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

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.


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, when E denotes the space and ðx n Þ n∈ℕ stands for a basis in E, each element x ∈ E admits a (unique) decomposi-tion of the form ∑ ∞ n=1 a n ðxÞx n whereby for each n ∈ ℕ, a n 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][2][3][4][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][7][8][9][10][11][12][13][14].
Cannon and Whittaker [5,[15][16][17] studied the existence of basic sets of polynomials of one complex variable (as bases) in the classical spaces OðBðrÞÞ, and later, many authors considered such problem in the space Oð BðrÞ × BðrÞÞ, 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 fP n ðzÞg, z ∈ ℂ, in the closed ball BðrÞ, it means that the set forms a base for the class E of holomorphic functions regular in BðrÞ with norm given by Mðf , rÞ, 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 example OðBðrÞÞ. 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 Oð BðrÞÞ 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 ℂ n [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 vectors ðx n Þ n∈ℕ in a (NF)-space E with basis is itself a basis for E. This leads to the notion of basispreserving 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][27][28][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.

Preliminaries
In the sequel, P = ðP ij Þ, Q = ðQ ij Þ, and T = ðT ij Þ 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 ðV, bÞ will be used, where V is the vector space and b = fb n g +∞ n=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., φðb n Þ = c n for all n); ðV, bÞ and ðV, cÞ are called similar. This relation is written ðV, bÞ ≃ ðV, cÞ. Obviously, it is not necessarily true that if b and f are two bases for V then ðV, bÞ ≃ ðV, f Þ. If it is true however we speak of conjugate bases. If x = ðx n Þ is a sequence of vectors in V and for each k, the series converges, we simply write y = Px.

Köthe Sequence Spaces.
Let ω denote the space of all infinite sequence of complex numbers and let A = ða k n Þ be an infinite matrix of real nonnegative numbers such that a k n ≤ a k+1 n for all n, k and for each n, these exist k such that a k n ≠ 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: with the topology given by the seminorms k•k k . Note that P = fk•k: k ∈ ℕg is a sequence of norms on KðAÞ, which is moreover a system of norms on KðAÞ. Putting for each n ∈ ℕ, e n = ðδ kn Þ k∈ℕ , we cite the following important result proved by Pietsch in [29].

Remark 2.
It is well known that if ðV, xÞ is a (NF)-space and the topology is given by seminorms p k , then ðV, xÞ ≃ ðKðAÞ, eÞ, where A is given by the relation a k n = p k ðx n Þ. This makes it possible to use alternatively the seminorms p k , kφð:Þk k , or jφð:Þj 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][31][32]). Now, let ðE, P Þ be an (F)-space where it is henceforth assumed that the countable system of seminorms P = fp k : k ∈ ℕg is in fact a sequence of norms. We therefore write The following result given in [28,33] is useful in the sequel.
Theorem 3 (Banach's homeomorphism theorem). Let ðE, P Þ and ðF, QÞ 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). Let ðE, P Þ be an (F)space. Then, it is called a (NF)-space if for each k ∈ ℕ, there 2 Journal of Function Spaces exist ℓ ∈ ℕ and a sequences ðy n Þ n∈ℕ in E and ðL n Þ n∈ℕ in E ′ such that for each x ∈ E, where kL n k ℓ ′ = sup x∈E,kxk ℓ ≤1 jL n ðxÞj and if kL n k ℓ ′ = ∞, then ky n k k = 0. Now, we recall some functional theorems concerning (NF)-spaces with basis.
Theorem 7 (see [31]). Let E be a (NF)-space with a complete biorthogonal system ðx n , x ′ n Þ n∈ℕ : Then, x = ðx n Þ n∈ℕ is a basis for E if and only if for each k ∈ ℕ, there exist ℓ ∈ ℕ, and K k > 0 such that for each y ∈ E, Theorem 8 (Haslinger's criterion for a (NF)-space with basis [30]). Let ðE, xÞ be a (NF)-space with basis and let ðy n , y ′ n Þ n∈ℕ be a complete biorthogonal system in E. Then, ðy n Þ n∈ℕ is a basis in E if and only if for each k ∈ ℕ, there exist ℓ ∈ ℕ and K k > 0 such that for each j ∈ ℕ, 2.3. Nuclear Köthe Sequence Spaces. As we have seen that a Köthe sequence space KðAÞ is an (F)-space with basis e, we have also according to [29].

Remark 11.
(i) Notice that, although SðAÞ is always an (F)-space, it need not have a basis. Indeed, it is sufficient to consider A = ða k n Þ with a k n = 1 for all n, k ∈ ℕ (ii) Notice also that for this matrix A, the condition (6) needed for the nuclearity of KðAÞ is not satisfied. This might suggest that the nuclearity of KðAÞ could be related to the existence of a basis for SðAÞ. This is indeed the case as it was shown in [26,[30][31][32]36], which can be stated in the following theorem  This theorem implies immediately that if KðAÞ is nuclear, then ðKðAÞ, eÞ ≃ ðSðAÞ, eÞ.
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.

A Nuclearity Criterion for (F)-Spaces with Basis
In this section, we give a criterion stating under which conditions an (F)-space E with basis x = ðx n Þ n∈ℕ is nuclear. Let us recall that for an (F)-space E with x = ðx n Þ n∈ℕ , the associated Journal of Function Spaces linear functional x ′ n , n ∈ ℕ is bounded according to Schauder's theorem [28] which states that "if E is an (F)-space and ðx n Þ n∈ℕ is a basis for E, then ðx n Þ n∈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 ðk•k m Þ m∈ℕ of norms.
Theorem 13. Let E be an (F)-space with basis x = ðx n Þ n∈ℕ . Then, the following are equivalent: Proof. Let E 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 Since Conversely, assume that for each t ∈ ℕ, there exists s ∈ ℕ such that Let S be the following space: S = a = a n ð Þ n∈ℕ : a n ∈ ℂ for all n ∈ ℕ and a Then, clearly, S is a supremum space associated with A = ða k n Þ where a k n = kx n k k , hence a Fréchet space. By the condition assumed, it follows from Theorem 12 that S is nuclear. Now, consider the map ψ : S → E such that ψ : a n ð Þ n∈ℕ → 〠 ∞ n=0 a n x n : Obviously, ψ is linear, continuous, and injective. We now prove that ψ is also surjective, i.e., if y ∈ E with y = ∑ ∞ n=0 a n x n , then a = ða n Þ n∈ℕ ∈ S. Fix s ∈ ℕ, then as ∑ ∞ n=0 a n x n converges in F, lim n→∞ ja n jkx n k s < ∞ whence b s = sup n∈ℕ a n j j x n k k s < +∞: Now, let t ∈ ℕ be chosen arbitrarily and let s ∈ ℕ such that ∑ ∞ n=0 ð∥x n ∥ t /∥x n ∥ s Þ < +∞. Hence, Consequently, a = ða n Þ n∈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 Theorem 13 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 ðx n Þ n∈ℕ satisfying (8), we put (2) As we have seen, SðAÞ is a nuclear supremum space which is isomorphic to E. Here, A = ða k n Þ with a k n = kx n k k (3) From the isomorphism ψ constructed in the proof above, it follows that if ðE, xÞ is a (NF)-space with basis ðx n Þ n∈ℕ , then the following isomorphisms hold: We thus obtain that the natural system of seminorms on E is equivalent to (i) the system of norms (ii) the system of norms 4 Journal of Function Spaces Note that in both cases, a n = x′ n ðyÞ, 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 Theorem 13.
ðr k Þ k∈ℕ being a strictly increasing sequence of positive numbers with 0 < r k < R, and lim k→∞ r k = R. One may take, for exam- As we saw before that ðOðBðRÞÞ, ξÞ is a Fréchet space with basis ξ = ðz n Þ n∈ℕ , although it is well known that ðOðBðRÞÞ, ξÞ is a (NF)-space (see [29]), its proof is not trivial.
As will be seen now, the criterion just proved yields the nuclearity of ðOðBðRÞÞ, ξÞ in an easy way. Indeed, as for each k ∈ ℕ and n ∈ ℕ, Taking k ∈ ℕ fixed, then for each ℓ > k, we have So, the criterion applies.
Remark 15. Notice that, as we previously mentioned, with A = ða k n Þ and a k n = sup jzj≤r k jz n j, whence the natural system of seminorms P = fk•k k : k ∈ ℕg and the system of seminorms P KðAÞ and P SðAÞ induced on ðOðBðRÞÞ, ξÞ are all equivalent. Here, where if f ∈ OðBðRÞÞ admits the Taylor series at the origin, then, for k ∈ ℕ, ½ f k = sup n∈ℕ jc n ja k n and k f k k,K = ∑ ∞ n=0 jc n ja k n .
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 each k ∈ ℕ and f ∈ OðBðRÞ, Thus, the equivalence between the systems of seminorms established above gives stronger results.
Again, ðr k Þ k∈ℕ is strictly increasing sequence of positive numbers with 0 < r k < R and lim k→∞ r k = R. As is well known, ðOðBðRÞ × BðRÞÞ, ξÞ is a Fréchet space. Moreover, as it is shown again by the Taylor series at the origin for any f ∈ OðBðRÞ × BðRÞÞ, the sequence ξ = ðu n v m Þ n,m∈ℕ is a basis for OðBðRÞ × BðRÞÞ. Although it is known that ðOðBðRÞ × BðRÞÞ, ξÞ 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, we obtain that, taking k ∈ ℕ fixed, for each ℓ > k, the nuclearity of this space is proved.

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 vectors ðx n Þ n∈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 5 Journal of Function 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 that E is a (NF)-space with basis ðx n , x′ n Þ n∈ℕ . Moreover, P = ðP ij Þ and Q = ðQ ij Þ stand for infinite matrices over ℂ and we denote the infinite identity matrix by I. In such a way, we formally write We start by stating the following result by Cnops and Abul-Ez [37].
Theorem 16 (uniqueness theorem). Suppose that ðy k Þ k∈ℕ is a basis for E. Then, P has two-sided inverse Q.
The following problem now arises: let E be a (NF)-space with basis ðx n Þ n∈ℕ and ðy n Þ n∈ℕ be a sequence in E satisfying the relations and formally, Under which condition on the matrices P and Q we conclude that ðy k Þ k∈ℕ is a basis for E? Remark 17. In fact, (31) and (32) do not assume that ðy k Þ k∈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 z k 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 OðBðRÞÞ, 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 sequence ðy k Þ k∈ℕ in E allow representations of the form (31) and (32). Then, a biorthogonal system ðy n , y ′ n Þ n∈ℕ forming a basis for E may be constructed if and only if Remark 19.
(1) From the isomorphism between ðE, ðx n Þ n∈ℕ Þ and SðAÞ and the derived equivalent system of norms on E (see also Theorems 13 and 12), we have that (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 (3) Some comments on the requirement that P has twosided 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 unique An example of such a ring R is the ring R up of all infinite upper-triangular matrices or the ring R lo of all infinite lower-triangular matrices. It is clear that if, e.g., P ∈ R up has all diagonal elements different from zero, then P has a two-sided inverse Q in R up 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 6 Journal of Function Spaces to the ring R. We address this phenomenon in the following example.
Example 3. Let R be the ring of all infinite row-finite matrices and P ∈ R given by P = ðP ij Þ ∞ i,j=1 , where that is, Then, P admits the two-sided inverse that is, : : ⋱ ⋱ : : Clearly, Q is not row-finite.
Remark 20. Applying P to the space ðOðBð1/2ÞÞ, ξÞ, Pξ is given by and using our criterion for a basis transform, p = Pξ is indeed a basis for OðBð1/2ÞÞ. These considerations therefore suggest that, when an infinite matrix P is given and we wish to use it as a basis transform in some (NF)-space ðE, xÞ, we check the following: 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 Q : QP = PQ = I and QP exists f g : The above remark leads to the following result.

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. Let α = ðα n Þ n∈ℕ be an increasing sequence of positive numbers and put for 0 < R < +∞ fixed, Then, calling A = ða k n Þ and associating with it the Köthe sequence space KðAÞ, it is well known that In the case R = ∞, we put a k n = e kα n ð44Þ and A = ða k n Þ: It is then well known that For an account of these results, we refer to [26]. In the case 0 < R < +∞, the space KðAÞ is denoted by KðAÞ = Λ R ð αÞ while the case R = ∞, the notation KðAÞ = Λ ∞ ðαÞ is used.
The question now arises to determine whether or not a given (NF)-space with basis is similar to a space of the type ðΛ R ðαÞ, eÞ or ðΛ ∞ ðαÞ, eÞ, the latter spaces being called power series spaces. Now, we illustrate the problem by means of the following.

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 space Kðα, RÞ. To reach this end, let us introduce the following notations.
Suppose that R is finite (the case R = +∞ is treated in a similar way). [37]). Let ðE, xÞ be a (NF)-space with basis and let Kðα, RÞ be some power series space. Then, ðE, xÞ ≃ Kðα, RÞ if and only if (i) rðkÞ↑R (iii) RðkÞ < R for each k ∈ ℕ After having established which (NF)-spaces with basis are similar to a power sequence space (Theorem 22), 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 Kðα, RÞ with basis e = ðe t Þ t∈ℕ . Then, we know that for each m ∈ ℕ,

Theorem 22 (similarity theorem basis
In the sequel, we put r m = R −1/m (0 < R < +∞) or r m = e m . So, if P is an infinite matrix acting on e, then by a direct trans-lation of Theorem 18, the authors of [37] established the following fundamental results.

Theorem 23 (basis criterion). Let Kðα, RÞ be a power sequence space and let P be an infinite matrix. Then, Pe is a basis for Kðα, RÞ if and only if
(i) there exists an infinite matrix Q such that PQ = QP = I (ii) for each m ∈ ℕ, there exist ℓ ∈ ℕ and K m > 0 such that for all n, k, t ∈ ℕ, Definition 24 (see [37]). Let P be an infinite matrix which has two-sided inverse Q and let r > 0 be fixed. Then, for n ∈ ℕ, we put and call if all J n ðP, rÞ are finite, and +∞ otherwise. From Theorem 23, the following result was deduced in [37].
Theorem 25 (criterion for basis transforms [37]). Let Kðα, RÞ be a power sequence space and let P be an infinite matrix with two-sided inverse Q. Then, P determines a basis transform in Kðα, RÞ if and only if for all 0 < 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 This leads to the following important result which was shown in [37].

Theorem 27 (general criterion for basis transforms [37]). Let
Kðα, RÞ 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 Kðα, RÞ if and only if for all 0 < r < R, Remark 28. It should be noted that Cannon [15] gave a criterion of the form in (54) for certain spaces of holomorphic 8 Journal of Function Spaces 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.

Cannon-Whittaker's Criterion Revisited.
Consider the space ðOðBðRÞÞ, ξÞ. 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 each 0 < r < R, the following entities were introduced [40]: The Cannon sum λ n ðP, rÞ ≤ ∑ k,r jQ nk jkP k k r for the Cannon bases. As for the non-Cannon base (general base), it introduced the Cannon sum in the form: Consequently, for the corresponding Cannon functions

A Refinement of Cannon-Whittaker Criterion for
Effectiveness. Cannon proved that the set of polynomials determined by Pξ is effective in |z | <R if and only if κðP, rÞ < R for all 0 < r < R and that for Cannon sets λðP, rÞ = κðP, rÞ. When we look at J n ðP, rÞ in the case under consideration, it is clear that for all n ∈ ℕ and 0 < r < R, J n P, r ð Þ≤ F n P, r ð Þ≤ λ n P, r ð Þ, J P, r ð Þ≤ κ P, r ð Þ≤ λ P, r ð Þ: ð58Þ Besides the fact that Cannon only proved the effectiveness of the set Pξ under the condition κðP, rÞ < R for all 0 < r < R while we pointed out that JðP, rÞ < R for all 0 < r < R implies that Pξ is a basis, it obviously follows from λðP, rÞ < R for all 0 < r < R that JðP, rÞ < R for all 0 < r < R. Hence, the given condition here using JðP, rÞ is weaker than the one obtained by Cannon using λðP, rÞ in the case of socalled Cannon sets (i.e., sets Pe for which λðP, rÞ = κðP, rÞ). There is even more to say. Since we can also use ZðP, rÞ to establish whether or not Pξ is a basis for OðBðRÞÞ and since clearly for all n ∈ ℕ and 0 < r < R, J n P, r ð Þ≤ F n P, r ð Þ≤ λ n P, r ð Þ≤ Z n P, r ð Þ, ð59Þ whence J P, r ð Þ≤ κ P, r ð Þ≤ λ P, r ð Þ≤ Z P, r ð Þ: We may conclude that λðP, rÞ 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 of ðOðBð1ÞÞ, ξÞ, the space of holomorphic functions in the unit ball.

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,

Journal of Function Spaces
This set fP n ðzÞg n≥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 jQ nk j and jP kt j, we may have Applying the well known Stirling's formula n!~ffi ffiffiffiffiffiffiffiffiffi 2πn n p e −n as n → ∞ to the combinatorics coefficients we can show after some calculations that the above series in (64) converges well only for r ≤ 1. Consequently, ZðP, rÞ ≤ r, r ≤ 1. Hence, fP n ðzÞg n≥0 is a basis for OðBð1ÞÞ and in the terminology of Cannon-Whittaker is effective in unit disc Bð1Þ. The material given in Sections 3 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.

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 ðx ′ k Þ k∈ℕ if and only if ðx ′ k Þ k∈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 ðF s Þ s∈ℕ such that for each s ∈ ℕ, the unit ball of F s is contained in the unit ball of F s+1 and is compact in F s+1 : Several other characterizations of Silva spaces may be given. We mention here the following: (i) A locally convex space F is a Silva space if and only if F is the strong dual of a Fréchet-Schwartz space E: If ðk:k s Þ s∈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 ðE k:k s ′ ′ Þ s∈ℕ whereby for each s ∈ ℕ, E k:k s ′ ′ is the linear hull of the polar of the closed k:k s -unit ball (see again [27]) (ii) A locally convex space F 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, if F = E β ′ 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, let F be a Silva space and let ðx n Þ n∈ℕ be a basis in F: Then, by (ii) and the continuity theorem (see [28], Theorem 14.2.5), ðx n Þ n∈ℕ is already a Schauder basis in F: (1) As an inductive limit of Banach spaces ðF s Þ s∈ℕ , we have the following and we can describe the topology on F by the norms k:k s ′ of the spaces F s . This means that k:k s ′ is defined on F s for x ∈ F s : kxk s ′ ≤ kxk s+1 ′ : In Silva spaces, we have the following criterion of convergence x n → x if and only if (a) there exist s such that x n ∈ F s for all n and x ∈ F s (b) x n → k:k s ′ x (2) Every nuclear Fréchet space is a Fréchet-Schwartz space so the dual of such spaces are Silva spaces (3) The spaces OðΩÞ 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

Series Representation in Silva Spaces
As a Silva space is an F + -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]). Let F = L s∈ℕ F s be a Silva space and let ðx n Þ n∈ℕ be a sequence in F with x n ≠ 0 for all n ∈ ℕ: For s ∈ ℕ fixed, suppose that each x ∈ F s is represented in F by a series of the form ∑ n∈ℕ α n x n : Then, there exists σ ∈ ℕ (depending only on s) such that for each x ∈ F s , x = ∑ n∈ℕ α n x n in F σ :

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Moreover, calling U s the space of sequences ðα n Þ n∈ℕ in ℂ such that the series ∑ n∈ℕn α n x n converges in F σ to some element of F s and putting for each ðα n Þ n∈ℕ ∈ U s , we have that ðU s , k:k U s Þ is a Banach space. Finally, the linear mapping u : U s → F s given by u ððα n Þ n∈ℕ Þ = ∑ n∈ℕ α n x n is a continuous surjection. Now, suppose that ðx n Þ n∈ℕ is a basis for the Silva space, then clearly the mapping u : U s → F s is a continuous bijection, whence by Banach's homeomorphism theorem 3, u is bicontinuous. This leads to the following: Theorem 31. Let F = L s∈ℕ F s be a Silva space with basis ðx n Þ n∈ℕ and let for each s ∈ ℕ, U s be defined as in Theorem 30. Then, U s and F s are linearly homeomorphic for all s ∈ ℕ : Corollary 32. Let F = L s∈ℕ F s be a Silva space, and ðx n Þ n∈ℕ be a basis for F, and let for s ∈ ℕ, σ ∈ ℕ be such that x ∈ F s admits in F σ the expansion x = ∑ n∈ℕ α n x n : If the continuous linear functional α n on F is not identically zero on F s , i.e., ð α n j F s ≠ 0Þ, then x n ∈ F σ : Proof. Suppose that for some n ∈ ℕ, α n j F s ≠ 0 and x n ∉ F σ : Then, taking y ∈ F s such that α n ðyÞ ≠ 0, we have that, as in F σ ,y = ∑ k∈ℕ α k x k , the partial sums S n ðyÞ = ∑ n i=1 α i ðyÞx i and S n−1 ðyÞ = ∑ n−1 i=1 α i ðyÞx i belong to F σ whence α n ðyÞx n = S n ðyÞ − S n−1 ðyÞ ∈ F σ , a contradiction.

Nuclear Silva Spaces with Basis
Let F be a nuclear Silva space. Then, as we saw in Section 6, F = E β ′ whereby E is a (NF)-space and E ≃ F β ′. In what follows, we therefore denote a basis for F (if it exists) by ðx k ′ Þ k∈ℕ and call ðx k Þ k∈ℕ the corresponding biorthogonal sequence in E, i.e., x ℓ ∈ E with x ℓ ðx k ′Þ = δ ℓk for all k, ℓ ∈ ℕ: Theorem 33. Let F = E β ′ be a (NS)-space and let ðx k ′ Þ k∈ℕ be a sequence of nonzero elements in F: Then, ðx k ′Þ k∈ℕ is a basis in F if and only if ðx k Þ k∈ℕ is a basis in E: Proof. If ðx k Þ k∈ℕ is a basis for E then E being a (NF)-space, the sequence ðx k ′Þ k∈ℕ is a basis in E β ′ = F (see, e.g., [28], Theorem 21.10.6). Conversely, suppose that ðx k ′ Þ k∈ℕ is a basis for F: Then, since ðx k ′ Þ k∈ℕ is a Schauder basis, the biorthogonal system ðx k , x k ′ Þ k∈ℕ exists. We prove that the biorthogonal system ðx k , x k ′ Þ k∈ℕ is complete, i.e., E = Spanfx k : k ∈ ℕg: Indeed, call L = Spanfx k : k ∈ ℕg and suppose that L ≠ E: Then, if x ∈ E \ L, by the Hahn-Banach theorem, there ought to exist x ′ ∈ E ′ such that x ′ ðxÞ = 1 and x ′ ðLÞ = f0g: But, as x ′ = ∑ ∞ k=1 x k ðx ′ Þx k ′ and x k ðx ′ Þ = 0 for all k ∈ ℕ, x ′ = 0, thus yielding a contradiction. Now, we prove that the complete biorthogonal system ðx k , x k ′Þ k∈ℕ satisfies Haslinger's criterion [30], i.e., for all s ∈ ℕ, there exists σ ∈ ℕ such that To this end, take s ∈ ℕ fixed. On the one hand, by the representation of seminorms, we have for each k ∈ ℕ that On the other hand, in view of Theorem 31, there exists σ ∈ ℕ and a corresponding space U s such that U s is linearly homeomorphic to F s : Denoting again by u this isomorphism, we may thus find C > 0 such that for all x′ ∈ F s , Hence, for all x ′ ∈ F s and k ∈ ℕ, Consequently, By virtue of Haslinger's criterion (see [30]), ðx k Þ k∈ℕ is a basis for E: Remark 34. The preceding theorem shows that if E is a (NF)space, then each basis in F = E β ′ is the dual basis of a basis in E: In view of [28], Theorem 21.10.6, if ðx k Þ is absolute, so is ðx k ′Þ: But every basis in E β ′ is absolute, so we obtain that each basis in a (NS)-space F is an absolute basis. This duality also leads to the following.

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 ðx k Þ k∈ℕ , the following are equivalent: (i) E is nuclear (ii) For each p ∈ P , there exists q ∈ P such that

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In this section, a criterion for the nuclearity of Silva spaces F with basis is proved whereby only the sequence of norms of the defining Banach spaces F s 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. Let F = E β ′ be a Silva with basis ðx k ′Þ k∈ℕ : Then, F is nuclear if and only if for each s ∈ ℕ, there exists σ ∈ ℕ such that Proof. Let F be nuclear. Then, E is a (NF)-space with basis ðx k Þ k∈ℕ (see Theorem 30) whence for each s ∈ ℕ, there exists σ ∈ ℕ such that (see, e.g., [32]) Moreover, putting A = ða s k Þ k,s∈ℕ with a s k = kx k k s , E is linearly homeomorphic to the nuclear Köthe sequence space K ðAÞ and so the topology on E is also determined by the defining sequence of norms ½: s with (see, e.g., [26]). Taking duals, we thus have that whereby for each t ∈ ℕ and x′ = ðE′, ½• t ′Þ, Notice that in particular for each k ∈ ℕ, Combining (75) and (79), we thus obtain that for each t ∈ ℕ, there exists τ ∈ ℕ such that Now, let s ∈ ℕ be fixed. Then, there exist t ∈ ℕ and K * s > 0 such that for all k ∈ ℕ, while for that t ∈ ℕ, there ought to exist τ ∈ ℕ such that (80) holds. However, for this τ, there exist σ ∈ ℕ and K τ > 0 such that for all k ∈ ℕ, Consequently, we obtain that for each s ∈ ℕ, there exists σ ∈ ℕ such that (74) holds.
Conversely, suppose that for each s ∈ ℕ, there exists σ ∈ ℕ such that (74) holds. Calling for each k, s ∈ ℕ, and putting A = ða s k Þ, we have that the Köthe sequence space KðAÞ is a (NF)-space, whence its topology is also determined by the sequence of norms k•k s , s ∈ ℕ, with Its dual KðAÞ′ is thus given by and of course, KðAÞ β ′ is a nuclear Silva space. Now, define B : F → ω by Then, we claim that BðFÞ = KðAÞ ′ : Indeed, if y ∈ KðAÞ ′ , then there exists s ∈ ℕ such that ∑ k∈ℕ ð|α k |/a s k Þ whence ∑ k∈ℕ jα k jkx k ′ k s ′ < ∞ or ∑ k∈ℕ α k x k ′ converges absolutely in F s and so ∑ k∈ℕ α k x k ′ ∈ F: This implies that KðAÞ′ ⊂ BðFÞ: Now, let x′ ∈ F admit the series representation x′ = ∑ k∈ℕ α k ðx ′ Þx k ′ : Then, there ought to exist s ∈ ℕ such that x ′ = ∑ k∈ℕ α k ðx ′ Þx k ′ ∈ F s , the convergence being valid in F s whence sup k∈ℕ jα k ðx′Þjkx k ′k s ′ = K < ∞: But for this s ∈ ℕ, there exists by assumption σ ∈ ℕ such that (74) holds.
We claim that ½Bðx′Þ σ exists. Indeed, ð88Þ By virtue of the open mapping theorem, F and KðAÞ ′ are linearly homeomorphic whence F is nuclear.
Remark 36. Another possible approach to the proof of Theorem 35 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 KðAÞ, 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 ðO A ðRe z < AÞ, ζÞ of holomorphic functions. Let λ = ðλ n Þ n∈ℕ be a fixed strictly increasing sequence of positive real numbers. With each sequence ða n Þ n∈ℕ such that lim n→∞ sup ðja n j/λ n Þ ≤ −A, we associate the function a n e zλ n , z ∈ ℂ: Take ε > 0 arbitrary chosen and consider the half-plane Then, for each n ∈ ℕ and z ∈ A ε , a n j j e zλ n ≤ a n j je Aλ n e −ελ n : But, in virtue of the assumption, lim n→∞ sup log a n j j we find that a n j je Aλ n = e log a n j j+Aλ , ð94Þ whence there exist C > 0 such that sup n∈ℕ a n j je Aλ n ≤ C: Consequently, whence the series defining f is normally convergent on each A ε , ε > 0: Consequently, f ∈ O A ðRe z < AÞ: We call O A ðRe z < AÞ the subspace of OðRe z < AÞ consisting of the elements f just defined and provide O A ðRe z < AÞ with the system P of seminorms p k with Then, it was proved by Kamthan that ðO A ðRe z < AÞ, P Þ is a Fréchet space. From the definition itself of the elements f in O A , it follows that the sequence of functions ζ = ðe zλ n Þ n∈ℕ is a basis for the space ðO A ðRe z < AÞ, P Þ. Then, we claim that ðO A ðRe z < AÞ, ζÞ is a (NF)-space. Indeed, from the definition itself of O A ðRe z < AÞ, it follows that ζ is a basis for it.
Having that ðO A ðRe z < AÞ, ζÞ 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, Therefore, by our criterion (Theorem 35), the nuclearity is proved.
Example 8. Consider the space ðOð S R Þ, ξÞ of holomorphic functions in two complex variables z, w, provided with the countable system P of seminorms p k where and S r is the closed hypersphere defined by sup S r k = z, w ∈ ℂ : z j j 2 + w j j 2 ≤ r 2 Again, ðr k Þ k∈ℂ is a strictly increasing sequence of positive numbers with 0 < r k < R and lim k→∞ r k = R. As is well known, ðOð S R Þ, ξÞ 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 13 Journal of Function Spaces ∈ Oð S R Þ, the sequence ξ = z m w n ð Þ m,n∈ℕ ð101Þ is a basis for Oð S R Þ. Although it is known that ðOð S R Þ, ξÞ 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]) where σ m,n is given by Then, we obtain that, taking k ∈ ℕ fixed, for each ℓ > K, the nuclearity of this space is proved.

General Remarks and Comments
The properties of series of the form ∑ ∞ i=0 c i P i ðzÞ, z ∈ ℂ where P i ðzÞ, i = 0, 1, ⋯ are prescribed polynomials and e i 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][16][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 consid-ered 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

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there is no conflict of interest.