF)-space if for each k∈ℕ, there exist ℓ∈ℕ and a sequences ynn∈ℕ in E and Lnn∈ℕ in E′ such that for each x∈E,
(i)
x=.k∑n∈ℕLnxyn(ii)
∑n=1∞Lnℓ′ynk<+∞
whereLnℓ′=supx∈E,xℓ≤1Lnx and if Lnℓ′=∞, then ynk=0.Now, we recall some functional theorems concerning (NF)-spaces with basis.Theorem 5 (Dynin-Mitiagin [34, 35]).
LetE,x be a (NF)-space with basis. Then, for each k∈ℕ, there exist ℓ∈ℕ and Kk>0 such that, if for y∈E, y=∑n=1∞αnyxn, where αny is the coefficient function, then
(3)∑n=1∞αnyxnk<Kkyℓ.Corollary 6 (see [34, 35]).
Each base in a (NF)-space with basis is absolute.Theorem 7 (see [31]).
LetE be a (NF)-space with a complete biorthogonal system xn,x′nn∈ℕ. Then, x=xnn∈ℕ is a basis for E if and only if for each k∈ℕ, there exist ℓ∈ℕ, and Kk>0 such that for each y∈E,
(4)supn∈ℕx′nyxnk≤Kkyℓ.Theorem 8 (Haslinger’s criterion for a (NF)-space with basis [30]).
LetE,x be a (NF)-space with basis and let yn,y′nn∈ℕ be a complete biorthogonal system in E. Then, ynn∈ℕ is a basis in E if and only if for each k∈ℕ, there exist ℓ∈ℕ and Kk>0 such that for each j∈ℕ,(5)supn∈ℕy′nxjxnk≤Kkxjℓ.


## 2.3. Nuclear Köthe Sequence Spaces

As we have seen that a Köthe sequence spaceKA is an (F)-space with basis e, we have also according to [29].Theorem 9. (see [29]).
LetKA be a Köthe sequence space. Then, KA is nuclear (and hence a (NF)-space with basis) if and only if for each k∈ℕ, there exists ℓ∈ℕ such that
(6)∑n∈ℕankanℓ<+∞.Now, suppose thatKA is a Köthe sequence space and put
(7)SA=ξ∈ω:supn∈ℕξnank<+∞,ξk=supn∈ℕξnank,k∈ℕ,ξ∈SA.Then, clearly,Q=∙k:k∈ℕ is a set of norms on SA. We call SA the supremum space associated with A. Then, we haveTheorem 10 (see [29]).
SA,P is an (F)-space.Remark 11.
(i)
Notice that, althoughSA is always an (F)-space, it need not have a basis. Indeed, it is sufficient to consider A=ank with ank=1 for all n,k∈ℕ(ii)
Notice also that for this matrixA, the condition (6) needed for the nuclearity of KA is not satisfied. This might suggest that the nuclearity of KA could be related to the existence of a basis for SA. This is indeed the case as it was shown in [26, 30–32, 36], which can be stated in the following theoremTheorem 12.
(i)
KA is nuclear if and only if SA is nuclear(ii)
KA and SA are homeomorphic(iii)
The standard basise=enn∈ℕ of KA is also a basis for SA, and thus, SA is a (NF)-space with basisThis theorem implies immediately that ifKA is nuclear, then KA,e≃SA,e.


## 2.4. Important Remarks

For (F)-spaces, and even for (NF)-spaces, the existence of a basis is not always guaranteed, as was shown in [34, 35]. The Haslinger’s criterion mentioned previously in Theorem 8 is essential for our study, namely, to establish whether or not an infinite matrix P determines a basis transform in either (NF)-space or (NS)-space with basis. That is what we strive to achieve in the sequel. This will give a refinement of the criteria of effectiveness problem in the sense of Cannon-Whittaker theory on basis of polynomials.Beforehand, we give a fruitful study concerning a criterion for an (F)-space to be nuclear showing by supporting examples that the provided criterion is attainable.


## 3. A Nuclearity Criterion for (F)-Spaces with Basis

In this section, we give a criterion stating under which conditions an (F)-spaceE with basis x=xnn∈ℕ is nuclear. Let us recall that for an (F)-space E with x=xnn∈ℕ, the associated linear functional x′n, n∈ℕ is bounded according to Schauder’s theorem [28] which states that “if E is an (F)-space and xnn∈ℕ is a basis for E, then xnn∈ℕ is a Schauder’s basis.”As usual, it is tacitly understood that the topology of the (F)-space is determined by a countable sequence∙mm∈ℕ of norms.Theorem 13.
LetE be an (F)-space with basis x=xnn∈ℕ. Then, the following are equivalent:
(i)
E is nuclear(ii)
For eacht∈ℕ, there exists s∈ℕ such that(8)∑n∈ℕxntxns<+∞.Proof.
LetE be a nuclear space. Then, by Dynin-Mitiagin theorem 5, it follows that for all t∈ℕ, there ought to exist s∈ℕ such that
(9)∑n=0∞xn′sxnt<∞.
Sincexn′xn=1, we have 1≤∥xn′∥s′∥xn∥s or ∥xn′∥s′≥1/∥xn∥s. Consequently,
(10)∑n=0∞xntxns<+∞.Conversely, assume that for eacht∈ℕ, there exists s∈ℕ such that
(11)∑n=0∞xntxns<+∞.LetS be the following space:
(12)S=a=ann∈ℕ:an∈ℂforalln∈ℕandak=∑n=0∞anxnk<+∞forallk∈ℕ.Then, clearly,S is a supremum space associated with A=ank where ank=xnk, hence a Fréchet space. By the condition assumed, it follows from Theorem 12 that S is nuclear.Now, consider the mapψ:S→E such that
(13)ψ:ann∈ℕ→∑n=0∞anxn.Obviously,ψ is linear, continuous, and injective. We now prove that ψ is also surjective, i.e., if y∈E with y=∑n=0∞anxn, then a=ann∈ℕ∈S. Fix s∈ℕ, then as ∑n=0∞anxn converges in F, limn→∞anxns<∞ whence
(14)bs=supn∈ℕanxns<+∞.Now, lett∈ℕ be chosen arbitrarily and let s∈ℕ such that ∑n=0∞∥xn∥t/∥xn∥s<+∞. Hence,
(15)∑n=0∞anxnt=∑n=0∞anxnsxntxns≤bs∑n=0∞xntxns<+∞.Consequently,a=ann∈ℕ∈S. By virtue of Banach’s homeomorphism theorem 3 (see [28]), ψ is bicontentious and so E is homeomorphic to S or in other words E is nuclear.Remark 14.
(1)
Notice that Theorem13 provides a relatively simple tool for determining the nuclearity of an (F)-space with basis. In what follows, for an (F)-space E having a basis xnn∈ℕ satisfying (8), we put(16)SA=a=ann∈ℕ:an∈ℂandforallk∈ℕ,ak=∑n=0∞anxnk<+∞(2)
As we have seen,SA is a nuclear supremum space which is isomorphic to E. Here, A=ank with ank=xnk(3)
From the isomorphismψ constructed in the proof above, it follows that if E,x is a (NF)-space with basis xnn∈ℕ, then the following isomorphisms hold:(17)E,x≃KA,e≃SA,e
We thus obtain that the natural system of seminorms onE is equivalent to
(i)
the system of norms(18)yk=∑n∈ℕanank,k∈ℕ(ii)
the system of norms(19)yk=supn∈ℕanank,k∈ℕ
Note that in both cases,an=x′ny, n∈ℕ.
(4)
A nuclearity criterion for the more general case of locally convex spaces having an equicontinuous Schauder basis was proved by Kamthan in [32]. Our Theorem 13 is a special case of his; however, since we are working in (F)-spaces, the proof can be done in a rather easy way. For the case of Köthe spaces, we also refer to [36].
Now, we illustrate the usefulness of the criterion obtained in Theorem13.Example 1.
Consider the spaceOBR of holomorphic functions in the open disk BR provided with the countable system P of seminorms pk where
(20)pkf=supz≤rkfz,z∈ℂ,rkk∈ℕ being a strictly increasing sequence of positive numbers with 0<rk<R, and limk→∞rk=R. One may take, for example, rk=R−1/k (assuming tacitly that R>1, if R≤1, then similar choices may, of course, be made) if R is finite and rk=k if R is infinite.As we saw before thatOBR,ξ is a Fréchet space with basis ξ=znn∈ℕ, although it is well known that OBR,ξ is a (NF)-space (see [29]), its proof is not trivial.As will be seen now, the criterion just proved yields the nuclearity ofOBR,ξ in an easy way. Indeed, as for each k∈ℕ and n∈ℕ,
(21)znk=supz≤rkzn=rkn.Takingk∈ℕ fixed, then for each ℓ>k, we have
(22)∑n=0∞znkznℓ=∑n=0∞rknrℓn=∑n=0∞rkrℓn<+∞.So, the criterion applies.Remark 15.
Notice that, as we previously mentioned,(23)OBR,ξ≃KA,e≃SA,e,with A=ank and ank=supz≤rkzn, whence the natural system of seminorms P=∙k:k∈ℕ and the system of seminorms PKA and PSA induced on OBR,ξ are all equivalent. Here,
(24)PKA=∙k,K:k∈ℕ,PSA=.k:k∈ℕ,where if f∈OBR admits the Taylor series at the origin,
(25)fz=∑n=0∞cnzn,then, for k∈ℕ, fk=supn∈ℕcnank and fk,K=∑n=0∞cnank.Notice that we already obtain directly, and this by using Cauchy’s inequality and the triangle inequality, respectively, the following comparison between the systems of seminorms under consideration for eachk∈ℕ and f∈O(BR,(26)fk≤fk≤fk,K.Thus, the equivalence between the systems of seminorms established above gives stronger results.Example 2.
Consider the spaceOBR×BR of holomorphic functions in two complex variables in the open polydisc BR×BR, provided with the countable system P of seminorms pk where
(27)pkf=sup∣u∣≤rk,∣v∣≤rkfu,v.Again,rkk∈ℕ is strictly increasing sequence of positive numbers with 0<rk<R and limk→∞rk=R. As is well known, OBR×BR,ξ is a Fréchet space. Moreover, as it is shown again by the Taylor series at the origin for any f∈OBR×BR, the sequence ξ=unvmn,m∈ℕ is a basis for OBR×BR. Although it is known that OBR×BR,ξ is a (NF)-space, our criterion will yield the nuclearity of it in a very simple way. Indeed, since for each k∈ℕ and n,m∈ℕ,
(28)unvmk=sup∣u∣≤rk,∣v∣<rkunvm=rkn+m,we obtain that, taking k∈ℕ fixed, for each ℓ>k,
(29)∑n=0∞∑m=0∞unvmkunvmℓ=∑n=0∞∑m=0∞rkrℓn+m=rℓrℓ−rk2<+∞,the nuclearity of this space is proved.


## 4. Overview on Basis Transforms in (NF)-Spaces and the General Theory of Effectiveness

### 4.1. Basis Transforms and Effectiveness Phenomena

In this section, we provide a general overview of the notion of basis transforms in (NF)-spaces from which a criterion is obtained. This criterion shows that under which conditions, a sequence of vectorsxnn∈ℕ in a (NF)-space E with basis is itself a basis for E. This leads to the notion of basis-preserving homeomorphisms between (NF)-spaces with basis. The given results are then applied to the case of power series spaces giving a general criterion for basis transforms which improve and refine that one of Cannon-Whittaker on the phenomena of effectiveness.The idea behind this study is to link the theory of basis transforms in some locally convex space with the theory of Cannon-Whittaker on polynomial bases. In the meantime, we improve and refine the effectiveness phenomena which represents the core of Whittaker’s theory.It is worth mentioning that in the classical treatment of the subject of polynomial bases as was introduced by Whittaker and Cannon [17], the methods for establishing effectiveness depend on the region of effectiveness and on the class of functions for which the base is effective. The first attempt at some uniformity among the different methods was made by Newns, who gave in [4] a topological approach leading to a general theory of effectiveness. Our approach is entirely different depending on functional analytical methods and the basis transforms being performed by means of infinite matrices. In the Cannon-Whittaker theory, the main tool they used depends essentially on assuming the row-finite matrices of coefficients and operators.In what follows, it is thus understood thatE is a (NF)-space with basis xn,x′nn∈ℕ. Moreover, P=Pij and Q=Qij stand for infinite matrices over ℂ and we denote the infinite identity matrix by I. In such a way, we formally write
(30)yk=∑n∈ℕPknxn,k∈ℕ.We start by stating the following result by Cnops and Abul-Ez [37].Theorem 16 (uniqueness theorem).
Suppose thatykk∈ℕ is a basis for E. Then, P has two-sided inverse Q.The following problem now arises: letE be a (NF)-space with basis xnn∈ℕ and ynn∈ℕ be a sequence in E satisfying the relations
(31)xn=∑j∈ℕQnjyjand formally,
(32)yk=∑n∈ℕPkn∑j∈ℕQnjyj.Under which condition on the matricesP and Q we conclude that ykk∈ℕ is a basis for E?Remark 17.
In fact, (31) and (32) do not assume that ykk∈ℕ is a basis. An example is given by Faber polynomials which are valid in a noncircular domain, see [38]. Since the nth Faber polynomials is of degree n, it is possible to write each zk as a linear combination (even a finite one) of Faber polynomials. However, it is well known that Faber polynomials in general do not form a basis for OBR, but restricted conditions should be considered, see the work of Newns [4]. The answer of the above question is provided by the following interesting results by Cnops and Abul-Ez [37].Theorem 18 (basis transforms in (NF)-spaces).
Let the sequenceykk∈ℕ in E allow representations of the form (31) and (32). Then, a biorthogonal system yn,y′nn∈ℕ forming a basis for E may be constructed if and only if
(B.1)PQ=QP=I
(B.2) For eachm∈ℕ, there exist ℓ∈ℕ, Km>0 such that for all k,n∈ℕ(33)Qnkykm≤Kmxnℓ.Remark 19.
(1)
From the isomorphism betweenE,xnn∈ℕ and SA and the derived equivalent system of norms on E (see also Theorems 13 and 12), we have that (B.2) is equivalent to
(B′.2) For each m∈ℕ, there exist ℓ∈ℕ and Km>0 such that for all n,k,t∈ℕ(34)QnkPktxtm≤Kmxnℓ(2)
On the other hand, using the sum-norm and applying the Dynin-Mitiagin theorem, we deduce that (B.2) and (B.2)′ are equivalent to
(B′′.2) For each m∈ℕ, there exist ℓ∈ℕ and K′m>0 such that for all n∈ℕ(35)∑k∈ℕ∑t∈ℕQnkPktxtm≤K′mxnℓ(3)
Some comments on the requirement thatP has two-sided inverse Q. It is of course possible that P has several two-sided inverses. But, if P is taken to belong to some class of infinite matrices which forms a ring R under the classical rules of addition and multiplication for matrices and moreover I∈R, then, if P has a two-sided inverses Q∈R, Q is uniqueAn example of such a ringR is the ring Rup of all infinite upper-triangular matrices or the ring Rlo of all infinite lower-triangular matrices. It is clear that if, e.g., P∈Rup has all diagonal elements different from zero, then P has a two-sided inverse Q in Rup which is therefore unique (see [39]). Nevertheless, it may happen that although the matrix P we are considering belongs to some ring R of infinite matrices, P admits a two-sided inverse which does not belong to the ring R. We address this phenomenon in the following example.Example 3.
LetR be the ring of all infinite row-finite matrices and P∈R given by P=Piji,j=1∞, where
(36)Pii=1,i∈ℕ,Pi,i+1=−1,i∈ℕ,Pij=0,j≠iorj≠i+1,that is,
(37)P=1−10⋯0001−1⋯00001−1⋯0::⋱⋱⋱:00⋯01−100⋯001.
Then,P admits the two-sided inverse Q=Qiji,j=1∞ with
(38)Qij=0ifj<i,1ifj≥i,that is,
(39)Q=111⋯11011⋯11001⋯11::⋱⋱::00⋯01100⋯001.
Clearly,Q is not row-finite.Remark 20.
ApplyingP to the space OB1/2,ξ, Pξ is given by
(40)pkz=zk−zk+1,k=0,1,2,⋯,and using our criterion for a basis transform, p=Pξ is indeed a basis for OB1/2.
These considerations therefore suggest that, when an infinite matrixP is given and we wish to use it as a basis transform in some (NF)-space E,x, we check the following:
(i)
The existence of an infinite matrixQ which is a two-sided inverse of P, i.e., (B.1)(ii)
The criterion (B.2) or (B′.2) or (B′′.2) holds
Suppose (i) is fulfilled. Then, the following situations may occur:
(i.1)Q is the unique two-sided inverse of P; then, (ii) above should be verified.
(i.2)P admits several two-sided inverses. If P defines a basis transform, there is a unique two-sided inverse Q such that Qp exists and Q satisfies (B.2). If P does not define a basis transform, the set of inverses
(41)Q:QP=PQ=IandQPexists.
The above remark leads to the following result.Corollary 21.
LetP be an infinite matrix having a two-sided inverse Q such that Qp converges. Then, P defines a basis transform if and only if the pair P,Q satisfies conditions (B.2) (equivalently (B′.2) or (B′′.2)).


### 4.2. Power Series Spaces

A class of Köthe sequence spaces which will be of special interest to us is that of so-called power series spaces. Letα=αnn∈ℕ be an increasing sequence of positive numbers and put for 0<R<+∞ fixed,
(42)akn=Re−1/kαn.Then, callingA=ank and associating with it the Köthe sequence space KA, it is well known that
(43)"KAisnuclear⇔limn→∞lognαn=0."In the caseR=∞, we put
(44)ank=ekαnand A=ank. It is then well known that
(45)"KAisnuclear⇔limsupn→∞lognαn<+∞."For an account of these results, we refer to [26]. In the case 0<R<+∞, the space KA is denoted by KA=ΛRα while the case R=∞, the notation KA=Λ∞α is used.Notice that, puttingrk=Re−1/k (0<R<+∞) or rk=ekαn (R=∞), then rk↑R or rk↑∞.The question now arises to determine whether or not a given (NF)-space with basis is similar to a space of the typeΛRα,e or Λ∞α,e, the latter spaces being called power series spaces.Now, we illustrate the problem by means of the following.Example 4.
The spaceOBR,ξ, 0<R<+∞. Define for each k∈ℕ and f∈OBR,
(46)fk=supz≤Re−1/kfz.Aslimn→∞logn/n=0, we can take α=αnn∈ℕ with αn=n for all n∈ℕ. Consequently, OBR,ξ≃ΛRℕ,e.Example 5 (the spaceOBℂ,ξ.
In this caseR=∞ and we may describe the topology on Oℂ by the system of seminorms P=∙k:k∈ℕ where for each k∈ℕ and f∈Oℂ,
(47)fk=supz≤ekfz.Hence, for eachn∈ℕ, znk=supz≤ekzn=enk.Assupn∈ℕlogn/n<+∞, we can take α=αnn∈ℕ with αn=n for all n∈ℕ. Consequently, Oℂ,ξ≃Λ∞ℕ,e.In the sequel, we denote for convenienceΛRα,e≃Kα,R, the associated supremum space will be denoted by Sα,R.


### 4.3. Similarity Theorem

The following criterion will completely answer the equation we mentioned above, namely, to know under which conditions a (NF)-space with basis is similar to some power series spaceKα,R. To reach this end, let us introduce the following notations.Given the basisx=xnn∈ℕ of E and the sequence α=αnn∈ℕ determining Kα,R, we put for each k∈ℕRk=limsupn→∞xnk1/αn and rk=liminfn→∞xnk1/αn. Clearly, rk≤Rk for each k∈ℕ.Suppose thatR is finite (the case R=+∞ is treated in a similar way).Theorem 22 (similarity theorem basis [37]).
LetE,x be a (NF)-space with basis and let Kα,R be some power series space. Then, E,x≃Kα,R if and only if
(i)
rk↑R(iii)
Rk<R for each k∈ℕAfter having established which (NF)-spaces with basis are similar to a power sequence space (Theorem22), we now aim to apply the criterion obtained for basis transforms (Theorem 18) to the case of power sequence spaces. To do this, consider the power sequence space Kα,R with basis e=ett∈ℕ. Then, we know that for each m∈ℕ,
(48)etm=Re−1/mαtfor0<R<∞,emαtforR=∞.In the sequel, we putrm=R−1/m (0<R<+∞) or rm=em. So, if P is an infinite matrix acting on e, then by a direct translation of Theorem 18, the authors of [37] established the following fundamental results.Theorem 23 (basis criterion).
LetKα,R be a power sequence space and let P be an infinite matrix. Then, Pe is a basis for Kα,R if and only if
(i)
there exists an infinite matrixQ such that PQ=QP=I(ii)
for eachm∈ℕ, there exist ℓ∈ℕ and Km>0 such that for all n,k,t∈ℕ,(49)QnkPktrmαt≤Kmrℓαn.Definition 24 (see [37]).
LetP be an infinite matrix which has two-sided inverse Q and let r>0 be fixed. Then, for n∈ℕ, we put
(50)JnP,r=supk,tQnkPktrαtand call
(51)JP,r=limn→∞JnP,r1/αnif all JnP,r are finite, and +∞ otherwise.
From Theorem23, the following result was deduced in [37].Theorem 25 (criterion for basis transforms [37]).
LetKα,R be a power sequence space and let P be an infinite matrix with two-sided inverse Q. Then, P determines a basis transform in Kα,R if and only if for all 0<r<R,
(52)JP,r<R.Remark 26.
We saw before that the criterion (B′.2) is equivalent to (B′′.2). This of course prompts the introduction of the following entities: given r>0, we put
(53)ZnP,r=∑k∑tQnkPktrαt,(54)ZP,r=limn→∞ZnP,r1/αn.
This leads to the following important result which was shown in [37].Theorem 27 (general criterion for basis transforms [37]).
LetKα,R be a power sequence space and let P be an infinite matrix having a two-sided inverse Q. Then, P determines a basis transforms in Kα,R if and only if for all 0<r<R,
(55)ZP,r<R.Remark 28.
It should be noted that Cannon [15] gave a criterion of the form in (54) for certain spaces of holomorphic functions, although he did not prove that P defines a basis transform but only that every function can be represented by a series in Pξ.


### 4.4. Cannon-Whittaker’s Criterion Revisited

Consider the spaceOBR,ξ. In [15], Cannon considered an infinite matrix P of the following type:
(i)
P is row-finite(ii)
P possesses a row-finite two-sided inverses Q [5]For each0<r<R, the following entities were introduced [40]:The Cannon sumλnP,r≤∑k,rQnkPkr for the Cannon bases. As for the non-Cannon base (general base), it introduced the Cannon sum in the form:
(56)FnP,r=supt1,t2supz≤r∑k=t1t2Qnk∑kPkszs.Consequently, for the corresponding Cannon functions(57)λP,r=limsupn→∞λnP,r1/n,κP,r=limsupn→∞FnP,r1/n.


### 4.5. A Refinement of Cannon-Whittaker Criterion for Effectiveness

Cannon proved that the set of polynomials determined byPξ is effective in ∣z∣<R if and only if κP,r<R for all 0<r<R and that for Cannon sets λP,r=κP,r. When we look at JnP,r in the case under consideration, it is clear that for all n∈ℕ and 0<r<R,
(58)JnP,r≤FnP,r≤λnP,r,JP,r≤κP,r≤λP,r.Besides the fact that Cannon only proved the effectiveness of the setPξ under the condition κP,r<R for all 0<r<R while we pointed out that JP,r<R for all 0<r<R implies that Pξ is a basis, it obviously follows from λP,r<R for all 0<r<R that JP,r<R for all 0<r<R. Hence, the given condition here using JP,r is weaker than the one obtained by Cannon using λP,r in the case of so-called Cannon sets (i.e., sets Pe for which λP,r=κP,r). There is even more to say. Since we can also use ZP,r to establish whether or not Pξ is a basis for OBR and since clearly for all n∈ℕ and 0<r<R,
(59)JnP,r≤FnP,r≤λnP,r≤ZnP,r,whence
(60)JP,r≤κP,r≤λP,r≤ZP,r.We may conclude thatλP,r may also be used in the case of non-Cannon sets. Of course, it should also be stressed that the matrices P we are considering need not be row-finite.Let us illustrate the previous observations in the case ofOB1,ξ, the space of holomorphic functions in the unit ball.Example 6.
Consider the function1−z−1=∑j=0∞zj∈OB1. Then, we claim that set 1/1−z,z,z2,⋯,zn,⋯ is a basis forOB1. Indeed, the associated infinite matrix P is given by
(61)P=111⋯110100⋯00010⋯0::⋱⋱0:0⋯00100⋯0001.
Note thatP is upper-triangular and has diagonal elements 1. Hence, it possesses a two-sided inverse Q given by
(62)Q=1−1−1⋯−1−10100⋯00010⋯0::⋱⋱⋯:0⋯00100⋯0001.
Takingα=αn=ℕ, a straightforward calculation yields for each 0<r<1 that
(i)
J0P,r=supt∈ℕ,t≥1rt=1(ii)
JnP,r=rn,n≥1
Consequently,JP,r=limsupn→∞JnP,r1/n=1, so that JP,r<1 for each 0<r<1. Hence, Pξ is a basis for the space OB1.


## 4.1. Basis Transforms and Effectiveness Phenomena

In this section, we provide a general overview of the notion of basis transforms in (NF)-spaces from which a criterion is obtained. This criterion shows that under which conditions, a sequence of vectorsxnn∈ℕ in a (NF)-space E with basis is itself a basis for E. This leads to the notion of basis-preserving homeomorphisms between (NF)-spaces with basis. The given results are then applied to the case of power series spaces giving a general criterion for basis transforms which improve and refine that one of Cannon-Whittaker on the phenomena of effectiveness.The idea behind this study is to link the theory of basis transforms in some locally convex space with the theory of Cannon-Whittaker on polynomial bases. In the meantime, we improve and refine the effectiveness phenomena which represents the core of Whittaker’s theory.It is worth mentioning that in the classical treatment of the subject of polynomial bases as was introduced by Whittaker and Cannon [17], the methods for establishing effectiveness depend on the region of effectiveness and on the class of functions for which the base is effective. The first attempt at some uniformity among the different methods was made by Newns, who gave in [4] a topological approach leading to a general theory of effectiveness. Our approach is entirely different depending on functional analytical methods and the basis transforms being performed by means of infinite matrices. In the Cannon-Whittaker theory, the main tool they used depends essentially on assuming the row-finite matrices of coefficients and operators.In what follows, it is thus understood thatE is a (NF)-space with basis xn,x′nn∈ℕ. Moreover, P=Pij and Q=Qij stand for infinite matrices over ℂ and we denote the infinite identity matrix by I. In such a way, we formally write
(30)yk=∑n∈ℕPknxn,k∈ℕ.We start by stating the following result by Cnops and Abul-Ez [37].Theorem 16 (uniqueness theorem).
Suppose thatykk∈ℕ is a basis for E. Then, P has two-sided inverse Q.The following problem now arises: letE be a (NF)-space with basis xnn∈ℕ and ynn∈ℕ be a sequence in E satisfying the relations
(31)xn=∑j∈ℕQnjyjand formally,
(32)yk=∑n∈ℕPkn∑j∈ℕQnjyj.Under which condition on the matricesP and Q we conclude that ykk∈ℕ is a basis for E?Remark 17.
In fact, (31) and (32) do not assume that ykk∈ℕ is a basis. An example is given by Faber polynomials which are valid in a noncircular domain, see [38]. Since the nth Faber polynomials is of degree n, it is possible to write each zk as a linear combination (even a finite one) of Faber polynomials. However, it is well known that Faber polynomials in general do not form a basis for OBR, but restricted conditions should be considered, see the work of Newns [4]. The answer of the above question is provided by the following interesting results by Cnops and Abul-Ez [37].Theorem 18 (basis transforms in (NF)-spaces).
Let the sequenceykk∈ℕ in E allow representations of the form (31) and (32). Then, a biorthogonal system yn,y′nn∈ℕ forming a basis for E may be constructed if and only if
(B.1)PQ=QP=I
(B.2) For eachm∈ℕ, there exist ℓ∈ℕ, Km>0 such that for all k,n∈ℕ(33)Qnkykm≤Kmxnℓ.Remark 19.
(1)
From the isomorphism betweenE,xnn∈ℕ and SA and the derived equivalent system of norms on E (see also Theorems 13 and 12), we have that (B.2) is equivalent to
(B′.2) For each m∈ℕ, there exist ℓ∈ℕ and Km>0 such that for all n,k,t∈ℕ(34)QnkPktxtm≤Kmxnℓ(2)
On the other hand, using the sum-norm and applying the Dynin-Mitiagin theorem, we deduce that (B.2) and (B.2)′ are equivalent to
(B′′.2) For each m∈ℕ, there exist ℓ∈ℕ and K′m>0 such that for all n∈ℕ(35)∑k∈ℕ∑t∈ℕQnkPktxtm≤K′mxnℓ(3)
Some comments on the requirement thatP has two-sided inverse Q. It is of course possible that P has several two-sided inverses. But, if P is taken to belong to some class of infinite matrices which forms a ring R under the classical rules of addition and multiplication for matrices and moreover I∈R, then, if P has a two-sided inverses Q∈R, Q is uniqueAn example of such a ringR is the ring Rup of all infinite upper-triangular matrices or the ring Rlo of all infinite lower-triangular matrices. It is clear that if, e.g., P∈Rup has all diagonal elements different from zero, then P has a two-sided inverse Q in Rup which is therefore unique (see [39]). Nevertheless, it may happen that although the matrix P we are considering belongs to some ring R of infinite matrices, P admits a two-sided inverse which does not belong to the ring R. We address this phenomenon in the following example.Example 3.
LetR be the ring of all infinite row-finite matrices and P∈R given by P=Piji,j=1∞, where
(36)Pii=1,i∈ℕ,Pi,i+1=−1,i∈ℕ,Pij=0,j≠iorj≠i+1,that is,
(37)P=1−10⋯0001−1⋯00001−1⋯0::⋱⋱⋱:00⋯01−100⋯001.
Then,P admits the two-sided inverse Q=Qiji,j=1∞ with
(38)Qij=0ifj<i,1ifj≥i,that is,
(39)Q=111⋯11011⋯11001⋯11::⋱⋱::00⋯01100⋯001.
Clearly,Q is not row-finite.Remark 20.
ApplyingP to the space OB1/2,ξ, Pξ is given by
(40)pkz=zk−zk+1,k=0,1,2,⋯,and using our criterion for a basis transform, p=Pξ is indeed a basis for OB1/2.
These considerations therefore suggest that, when an infinite matrixP is given and we wish to use it as a basis transform in some (NF)-space E,x, we check the following:
(i)
The existence of an infinite matrixQ which is a two-sided inverse of P, i.e., (B.1)(ii)
The criterion (B.2) or (B′.2) or (B′′.2) holds
Suppose (i) is fulfilled. Then, the following situations may occur:
(i.1)Q is the unique two-sided inverse of P; then, (ii) above should be verified.
(i.2)P admits several two-sided inverses. If P defines a basis transform, there is a unique two-sided inverse Q such that Qp exists and Q satisfies (B.2). If P does not define a basis transform, the set of inverses
(41)Q:QP=PQ=IandQPexists.
The above remark leads to the following result.Corollary 21.
LetP be an infinite matrix having a two-sided inverse Q such that Qp converges. Then, P defines a basis transform if and only if the pair P,Q satisfies conditions (B.2) (equivalently (B′.2) or (B′′.2)).


## 4.2. Power Series Spaces

A class of Köthe sequence spaces which will be of special interest to us is that of so-called power series spaces. Letα=αnn∈ℕ be an increasing sequence of positive numbers and put for 0<R<+∞ fixed,
(42)akn=Re−1/kαn.Then, callingA=ank and associating with it the Köthe sequence space KA, it is well known that
(43)"KAisnuclear⇔limn→∞lognαn=0."In the caseR=∞, we put
(44)ank=ekαnand A=ank. It is then well known that
(45)"KAisnuclear⇔limsupn→∞lognαn<+∞."For an account of these results, we refer to [26]. In the case 0<R<+∞, the space KA is denoted by KA=ΛRα while the case R=∞, the notation KA=Λ∞α is used.Notice that, puttingrk=Re−1/k (0<R<+∞) or rk=ekαn (R=∞), then rk↑R or rk↑∞.The question now arises to determine whether or not a given (NF)-space with basis is similar to a space of the typeΛRα,e or Λ∞α,e, the latter spaces being called power series spaces.Now, we illustrate the problem by means of the following.Example 4.
The spaceOBR,ξ, 0<R<+∞. Define for each k∈ℕ and f∈OBR,
(46)fk=supz≤Re−1/kfz.Aslimn→∞logn/n=0, we can take α=αnn∈ℕ with αn=n for all n∈ℕ. Consequently, OBR,ξ≃ΛRℕ,e.Example 5 (the spaceOBℂ,ξ.
In this caseR=∞ and we may describe the topology on Oℂ by the system of seminorms P=∙k:k∈ℕ where for each k∈ℕ and f∈Oℂ,
(47)fk=supz≤ekfz.Hence, for eachn∈ℕ, znk=supz≤ekzn=enk.Assupn∈ℕlogn/n<+∞, we can take α=αnn∈ℕ with αn=n for all n∈ℕ. Consequently, Oℂ,ξ≃Λ∞ℕ,e.In the sequel, we denote for convenienceΛRα,e≃Kα,R, the associated supremum space will be denoted by Sα,R.


## 4.3. Similarity Theorem

The following criterion will completely answer the equation we mentioned above, namely, to know under which conditions a (NF)-space with basis is similar to some power series spaceKα,R. To reach this end, let us introduce the following notations.Given the basisx=xnn∈ℕ of E and the sequence α=αnn∈ℕ determining Kα,R, we put for each k∈ℕRk=limsupn→∞xnk1/αn and rk=liminfn→∞xnk1/αn. Clearly, rk≤Rk for each k∈ℕ.Suppose thatR is finite (the case R=+∞ is treated in a similar way).Theorem 22 (similarity theorem basis [37]).
LetE,x be a (NF)-space with basis and let Kα,R be some power series space. Then, E,x≃Kα,R if and only if
(i)
rk↑R(iii)
Rk<R for each k∈ℕAfter having established which (NF)-spaces with basis are similar to a power sequence space (Theorem22), we now aim to apply the criterion obtained for basis transforms (Theorem 18) to the case of power sequence spaces. To do this, consider the power sequence space Kα,R with basis e=ett∈ℕ. Then, we know that for each m∈ℕ,
(48)etm=Re−1/mαtfor0<R<∞,emαtforR=∞.In the sequel, we putrm=R−1/m (0<R<+∞) or rm=em. So, if P is an infinite matrix acting on e, then by a direct translation of Theorem 18, the authors of [37] established the following fundamental results.Theorem 23 (basis criterion).
LetKα,R be a power sequence space and let P be an infinite matrix. Then, Pe is a basis for Kα,R if and only if
(i)
there exists an infinite matrixQ such that PQ=QP=I(ii)
for eachm∈ℕ, there exist ℓ∈ℕ and Km>0 such that for all n,k,t∈ℕ,(49)QnkPktrmαt≤Kmrℓαn.Definition 24 (see [37]).
LetP be an infinite matrix which has two-sided inverse Q and let r>0 be fixed. Then, for n∈ℕ, we put
(50)JnP,r=supk,tQnkPktrαtand call
(51)JP,r=limn→∞JnP,r1/αnif all JnP,r are finite, and +∞ otherwise.
From Theorem23, the following result was deduced in [37].Theorem 25 (criterion for basis transforms [37]).
LetKα,R be a power sequence space and let P be an infinite matrix with two-sided inverse Q. Then, P determines a basis transform in Kα,R if and only if for all 0<r<R,
(52)JP,r<R.Remark 26.
We saw before that the criterion (B′.2) is equivalent to (B′′.2). This of course prompts the introduction of the following entities: given r>0, we put
(53)ZnP,r=∑k∑tQnkPktrαt,(54)ZP,r=limn→∞ZnP,r1/αn.
This leads to the following important result which was shown in [37].Theorem 27 (general criterion for basis transforms [37]).
LetKα,R be a power sequence space and let P be an infinite matrix having a two-sided inverse Q. Then, P determines a basis transforms in Kα,R if and only if for all 0<r<R,
(55)ZP,r<R.Remark 28.
It should be noted that Cannon [15] gave a criterion of the form in (54) for certain spaces of holomorphic functions, although he did not prove that P defines a basis transform but only that every function can be represented by a series in Pξ.


## 4.4. Cannon-Whittaker’s Criterion Revisited

Consider the spaceOBR,ξ. In [15], Cannon considered an infinite matrix P of the following type:
(i)
P is row-finite(ii)
P possesses a row-finite two-sided inverses Q [5]For each0<r<R, the following entities were introduced [40]:The Cannon sumλnP,r≤∑k,rQnkPkr for the Cannon bases. As for the non-Cannon base (general base), it introduced the Cannon sum in the form:
(56)FnP,r=supt1,t2supz≤r∑k=t1t2Qnk∑kPkszs.Consequently, for the corresponding Cannon functions(57)λP,r=limsupn→∞λnP,r1/n,κP,r=limsupn→∞FnP,r1/n.


## 4.5. A Refinement of Cannon-Whittaker Criterion for Effectiveness

Cannon proved that the set of polynomials determined byPξ is effective in ∣z∣<R if and only if κP,r<R for all 0<r<R and that for Cannon sets λP,r=κP,r. When we look at JnP,r in the case under consideration, it is clear that for all n∈ℕ and 0<r<R,
(58)JnP,r≤FnP,r≤λnP,r,JP,r≤κP,r≤λP,r.Besides the fact that Cannon only proved the effectiveness of the setPξ under the condition κP,r<R for all 0<r<R while we pointed out that JP,r<R for all 0<r<R implies that Pξ is a basis, it obviously follows from λP,r<R for all 0<r<R that JP,r<R for all 0<r<R. Hence, the given condition here using JP,r is weaker than the one obtained by Cannon using λP,r in the case of so-called Cannon sets (i.e., sets Pe for which λP,r=κP,r). There is even more to say. Since we can also use ZP,r to establish whether or not Pξ is a basis for OBR and since clearly for all n∈ℕ and 0<r<R,
(59)JnP,r≤FnP,r≤λnP,r≤ZnP,r,whence
(60)JP,r≤κP,r≤λP,r≤ZP,r.We may conclude thatλP,r may also be used in the case of non-Cannon sets. Of course, it should also be stressed that the matrices P we are considering need not be row-finite.Let us illustrate the previous observations in the case ofOB1,ξ, the space of holomorphic functions in the unit ball.Example 6.
Consider the function1−z−1=∑j=0∞zj∈OB1. Then, we claim that set 1/1−z,z,z2,⋯,zn,⋯ is a basis forOB1. Indeed, the associated infinite matrix P is given by
(61)P=111⋯110100⋯00010⋯0::⋱⋱0:0⋯00100⋯0001.
Note thatP is upper-triangular and has diagonal elements 1. Hence, it possesses a two-sided inverse Q given by
(62)Q=1−1−1⋯−1−10100⋯00010⋯0::⋱⋱⋯:0⋯00100⋯0001.
Takingα=αn=ℕ, a straightforward calculation yields for each 0<r<1 that
(i)
J0P,r=supt∈ℕ,t≥1rt=1(ii)
JnP,r=rn,n≥1
Consequently,JP,r=limsupn→∞JnP,r1/n=1, so that JP,r<1 for each 0<r<1. Hence, Pξ is a basis for the space OB1.


## 5. Application: Chebychev Polynomials

As an actual application of the criterion of basis transforms which refines the analog one of Cannon-Whittaker, we deal with the set of Chebychev polynomials of the first kind (see [41]).Consider the Chebychev polynomials of the first kind, namely,(63)Pnz=∑k=0n/1n2kzn−2kz2−1k.This setPnzn≥0 forms a basic set in the sense of Whittaker [5] and also in the sense of Theorem 16. In view of the expression (53) and (54) and after having the two matrices Qnk and Pkt, we may have
(64)ZnP,r=liminfn→∞∑k=0n∑j=0n/2QnkPktrk−2jr2−1j.Applying the well known Stirling’s formulan!~2πnne−n as n→∞ to the combinatorics coefficients
(65)n2k=n!2k!n−2k!,we can show after some calculations that the above series in (64) converges well only for r≤1. Consequently, ZP,r≤r, r≤1. Hence, Pnzn≥0 is a basis for OB1 and in the terminology of Cannon-Whittaker is effective in unit disc B1.The material given in Sections3 and 4 opened the door to discuss the basis transforms in further functional spaces. In the following section, we investigate some counterpart results in nuclear Silva spaces with basis.


## 6. Nuclear Silva Spaces with Basis

For Fréchet spaces, and even for nuclear Fréchet spaces, the existence of a basis is not always guaranteed, as shown in the fundamental paper by Mitiagin-Zobin and Mitiagin [34, 35]. This problem had been treated by Cnops and Abul-Ez [37] and has been exhaustive demonstrated in Sections 3 and 4 above, where nuclearity criterion is given for Fréchet spaces with basis, as well as basis transforms in such spaces. Related aspects had been treated for the space of holomorphic functions [10] and hyperholomorphic functions [6] to give that the set of Bessel polynomials is a basis in a functional space, and consequently, we have a (NF)-space.It is worth mentioning that Abul-Ez [25] pointed out that the space of entire functions of finite growth order and type is a (NF)-space, as well as he studied the existence of basis in such space. In this section, we are going to show that if E is a (NF)-space and F=E′β is the corresponding Silva space, then F admits a basis x′kk∈ℕ if and only if x′kk∈ℕ is the dual basis of a basis in E. Moreover, a nuclearity criterion is given for a Silva space with basis.By definition (see [27], p. 264), a Silva space F is the inductive limit of a sequence of Banach spaces Fss∈ℕ such that for each s∈ℕ, the unit ball of Fs is contained in the unit ball of Fs+1 and is compact in Fs+1. Several other characterizations of Silva spaces may be given. We mention here the following:
(i)
A locally convex spaceF is a Silva space if and only if F is the strong dual of a Fréchet-Schwartz space E. If .ss∈ℕ is a defining sequence of seminorms on E, then F may also be considered as being the inductive limit of the sequence of Banach spaces E.s′′s∈ℕ whereby for each s∈ℕ,E.s′′ is the linear hull of the polar of the closed .s-unit ball (see again [27])(ii)
A locally convex spaceF is a Silva space if and only if F is a complete DF-Schwartz space in which each null sequence converges locally (see [28], Corollary 12.5.9, and [42])Moreover, ifF=Eβ′ is a Silva space, E being a Fréchet-Schwartz space, then F is nuclear if and only if E is nuclear (see, e.g., [28], Theorem 21.5.3).Now, letF be a Silva space and let xnn∈ℕ be a basis in F. Then, by (ii) and the continuity theorem (see [28], Theorem 14.2.5), xnn∈ℕ is already a Schauder basis in F.Remark 29.
(1)
As an inductive limit of Banach spacesFss∈ℕ, we have the following(66)F1⊂F2⊂F3⋯⊂Fn⊂⋯⊂Fand we can describe the topology on F by the norms .s′ of the spaces Fs. This means that .s′ is defined on Fs for x∈Fs:xs′≤xs+1′.
In Silva spaces, we have the following criterion of convergencexn→x if and only if
(a)
there exists such that xn∈Fs for all n and x∈Fs(b)
xn→.s′x(2)
Every nuclear Fréchet space is a Fréchet-Schwartz space so the dual of such spaces are Silva spaces(3)
The spacesOΩ of functions holomorphic (regular) in an arbitrary neighborhood of a compact set Ω are of this type. Since we are going to prove that having a basis in F is equivalent to having a basis in F′, we can transfer the basis criterion for nuclear Fréchet spaces to nuclear Silva spaces


## 7. Series Representation in Silva Spaces

As a Silva space is anF+-space in the sense of Newns, Theorem 3.2 of [4] and part of its proof may be reformulated as follows.Theorem 30 (see [4]).
LetF=Ls∈ℕFs be a Silva space and let xnn∈ℕ be a sequence in F with xn≠0 for all n∈ℕ.Fors∈ℕ fixed, suppose that each x∈Fs is represented in F by a series of the form ∑n∈ℕαnxn. Then, there exists σ∈ℕ (depending only on s) such that for each x∈Fs,x=∑n∈ℕαnxn in Fσ.Moreover, callingUs the space of sequences αnn∈ℕ in ℂ such that the series ∑n∈ℕnαnxn converges in Fσ to some element of Fs and putting for each αnn∈ℕ∈Us,
(67)αnn∈ℕUs=∑n∈ℕαnxns+maxℓ,t∑k=ℓtαkxkσ,we have that Us,.Us is a Banach space.Finally, the linear mappingu:Us→Fs given by uαnn∈ℕ=∑n∈ℕαnxn is a continuous surjection.Now, suppose thatxnn∈ℕ is a basis for the Silva space, then clearly the mapping u:Us→Fs is a continuous bijection, whence by Banach’s homeomorphism theorem 3, u is bicontinuous. This leads to the following:Theorem 31.
LetF=Ls∈ℕFs be a Silva space with basis xnn∈ℕ and let for each s∈ℕ,Us be defined as in Theorem 30. Then, Us and Fs are linearly homeomorphic for all s∈ℕ.Corollary 32.
LetF=Ls∈ℕFs be a Silva space, and xnn∈ℕ be a basis for F, and let for s∈ℕ,σ∈ℕ be such that x∈Fs admits in Fσ the expansion x=∑n∈ℕαnxn. If the continuous linear functional αn on F is not identically zero on Fs, i.e., αnFs≠0, then xn∈Fσ.Proof.
Suppose that for somen∈ℕ,αnFs≠0 and xn∉Fσ. Then, taking y∈Fs such that αny≠0, we have that, as in Fσ,y=∑k∈ℕαkxk, the partial sums Sny=∑i=1nαiyxi and Sn−1y=∑i=1n−1αiyxi belong to Fσ whence αnyxn=Sny−Sn−1y∈Fσ, a contradiction.


## 8. Nuclear Silva Spaces with Basis

LetF be a nuclear Silva space. Then, as we saw in Section 6, F=Eβ′ whereby E is a (NF)-space and E≃Fβ′. In what follows, we therefore denote a basis for F (if it exists) by xk′k∈ℕ and call xkk∈ℕ the corresponding biorthogonal sequence in E, i.e., xℓ∈E with xℓxk′=δℓk for all k,ℓ∈ℕ.Theorem 33.
LetF=Eβ′ be a (NS)-space and let xk′k∈ℕ be a sequence of nonzero elements in F. Then, xk′k∈ℕ is a basis in F if and only if xkk∈ℕ is a basis in E.Proof.
Ifxkk∈ℕ is a basis for E then E being a (NF)-space, the sequence xk′k∈ℕ is a basis in Eβ′=F (see, e.g., [28], Theorem 21.10.6). Conversely, suppose that xk′k∈ℕ is a basis for F. Then, since xk′k∈ℕ is a Schauder basis, the biorthogonal system xk,xk′k∈ℕ exists. We prove that the biorthogonal system xk,xk′k∈ℕ is complete, i.e., E=Spanxk:k∈ℕ¯.Indeed, callL=Spanxk:k∈ℕ¯ and suppose that L≠E. Then, if x∈E\L, by the Hahn-Banach theorem, there ought to exist x′∈E′ such that x′x=1 and x′L=0. But, as x′=∑k=1∞xkx′xk′ and xkx′=0 for all k∈ℕ,x′=0, thus yielding a contradiction.Now, we prove that the complete biorthogonal systemxk,xk′k∈ℕ satisfies Haslinger’s criterion [30], i.e., for all s∈ℕ, there exists σ∈ℕ such that
(68)supk∈ℕxk′σ′xks<+∞.To this end, takes∈ℕ fixed. On the one hand, by the representation of seminorms, we have for each k∈ℕ that
(69)xks=xks′′=supxk′s′=1xkx′.On the other hand, in view of Theorem31, there exists σ∈ℕ and a corresponding space Us such that Us is linearly homeomorphic to Fs. Denoting again by u this isomorphism, we may thus find C>0 such that for all x′∈Fs,
(70)xk′s′≥1Cu−1x′Us.Hence, for all x′∈Fs and k∈ℕ,
(71)xk′s′≥1Cxkx′xk′σ′=1Cxkx′xk′σ′.Consequently,(72)supk∈ℕxksxk′σ′≤C.By virtue of Haslinger’s criterion (see [30]), xkk∈ℕ is a basis for E.Remark 34.
The preceding theorem shows that ifE is a (NF)-space, then each basis in F=Eβ′ is the dual basis of a basis in E. In view of [28], Theorem 21.10.6, if xk is absolute, so is xk′. But every basis in Eβ′ is absolute, so we obtain that each basis in a (NS)-space F is an absolute basis. This duality also leads to the following.


## 9. A Nuclearity Criterion for Silva Spaces with Basis

As was shown in [32], for a vector space E provided with a system of seminorms P and having a Schauder basis xkk∈ℕ, the following are equivalent:
(i)
E is nuclear(ii)
For eachp∈P, there exists q∈P such that(73)∑k∈ℕpxkqxk<+∞.In this section, a criterion for the nuclearity of Silva spacesF with basis is proved whereby only the sequence of norms of the defining Banach spaces Fs is used. In such a way, a criterion for nuclearity is obtained which avoids the use of the system of seminorms defining the inductive limit topology on F.Theorem 35.
LetF=Eβ′ be a Silva with basis xk′k∈ℕ. Then, F is nuclear if and only if for each s∈ℕ, there exists σ∈ℕ such that
(74)∑k∈ℕxk′σ′xk′s′<+∞.Proof.
LetF be nuclear. Then, E is a (NF)-space with basis xkk∈ℕ (see Theorem 30) whence for each s∈ℕ, there exists σ∈ℕ such that (see, e.g., [32])
(75)∑k∈ℕxksxkσ<∞.Moreover, puttingA=aksk,s∈ℕ with aks=xks,E is linearly homeomorphic to the nuclear Köthe sequence space KA and so the topology on E is also determined by the defining sequence of norms .s with
(76)xs=∑k∈ℕxk′xxks,x∈E(see, e.g., [26]).Taking duals, we thus have that(77)F=Eβ′≃Ls∈ℕE′,∙s′≃Lt∈ℕE′,∙t′,whereby for each t∈ℕ and x′=E′,∙t′,
(78)x′t′=supk∈ℕ1xktx′xk.Notice that in particular for eachk∈ℕ,
(79)xk′t=1xkt.Combining (75) and (79), we thus obtain that for each t∈ℕ, there exists τ∈ℕ such that
(80)∑k∈ℕxk′τxk′t<+∞.Now, lets∈ℕ be fixed. Then, there exist t∈ℕ and Ks∗>0 such that for all k∈ℕ,
(81)Ks∗xk′s≥xk′t,while for thatt∈ℕ, there ought to exist τ∈ℕ such that (80) holds. However, for this τ, there exist σ∈ℕ and Kτ>0 such that for all k∈ℕ,
(82)Kτxk′τ≥xk′σ′.Consequently, we obtain that for eachs∈ℕ, there exists σ∈ℕ such that (74) holds.Conversely, suppose that for eachs∈ℕ, there exists σ∈ℕ such that (74) holds. Calling for each k,s∈ℕ,
(83)aks=1xk′s′ifxk′∈E′,∙s′=Fs,0ifxk′∉E′,∙s′=Fs,and putting A=aks, we have that the Köthe sequence space KA is a (NF)-space, whence its topology is also determined by the sequence of norms ∙s,s∈ℕ, with
(84)ξs=supk∈ℕξaks,ξ=ξkk∈ℕ∈KA.Its dualKA′ is thus given by
(85)KA′=y=akk∈ℕ∈ω:∃s∈ℕwithys=∑k∈ℕakaks<∞,and of course, KAβ′ is a nuclear Silva space.Now, defineB:F→ω by
(86)B∑n∈ℕαnx′xn′=αnx′n∈ℕ.Then, we claim thatBF=KA′.Indeed, ify∈KA′, then there exists s∈ℕ such that ∑k∈ℕ∣αk∣/aks whence ∑k∈ℕαkxk′s′<∞ or ∑k∈ℕαkxk′ converges absolutely in Fs and so ∑k∈ℕαkxk′∈F. This implies that KA′⊂BF.Now, letx′∈F admit the series representation x′=∑k∈ℕαkx′xk′. Then, there ought to exist s∈ℕ such that x′=∑k∈ℕαkx′xk′∈Fs, the convergence being valid in Fs whence supk∈ℕαkx′xk′s′=K<∞. But for this s∈ℕ, there exists by assumption σ∈ℕ such that (74) holds.We claim thatBx′σ exists. Indeed,
(87)Bx′σ=∑k∈ℕαkx′akσ=∑k∈ℕαkx′aksaksakσ≤K∑k∈ℕaksakσ<∞.Consequently,Bx′∈KA′ and so BF⊂KA′. Obviously, B is an isomorphism between F and KA′.Now, we show thatB−1 is continuous. Indeed, take y∈KA′. Then, there exists s∈ℕ such that
(88)ys=∑k∈ℕαkx′aks<∞.Puttingx′=B−1y, we have
(89)ys=∑k∈ℕαkxk′s′≥∑k∈ℕαkxk′s′=x′s′.By virtue of the open mapping theorem,F and KA′ are linearly homeomorphic whence F is nuclear.Remark 36.
Another possible approach to the proof of Theorem35 is one using the Grothendieck-Pietsch criterion for nuclearity for Köthe sequence spaces (see [29, 36]). Indeed, from our criterion, it follows that the basis considered is absolute and an explicit expression for the system of seminorms defining the topology of F can be given in terms of a Köthe sequence space. Applying the Grothendieck-Pietsch criterion then yields the nuclearity of the space F. On the other hand, if F is nuclear and has a basis, it follows from Theorem 33 that this basis is absolute whence F is linearly homeomorphic to some Köthe sequence space KA, which, by assumption upon F, is nuclear. Hence, the Grothendieck-Pietsch criterion holds which can then be translated into the criterion of the above Theorem 35. For the Grothendieck-Pietsch criterion, we refer to ([28], Theorem 21.6.2).Example 7.
In [32], Kamthan introduced the following Fréchet space OARez<A,ζ of holomorphic functions. Let λ=λnn∈ℕ be a fixed strictly increasing sequence of positive real numbers. With each sequence ann∈ℕ such that limn→∞supan/λn≤−A, we associate the function
(90)fz=∑n=1∞anezλn,z∈ℂ.Takeε>0 arbitrary chosen and consider the half-plane
(91)Aε=z∈ℂ:Rez≤A−ε.Then, for eachn∈ℕ and z∈Aε,
(92)anezλn≤aneAλne−ελn.But, in virtue of the assumption,(93)limn→∞suploganλn≤−A,we find that
(94)aneAλn=elogan+Aλ,whence there exist C>0 such that
(95)supn∈ℕaneAλn≤C.Consequently,(96)∑n∈ℕsupz∈Aεanezλn≤C∑n∈ℕe−ελn<+∞,whence the series defining f is normally convergent on each Aε,ε>0. Consequently, f∈OARez<A.We callOARez<A the subspace of ORez<A consisting of the elements f just defined and provide OARez<A with the system P of seminorms pkwith
(97)pkf=supRez≤A−1/kfz.Then, it was proved by Kamthan thatOARez<A,P is a Fréchet space. From the definition itself of the elements f in OA, it follows that the sequence of functions ζ=ezλnn∈ℕ is a basis for the space OARez<A,P. Then, we claim that OARez<A,ζ is a (NF)-space. Indeed, from the definition itself of OARez<A, it follows that ζ is a basis for it.Having thatOARez<A,ζ is a (NF)-space, then using the discussion in Section 6, it can be seen that it is a (NS)-space with basis. Now, let k∈ℕ be fixed and take any l∈ℕ with l>k. Then,
(98)∑n=1∞ezλnkezλnl=∑n=1∞eA−1/kλneA−1/lλn=∑n=1∞e1/l−1/kλn<∞.Therefore, by our criterion (Theorem35), the nuclearity is proved.Example 8.
Consider the spaceOS¯R,ξ of holomorphic functions in two complex variables z,w, provided with the countable system P of seminorms pk where
(99)pkf=supS¯rkfz,w,and S¯r is the closed hypersphere defined by
(100)supS¯rk=z,w∈ℂ:z2+w2≤rk2.Again,rkk∈ℂ is a strictly increasing sequence of positive numbers with 0<rk<R and limk→∞rk=R. As is well known, OS¯R,ξ is a Fréchet, and consequently, it can be proved that it is a Silva space. Moreover, as it was shown again by the Taylor series at the origin for any f∈OS¯R, the sequence
(101)ξ=zmwnm,n∈ℕis a basis for OS¯R. Although it is known that OS¯R,ξ is a (NF)-space and then a (NS)-space, our criterion will yield the nuclearity of it in a very simple way. Indeed, as for each k,m,n∈ℕ, it can be proved that (see [2])
(102)zmwnk=rkm+nσm,n,where σm,n is given by
(103)σm,n=m+n1/2m+nmm/2nn/2whenm,n>0,1whenm=0orn=0.Then, we obtain that, takingk∈ℕ fixed, for each ℓ>K,
(104)∑m=0∞∑n=0∞zmwnkzmwnℓ=∑m=0∞∑n=0∞rkrℓm+n=rℓrℓ−rkrℓ2<+∞,the nuclearity of this space is proved.


## 10. General Remarks and Comments

The properties of series of the form∑i=0∞ciPiz, z∈ℂ where Piz, i=0,1,⋯ are prescribed polynomials and ei chosen in a field K of scalars, widely differ according to the particular chosen polynomials. For example, the region of convergence (which is called the region of effectiveness) may be a circle (for Taylor series), an ellipse (for series of Legendre polynomials), and a half-plane (for Newton’s interpolation series). Whittaker [40], in his attempt to find the common properties exhibited by all these polynomials, introduced the notion of basic sets of polynomials. In his work [5], he defined the basic sets, basic series, and effectiveness of basic sets. In [15–17], Cannon obtained the necessary and sufficient condition for the effectiveness of basic sets for classes of functions of finite radii of regularity and entire functions. In the classical treatment of the subject of basic sets [5], the methods for establishing effectiveness depend on the region of effectiveness and the class of functions for which the set is effective.The first attempt at some uniformity among the different methods was made by Newns who gave in [4] a topological approach leading to a general theory of effectiveness. It is well known that a lot of classical function spaces are important examples of so-called nuclear Fréchet spaces, for example, spaces of null solutions of elliptic partial differential operators with constant coefficients such as the Cauchy-Riemann operator and the Laplace operator.On an abstract level, the problem of effectiveness of basic sets of polynomials in spaces of holomorphic functions as introduced by Cannon-Whittaker may be therefore considered as being related to the problem of the change of bases in nuclear Fréchet spaces as well as in other related spaces.In the present work, we show that general criteria for basis transforms are obtained for the nuclearity of Fréchet spaces with basis which are applied to characterize basis transforms in terms of infinite matrices in classes of nuclear Fréchet spaces. This study is considered to be a refinement of those given by Cannon, Whittaker, and Newns and all relevant generated topics.In such a way analog results are given concerning nuclear Silva spaces with bases. This might form a starting point for further investigations regarding the basis transforms in more general locally convex spaces or higher dimensional spaces with different domains of convergence. Finally, it will be expected in the forthcoming work to study basis transforms in(i)
spaces involving product bases, inverse bases, transpose bases, derived and integrated bases, etc.(ii)
spaces of entire functions having finite growth(iii)
spaces of holomorphic functions in Faber regions(iv)
spaces of several complex variables(v)
monogenic function spaces in the framework of Clifford analysis.



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