Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.


Introduction
In this article, the study of hyperseries in the non-Archimedean ring of Colombeau generalized numbers (CGN), as carried out in [29], is applied to the corresponding notion of hyper-power series.As we will see, this yields results which are more closely related to classical ones, such as, e.g. the equality ρ n∈ ρ N x n n! = e x that holds for all x ∈ ρ R where the exponential is moderate, i.e. if |x| ≤ log dρ −R for some R ∈ N. On the other hand, we will see that classical smooth but non-analytic functions, e.g.smooth functions with flat points, and Schwartz distributions like the Dirac delta, are now included in the related notion of generalized real analytic function (GRAF).This implies that necessarily we cannot have a trivial generalization of the identity theorem (see e.g.[22, Cor.1.2.6, 1.2.7]) but, on the contrary, only a suitable sufficient condition (see Thm. 40 below).The notion of generalized real analytic function hence reveals to be less rigid than the classical concept, by including a large family of non-trivial generalized functions (e.g.Dirac delta δ, Heaviside function H, but also powers δ k , k ∈ N, and compositions δ • δ, δ k • H h , H h • δ k , etc., for h, k ∈ N. Conversely, GRAF preserve a lot of classical results: they can be thought of as infinitely long polynomials f (x) = ρ n∈ σ N a n (x − c) n , with uniquely determined coefficient a n = f (n) (c)  n! , they can be added, multiplied, composed, differentiated, integrated term by term, are closed with respect to inverse function, etc.This lays the foundation for a potential interesting generalization of the Cauchy-Kowalevski theorem which is able to include many non-analytic (but generalized real analytic) generalized functions.
Concerning the theory of analytic Colombeau generalized functions, as developed in [26] for the real case and in [5,6,1,7,24,20,30] for the complex one, it is worth to mention that several properties have been proved in both cases: closure with respect to composition, integration over homotopic paths, Cauchy integral theorem, existence of analytic representatives, identity theorem on a set of positive Lebesgue measure, etc. (cf.[26,30] and references therein).On the other hand, even if in [30] it is also proved that each complex analytic Colombeau generalized functions can be written as a Taylor series, necessarily this result holds only in an infinitesimal neighborhood of each point.The impossibility to extend this property to a finite neighborhood is a general drawback of the use of ordinary series in a (Cauchy complete) non-Archimedean framework instead of hyperseries, as explained in details in [29].
We refer to [23] for basic notions such as the ring of Robinson-Colombeau, subpoints, hypernatural numbers, supremum, infimum and hyperlimits, and [29] for the notion of hyperseries as well as their notations and properties.Once again, the ideas presented in the present article can be useful to explore similar ideas in other non-Archimedean settings, such as [3,17,18,4,19,21,28].
2. hyper-power series and its basic properties 2.1.Definition of hyper-power series.In the entire paper, ρ and σ are two arbitrary gauges; only when it will be needed, we will assume a relation between them, such as σ ≤ ρ * or σ ≥ ρ * (see [29]).
A power series of real numbers is simply "a series of the form n∈N a n (x − c) n , where x, c, a n ∈ R for all n ∈ N".Actually, this (informal) definition allows us to consider only finite sums N n=0 a n (x − c) n , N ∈ N, and hence to evaluate whether convergence holds or not.A similar approach can be used for hyper-power series (HPS) if we think at the ρ R-module ρ σ R s of sequences for hyperseries exactly as the space where we can consider hyperfinite sums regardless of convergence.This is the idea to define the space ρ σ R x − c of formal HPS: Definition 1.Let x, c ∈ ρ R. We say (b n ) n ∈ ρ σ R x − c if and only if there exist (a nε ) n,ε ∈ R N×I and representatives [ Elements of ρ σ R x − c are called formal HPS because here we are not considering their convergence.In other words, a formal HPS is a hyper series (i.e. an equivalence class (b n ) n ∈ ρ σ R s in the space of sequences for hyperseries) of the form [a nε • (x ε − c ε ) n ] s .Remark 2.

(i)
We explicitly note that x−c is not an indeterminate, like in the case of formal power series R x , but a generalized number of ρ R. For example, in Lem. 10 below, we will prove that if x − c = y − d, then ρ σ R x − c = ρ σ R y − d .(ii) On the contrary with respect to the case of real numbers, being a formal HPS, i.e. an element of ρ σ R x − c , depends on the interplay of the two gauges ρ and σ: take e.g. a n = 1 n 2 and x − c = 2, so that for all N ∈ σ N we have n ∼ log(N ).Therefore, taking e.g.σ ε = exp − exp 1 ρε and N ε := int (σ ε ), we have that (log N ε ) / ∈ R ρ and hence we cannot even consider hyperfinite sums of this form.Informally stated, for this gauge σ, we have that ρ n∈ σ N 1 n 2 2 n is not a formal HPS, i.e. even before considering its convergence or not, we cannot compute σ-hyperfinite sums and get a number in ρ R. (iii) In [29], we proved that if x is finite, then x n n! s ∈ ρ σ R x is a formal HPS for all gauges ρ, σ.In Sec.14, we will prove that (dρ −1 ) n n! s / ∈ ρ ρ R dρ −1 ; on the other hand, we will also see that if x ≤ log dρ −N and dσ Q ≤ dρ N for some Q ∈ N, then x n n! s ∈ ρ σ R x is a formal HPS.The previous Def. 1 sets immediate problems concerning independence of representatives: every time we start from n converges or not.On the other hand, we also have to prove that it is well-defined, i.e. that taking different representatives [29,Thm. 4]) and hence, in general, the operation The problem can also be addressed differently: what notion of equality do we have to set on a suitable subring of R N×I so as to have independence on representatives?This notion of equality naturally emerges in proving that the following definition of radius of convergence is well-defined (see Lem. 4).What subring we need to consider arises from the idea to include yields a non ρ-moderate net (for example for ε ∈ L ⊆ 0 I) because this case would intuitively identify a radius of convergence larger than any infinite number in ρ R: ∞} be the extended real number system with the usual (partially defined) operations.We set ρ R := R I / ∼ ρ , where for arbitrary (x ε ), (y ε ) ∈ R I , as usual we define In ρ R, we can also consider the standard order relation Note that ρ R \ {−∞}, +, ≤ is an ordered group but, since we are considering arbitrary nets R I , the set ρ R is not a ring: e.g.+∞ • 0 is still undefined and is the ring of weakly ρ-moderate nets, and (in this case, we say that these two nets are strongly ρ-equivalent In the following lemma, we prove that rad (a n ) c is well-defined: ) and similarly define rε using ānε .Then (r ε ) ∼ ρ (r ε ), and hence Proof.For all ε ∈ I and all n ∈ N >0 , we have The binomial formula yields (x + y) ≤ x 1/n + y 1/n n for all x, y ∈ R ≥0 , so that 3), for all q ∈ N and for ε small we have ∀n ∈ N : Inverting the role of (a nε ) n,ε and (ā nε ) n,ε we finally obtain i.e. the net (a nε ) ε is ρ-moderate.This is the main motivation to consider the exponent "−R" in (2.2) (recall that in our notation 0 ∈ N): without the term "−R", the only possibility to have (a which is an unnecessary limitation.Similarly, we can motivate why we are considering the quantifier "∀n ∈ N" in the same formula (instead of, e.g., "∃N ∈ N ∀n ∈ N ≥N ").The proof of the next Lem.10 will motivate why in (2.2) we consider the uniform property "∀ 0 ε ∀n ∈ N" and not "∀n for all n ∈ N. Similarly (n!) n∈N is not weakly ρ-moderate and hence our theory does not apply to a "hyperseries" of the form On the other hand, in Lem.7.(i) we will show that, as a consequence of considering only weakly moderate coefficients, the radius of convergence of our hyperseries is always strictly positive.
Let i b R be the embedding of Schwartz distributions into generalized smooth functions (GSF) defined by µ and by the infinite number b ∈ ρ R (see e.g.[13]).The Schwartz's Paley-Wiener theorem implies that µ is an entire function and we know that if dρ [13]).For n ∈ N, we have This inequality shows that ∈ ρ R c and motivates our definition of weakly ρ-moderate nets.The corresponding radius of convergence is , and assume that for all ε there exists r ε := lim n→+∞ |a nε | 1/n −1 such that r := [r ε ] ∈ ρ R. Then from [23,Thm. 28], for some gauge σ ≤ ρ we have ρ lim n∈ σ N |a n | 1/n = 1 r and r = rad (a n ) c ∈ ρ R. In Cor. 17, we will see the relationship between our definition of radius of convergence and the least upper bound of all the radii where the HPS converges.
In the following lemma, we show that ρ R c is a ring: Lemma 6.With pointwise operations, ρ R c is a quotient ring.
Proof.Actually, the result follows from [14,Thm. 3.6] because the set ∈ N} is an asymptotic gauge with respect to the order (n, ε) ≤ (n, ε) if and only if ε ≤ ε.However, an independent proof follows the well-known lines of the corresponding proof for the ring ρ R, and depends on the following properties of B: The following lemma represents a useful tool to deal with the radius of convergence.It essentially states that the radius of convergence equals +∞ on some subpoint, or it is moderate on some subpoint or it is greater than any power dρ −P .
Assume that for all L ⊆ 0 I, the following implication holds Then ∀ 0 ε : P {r ε }, i.e. the property P {r ε } holds for all sufficiently small ε. (v) If q ∈ ρ R and q < r, then ∃s ∈ ρ R : q < s ≤ r.
(iii): Since we assume that r < +∞, without loss of generality we can take r ε < +∞ for all ε.We also assume that (a) is false, i.e. r ≤ M dρ − P for some P ∈ N and some M ⊆ 0 I.We first prove (b.1): take ε ∈ P ∈N r > ρ −P , then r ε > ρ −P ε for all P ∈ N, so that r ε = +∞ for P → +∞, and this is not possible.
We also note that r ≤ ρ −P ⊆ r ≤ ρ −Q for all Q ≥ P .From M ⊆ 0 I and r ≤ M dρ − P , we have (0, ε 0 ] ∩ M ⊆ r ≤ ρ −( P +1) ⊆ 0 I, and hence definition (2.5) yields P m ∈ N and also proves (b.2).For all P ∈ N <Pm , we hence have r ≤ ρ −P ⊆ 0 I, i.e. (0, ε P ] ⊆ r > ρ −P for some ε P .This implies dρ −P ≤ r and proves (b.3).Finally, if P m = 0 and . If L c 0 ⊆ 0 I, then (0, ε 0 ] ⊆ L 0 for some ε 0 , i.e. r ≤ 1. (iv): By contradiction, assume that ¬P {r ε } for all ε ∈ L and for some L ⊆ 0 I.As usual, we assume that all the results we proved for ρ R can also be similarly proved for the restriction ρ R| L .From (ii) for ρ R| L , we have r < L +∞ or r = K +∞ for some K ⊆ 0 L. The second case implies r > L dρ −Q for all Q ∈ N. Since K ⊆ 0 I, we can apply the second alternative in the implication (2.6) to get ∀ 0 ε ∈ K : P {r ε }, which gives a contradiction because K ⊆ L. We can hence consider the first case r < L +∞ and apply the subcase (a), i.e. r > L dρ −P for all P ∈ N, and we hence proceed as above applying the second alternative of the implication (2.6).In the remaining subcase, we can use (b.2) (with L instead of I).This yields L Pm ⊆ 0 L and r ≤ LP m dρ −Pm .Since L Pm ⊆ 0 I, we can apply the first alternative in the implication (2.6) to get once again a contradiction.
Explicitly note the meaning of Lem.7.(iv): on an arbitrary subpoint r| L of the radius of convergence r = rad (a n ) c , we have to consider only two cases: either r| L is ρ-moderate or it is greater than any power dρ −Q (the latter case including also the case r| L = +∞); if in both cases we are able to prove the property P {r ε } for ε ∈ L sufficiently small, then this property holds for all ε sufficiently small.

Set of convergence.
Even if the radius of convergence of the exponential hyperseries is rad for some R ∈ N: in other words, the constraint to get a ρ-moderate number implies that even if The following definition of set of convergence closely recalls the definition of GSF: and such that there exist representatives Note that condition (ii) is necessary because in (iii) we use a HPS; on the other hand, conditions (iii) and (iv) state that the function is a GSF defined by the net of smooth functions As for GSF, see [13,Thm. 16], condition (iv) will be useful to prove that we have independence from representatives of x in all the derivatives.In Cor. 25, we will see that under very general assumptions and if σ ≤ ρ * , condition (iv) can be omitted.
In Sec. 14 we will show that log dρ −1 ∈ ρ σ conv 1 n! c , 0 (the set of convergence of the exponential HPS at the origin), but dρ , and because of this property without loss of generality we will frequently assume c = 0.
We also note that condition (iii) states that the hyperseries ρ n∈ σ N a n (x − c) n converges, and it does exactly to the generalized number It is hence natural to wonder whether it is possible that it converges to some different quantity.This is the problem of the relation between hyperlimit and ε-wise limit: which has been already addressed in [29,Thm. 12,Thm. 13].Intuitively speaking, if the gauge (σ ε ) is not sufficiently small, and hence the infinite nets (σ −N ε ) are not sufficiently large, it can happen that ni(N ) ε → +∞ as ε → 0 only very slowly, whereas the ε-wise limit could require N → +∞ at a greater speed to converge.This can be stated more precisely in the following way: x−c be a formal HPS and assume that +∞ n=0 a nε (x ε − c ε ) n < +∞ for ε small.Then, for all q ∈ N and for all ε small, we can find N q ε ∈ N such that However, only if As expected, for HPS the set of convergence is never a singleton: Proof.From Thm. 7.(i), we have r := rad (a n ) c ≥ dρ q1 for some q 1 ∈ N. We also have 2).Assume that |x − c| < dρ q : we want to find q ∈ N ≥q1 so that x ∈ ρ σ conv ((a n ) c , c).To prove property Def.8.(ii), for N ε , M ε ∈ N and for ε small, we estimate Therefore, taking q = max(1 + Q, q 1 ), we get and this proves Def.8.(ii).Similarly, we have Since ρ lim M∈ σ N dρ M+1 = 0, this proves Def.8.(iii).Finally, for all k ∈ N >0 and all representatives [x ε ] = x, we have and hence In the last step we used q ≥ Q+1 and the binomial series We can now prove independence from representatives both in Def. 8 and in Def.1: The nets (a nε ) n,ε , (x ε ) and (c ε ) also satisfy all the conditions of Def. 8 of set of convergence. (2.10) Since ẑ = 0 , we also have for fixed arbitrary p, q ∈ N. We first consider the second summand in (2.10): Since s ∈ ρ R, we can take p ∈ N sufficiently large so that ρ p ε s ε < 1.This implies Thereby, for q → +∞, this summand defines a negligible net.For the first summand of (2.10), we can use the mean value theorem to get (2.12) In the case M ε < +∞ for all ε and k = 0, this proves that [a nε • y n ε ] s ∈ ρ σ R s because (â nε ) n,ε and (ŷ ε ) satisfy Def.8.(ii).In the case M ε = +∞ and N ε = 0 = k, it also proves the moderateness of +∞ n=0 a nε y n ε , i.e. the implicit moderateness requirement of Def.8.(iii).Finally, for k > 0, property (2.12) also shows that Def.8.(iv) also holds for (a nε ) and (y ε ) because of (2.8).We can also apply (2.12)This proves claim (ii) and hence also Def. 8.(iii) because 2.4.Examples.We start studying geometric hyperseries, which in general are convergent HPS if σ ≤ ρ * : Example 11 (Geometric hyperseries).Assume that x ∈ (−1, 1) ⊆ ρ R. We have: x is another representative and k ∈ N >0 , then −1 < xε < 1 for ε small, and from (2.8) we get a sufficient condition ensuring the convergence of any geometric hyperseries with |x| < 1.However, we already used (see e.g.Thm. 9) the convergence of the geometric hyperseries ρ n∈ σ N dρ n = 1 1−dρ for all gauges ρ, σ.More generally, exactly as proved in [29,Example 8], it is easy to see that Example 12 (A smooth function with a flat point).Consider the GSF corresponding to the ordinary smooth function f (x) := e −1/x if x ∈ R >0 0 otherwise .It is not hard to prove that |f (x)| ≤ |x| q for all x ≈ 0 and all q ∈ N. Thereby, f (x) = 0 for all x such that |x| ≤ dρ r for some r ∈ R >0 .Therefore, we trivially have f (x) = ρ n∈ σ N 0 • x n only for all x in this infinitesimal neighborhood of 0. On the other hand, ρ σ conv ((0 The GSF f is therefore a candidate to be a GRAF, but not an entire GRAF. Example 13 (A nowhere analytic smooth function).A classical example of an infinitely differentiable function which is not analytic at any point is , where 2 N := {2 n | n ∈ N}.Since for all x = π p q , with p ∈ N and q ∈ 2 N and for all n ∈ 2 N , n ≥ 4, n > q, we have i.e. they are not coefficients for a HPS.
Example 14 (Exponential).We clearly have 1 n! c ∈ ρ R c and rad 1 n! c = +∞, i.e. we have coefficients for an HPS with infinite radius of convergence.Set for ε small, and this shows that x n n! s ∈ ρ σ R x , i.e. for all x ∈ C, we have a formal HPS.We finally want to prove that and hence Similarly, we can consider trigonometric functions whose set of convergence is the entire ρ R.
Example 15 (Dirac delta).In Rem.5.(v), we already proved that δ (n) (0) A different way to include a large class of examples is to use the characterization Thm.37 by factorial growth of derivatives of GRAF.
When we say that a HPS ρ n∈ σ N a n (x−c) n is convergent, we already assume that its coefficients are correctly chosen and that the point x is in the set of convergence, as stated in the following Definition 16.We say that c).In all the previous examples, we recognized that dealing with HPS is more involved than working with ordinary series, where we only have to check that the final result is a convergent series "of the form" ∞ n=0 a n (x − c) n .On the contrary, for HPS we have to control the following steps: 1) We have to check that the net (a nε ) n,ε defines coefficients for HPS (Def.3.(ii)), i.e. that . This allows us to talk of the radius of convergence rad (a n ) c and of the set of convergence ρ σ conv ((a n ) c , c) (Def.8).Because of Thm. 9, this set is always non-trivial but in general is not an interval, like the case of the exponential function clearly shows.This step already allows us to say that the HPS ρ n∈ σ N a n (x − c) n is convergent, i.e.Def.16, if x ∈ ρ σ conv ((a n ) c , c). 2) At this point, we can study the set of convergence, e.g. to arrive at an explicit form ).This depends mainly on three conditions: a) For all x ∈ C, we must have a formal HPS (Def. 1) because this allows us to talk of any hyperfinite sum Here, the main step is to prove that the net b) For all x ∈ C, we have to check Def.8.(iii), i.e. the equality: ) Note that from Example 14, we have that the least upper bound of n! is a convergent HPS (2.17) does not exist in ρ R, whereas Def. 3 yields the value rad 1 n! c = +∞.Therefore, Def. 3 allows us to consider the exponential HPS even if the supremum of (2.17) does not exist.It remains an open problem whether r = rad (a n ) c , at least if the least upper bound (2.16), or the corresponding sharp supremum, exists.
We now study absolute convergence of HPS, and sharply boundedness of the summands of a HPS.We first show that the hypersequence (a n (x − c) n ) n∈ σ N of the terms of a HPS is sharply bounded: (2.18) We recall that because of the definition of formal HPS (Def. 1) and [29, Lem.7] the term a n (x − c) n ∈ ρ R is well-defined for all n ∈ σ N. The previous proof is essentially the generalization in our setting of the classical one, see e.g.[22].However, property (2.18) does not allow us to apply the direct comparison test [29,Thm. 22].Indeed, let us imagine that we only prove |a n x n | < Kh n , with h < 1, for all n ∈ σ N and with K coming from (2.18); as we already explained in [29,Sec. 3.3], this would imply ∀n ∈ N ∃ε 0n ∀ε ≤ ε 0n : |a nε x n ε | ≤ K ε h n ε , and the dependence of ε 0n from n ∈ N is a problem in estimating inequalities of the form [29].A solution of this problem is to consider a uniform property of n ∈ N with respect to ε: (2.20) Remark 20.

(i)
The adverb eventually clearly refers to the validity of the uniform inequality in (2.20)only for n sufficiently large.(ii) If for ε small, the series  The last example also shows that property (2.20) does not hold for all point x ∈ ρ σ conv ((a n ) c , c).However, it always holds for any c if x is sufficiently near to c: Lemma 21.Let (a n ) c ∈ ρ R c and c ∈ ρ R, then there exists σ ∈ R >0 such that for all x ∈ B σ (c), the sequence of summands (a n (x − c) n ) n∈N is eventually ρ R-bounded in ρ R c .
Proof.Using the same notation as above, since (a The following result is a stronger version of the previous Lem.18, and allow us to apply the dominated convergence test: i.e. for all representatives (a n Even if the case of the exponential HPS (see Example 14) shows that in general the set of convergence is not an interval, it has very similar properties, at least if the gauge σ is sufficiently small: Theorem 23.Let σ ≤ ρ * and ρ n∈ σ N a n (x − c) n be a convergent HPS whose sequence of summands (a n (x − c) n ) n∈N is eventually ρ R-bounded in ρ R c .Then for all x ∈ B |x−c| (0) we have: The HPS converges absolutely at x, and hence uniformly on every functionally ) is strongly connected, i.e. it is not possible to write it as union of two non empty strongly disjoint sets, i.e. such that (a) By the direct comparison test [29,Thm. 22], the HPS ρ n∈ σ N a n x n converges absolutely because ρ n∈ σ N Kh n converges since σ ≤ ρ * and h < 1.Finally, [13,Thm. 74] yields that pointwise convergence implies uniform convergence on functionally compact sets.This proves (i).
For (iii), it suffices to consider that Finally, if [x ε ] = x and k ∈ N >0 , we have where we used Lem.22, and hence Def.8.(iv) also holds.
(ii): For s := |x|−|x| > 0 and x ∈ B s (x), we have |x| ≤ |x−x|+|x| < s+|x| = |x|, and hence x ∈ ρ σ conv ((a n ) c , c) from (i). (iii): Setting x := y + t(x − y), we have y ≤ x ≤ x.We can use trichotomy law [23,Lem. 7.(iii)] to distinguish the cases y = L 0 or y > L 0 or y > L 0 for L ⊆ 0 I.The latter has to be subdivided into the sub-cases x > M 0 or x = M 0 or x < M 0 with M ⊆ 0 L, i.e. using [23,Lem. 7.(iii)] for the ring ρ R| L .Finally, the latter of these sub-cases has to be further subdivided into x > K y or x < K y or x = K y with K ⊆ 0 M .In all these cases we can prove Def. 8 in the corresponding co-final set.
(iv): By contradiction, if a ∈ A and b ∈ B, then In spite of Thm.23.(ii), it remains open the problem whether the set of convergence is always a sharply open set or not.Using the previous theorem, this problem depends, for each point x in the set of convergence, on the existence of a point x satisfying its assumptions.However, that such a point x in this case does not exist.
Corollary 24.Let σ ≤ ρ * and let R be the set of all the numbers of the form s = |x − c| for some x ∈ ρ R satisfying: ), the HPS ρ n∈ σ N a n (x − c) n converges absolutely for all x ∈ B r (c) and uniformly on every functionally compact Proof.Without loss of generality, we assume c = 0, and let x ∈ B r (c).Since |x| < r, by the definition of sharp supremum, (see [23]) there exist s = |x| such that |x| < |x| ≤ r and such that (i) and (ii) hold.The conclusions then follow by Thm.23.
Property Thm.23.(i) can also be written as a characterization of the set of convergence: 2.6.Algebraic properties of hyper-power series.In this section, we extend to HPS the classical results concerning algebraic operations and composition of power series.
Theorem 26.Assume that (2.25) (ii) The sum of these HPS is a convergent HPS with and (iii) For all x ∈ B |x−c| (c), the product of these HPS converges to their Cauchy product: which is still a convergent HPS with radius of convergence greater or equal to min(rad is invertible, and recursively define (for ε small) d 0ε := a0ε b0ε , Then coefficients (d n ) c ∈ ρ R c define a convergent HPS with radius of convergence greater or equal to min(rad Proof.Equalities (2.25) and (2.26) follow directly from analogous properties of convergent hyperlimits, i.e. [23,Sec. 5.2].All the inequalities concerning the radius of convergence can be proved in the same way from analogous results of the classical theory, because of Def..(iii)3.For example, from Def. 8.(iii) we have that both the ordinary series To prove (2.27) (assuming that x lies in the convergence set of the product HPS, see below), from Lem. 23 we have that both the series converge absolutely because x ∈ B |x−c| (c).We can hence apply the generalization of Mertens' theorem to hyperseries (see [29,Thm. 37]).To complete the proof of (iii), we start by showing that the terms ( (2.30) Without loss of generality we can assume We have where R := R 1 + R 2 and for a suitable Q ∈ N (that can be chosen uniformly with respect to n ∈ N).Thereby, the product HPS has well-defined coefficients and hence a suitable set of convergence.Now, we want to show that x lies in this set of convergence.Since Def.8.(i) clearly holds and Def.8.(iii) follows from Mertens' Theorem (both [29,Thm. 37] and the classical version), it remains to prove that we actually have a formal HPS (Def.8.(ii)) and moderateness of derivatives (Def.8.(iv)).The latter follows by the general Leibniz rule for the k-th derivative of a product.For the former one, without loss of generality we can assume c = 0; let (M ε ), (N ε ) ∈ N σ , then for suitable Mε , Mε ∈ N σ and Nε , Nε ∈ N σ such that M ε = Mε + Mε and N ε = Nε + Nε , we have and thereby Def.8.(ii) follows.
(iv): To prove that (d n ) c ∈ ρ R c , without loss of generality, we can assume in (2.30) and (2.31) that (2.34) For n = 0, we have for all ε because Q > 0. For the inductive step, we assume (2.34) and use the recursive definition (2.28): which holds for ε small (independently by n).Finally, equality (2.29) can be proved as we did above for the product because x ∈ B |x−c| (c) and From this equality, it also follows Def.16.(ii) because the product of a non-moderate net (on a co-final set) by a moderate net cannot yield a moderate net.Finally, as above, moderateness of derivatives follows from Mertens' theorem and the k-th derivative of the quotient.
The following theorem concerns the composition of HPS: For ε small, we have 4 for the same ε and for all n ∈ N. Now, take ε small so that also we finally get which proves that (c nε ) n,ε defines coefficients for an HPS.To prove that x ∈ ρ σ conv ((c n ) c , c), we can proceed as follows: Def.8.(i) can be proved like in the classical case; Def.8.(ii) is a consequence of composition of polynomials if M ε < +∞ or it can be proved proceeding like in the case of composition of GSF if M ε = +∞: Def.8.(iii) and Def.8.(iv) can be proved like for GSF (see [13] and Thm.28 below).

Generalized real analytic functions and their calculus
A direct consequence of Def. 8 of set of convergence is the following Before defining the notion of GRAF, we need to prove that the derived HPS has the same set of convergence of the original HPS: | the opposite implication follows.The condition Def.8.(iv) about moderateness of derivatives for the original HPS clearly implies the analogue condition for the derived one.For the opposite inclusion, we can distinguish the case x = s c or |x − c| > 0, the former one being trivial.We have Thm. 28 motivates the following definition: Definition 30.Let σ ≤ ρ * and U be a sharply open set of ρ R, then we say that f is a GRAF on U (with respect to ρ, σ), and we write f We also say that f is entire at c if (iii) and (iv) hold.
(a) Clearly, if (a n ) c ∈ ρ R c , c ∈ ρ R, and we set f (x) = ρ n∈ σ N a n (x − c) n , then f is a GRAF on the interior points of the set of convergence ρ σ conv ((a n ) c , c).Vice versa, if f ∈ ρ σ GC ω (U, ρ R), then U is contained in the union of all the sharp interior sets int ( ρ σ conv ((a n ) c , c)), because of condition (i).(b) Example 15 shows that Dirac δ is entire at 0 but it is not at any c ∈ ρ R such that |c| ≥ s ∈ ρ R for some s.(c) Example 12 of a function f with a flat point shows that f is a GRAF, but if c = 0, then s ∈ ρ R >0 satisfying condition (i) is infinitesimal, whereas if c ≫ 0, then s ≫ 0 is finite, and these two types of set of convergence are always disjoint.
Corollary 32.Let σ ≤ ρ * , U ⊆ ρ R be a sharply open set and f ∈ ρ σ GC ω (U, ρ R), then also f ′ ∈ ρ σ GC ω (U, ρ R) and it can be computed with the derived HPS.Because of our definition Def. 8 of set of convergence, several classical results can be simply translated in our setting considering the real analytic function that defines a given GRAF.
for all k ∈ N.
Proof.From Cor. 32, we have f ).For x = c (which is always a sharply interior point because of Thm. 9) this yields the conclusion.
Corollary 34.Let σ ≤ ρ * , U be a sharply open set of ρ R, and f ∈ ρ σ GC ω (U, ρ R).Then for all c ∈ U the Taylor coefficients f The definition of 1-dimensional integral of GSF by using primitives, allows us to get a simple proof of the term by term integration of GRAF: Theorem 35.In the assumptions of the previous theorem, set for all the interior points x ∈ ρ σ conv ((a n ) c , c).Then F (x) = ´x c f (x) dx and F is a GRAF on the interior points of ρ σ conv ((a n ) c , c).
Proof.The proof that is a convergent HPS with the same set of convergence of f can be done as in Thm. 29, and hence F is a GRAF on the interior points of ρ σ conv ((a n ) c , c).The remaining part of the proof is straightforward by using Cor.32, so that F ′ (x) = f (x) and F (c) = 0. and using [13,Thm. 42,Def. 43].
We close this section by first noting that, differently with respect to the classical theory, if f (x) = ρ n∈ σ N a n (x−c) n for all x ∈ ρ σ conv ((a n ) c , c), and we take another point c ∈ ρ σ conv ((a n ) c , c), we do not have that (c − rad (a ).On the other hand, in the following result we show that ρ σ conv f , c , we have , c , we have Therefore, the usual proof, see e.g.[22], yields , which implies the conclusion.

Characterization of generalized real analytic functions, inversion and identity principle
The classical characterization of real analytic functions by the growth rate of the derivatives establishes a difference between GRAF and Colombeau real analytic functions: Proof.We prove that condition (4.1) is necessary.For c ∈ U , we have f s) for some s > 0 from Def. 30 and Thm.33.We first note that condition (4.1) can also be formulated as an inequality in ρ R c and as such it does not depend on the representatives involved.Therefore, from Thm. 28 and Thm.33, without loss of generality, we can assume that the given net (f ε ) is of real analytic functions satisfying f for some K = [K ε ] ∈ ρ R. Set s := 1 2 min(σ, s) ∈ R >0 and S := |x − c|, where x is any point such that s < |x − c| < σ, so that 0 < s S < 1 and from (4.2) we obtain For each [x ε ] ∈ B s (c), we have and hence from (4.3): which is our claim for C := K 1− s S and R := S 1 − s S .Note that, differently with respect to the case of Colombeau real analytic functions [26], not necessarily the constant 1 R is finite, e.g. if s ≈ S. We now prove that the condition is sufficient.
) and take x ∈ B s(c).We first prove the equality f For all ε, from Taylor's formula for the smooth f ε , we have for some t ε ∈ [0, 1] R and for sε < s ε , we can apply (4.1) and get ∀ 0 ε ∀n ∈ N :  n! (x − c) n converges; moreover from (4.1) we also have ∀ 0 ε ∀n ∈ N : As we have already noted in this proof, differently with respect to the definition of Colombeau real analytic function [26], we have that, generally speaking, 1 R ∈ ρ R is not finite.For example, for f = δ at c = 0, we have , where µ (n) (x ε ) ≤ ´β =: C and hence 1 R = b which is an infinite number.Thereby, in the particular case when 1 R is finite, f is a Colombeau real analytic function in a neighborhood of c. Vice versa, any Colombeau real analytic function and any ordinary real analytic function are GRAF.
This characterization also yields the closure of GRAF with respect to inversion.We first recall that the local inverse function theorem holds for GSF, see [10].Therefore, if f ∈ ρ σ GC ω (U, ρ R) ⊆ ρ GC ∞ (U, ρ R) and at the point x 0 ∈ U the derivative f ′ (x 0 ) is invertible, we can find open neighborhoods of x 0 ∈ X ⊆ U and of y 0 := f (x 0 ) ∈ Y such that f | X : X → Y is invertible, (f | X ) −1 ∈ ρ GC ∞ (Y, X) and f ′ (x) is invertible for all x ∈ X.
Theorem 38.If σ ≤ ρ * and we use notations and assumptions introduced above, then (f | X ) −1 ∈ ρ σ GC ω (Y, X).Proof.For simplicity, set g := (f | X ) −1 and h(x) := 1 f ′ (x) for all x ∈ X, so that g ′ (y) = h[g(y)] for all y ∈ Y .From Cor. 32 and Thm.26, we know that h is a GRAF.Therefore, Thm.37 yields ∀ 0 ε ∀n ∈ N : h  ε (y ε ) ≤ j!(−1) j−1 1/2 j (2Cε) j R j−1 ε for all j ∈ N >0 , and hence g ∈ ρ σ GC ω (U, ρ R) once again by Thm.37. Since δ is a GRAF, in general the identity principle does not hold for GRAF.From our point of view this is a feature of GRAF because it allows to include as GRAF a large class of interesting generalized functions and hence pave the way to a more general related Cauchy-Kowalevski theorem.The following theorem clearly shows that the identity principle does not hold in our framework exactly because we are in a non-Archimedean setting: every interval is not connected in the sharp topology because the set of all the infinitesimals is a clopen set, see e.g.[9].For example, if f = δ and g = 0, the set int x ∈ ρ R | δ(x) = 0 ⊇ x ∈ ρ R | |x| ≫ 0 is clopen.Thereby, also ρ R \ int x ∈ ρ R | δ(x) = 0 is clopen, and we have If we assume that all the derivatives of f are finite and the neighborhoods of Def. 30 are also finite, then we can repeat the previous proof considering only standard points c ∈ R and radii r ∈ R >0 , obtaining the following sufficient condition: Theorem 40.Let U ⊆ ρ R be an open set such that U ∩ R is connected.Let f , g ∈ ρ σ GC ω (U, ρ R) be such that f | V ∩R = g| V ∩R for some nonempty subset V ⊆ U such that V ∩ R is open in the Fermat topology, i.e. ∀x ∈ V ∩ R ∃r ∈ R >0 : B r (x) ⊆ V ∩ R.
Finally, assume that all the following quantities are finite: The neighborhood length s in Def. 30 is finite for each c ∈ U ∩ R, (ii) ∀x ∈ U ∀n ∈ N : f (n) (x) and g (n) (x) are finite.
Then f | U∩R = g| U∩R .

2. 5 .
Topological properties of the set of convergence.The first consequence of our definition of convergent HPS Def.16 and radius of convergence Def. 3, is the following Lemma 17.Let ρ n∈ σ N a n (x − c) n be a convergent HPS.If the following least upper bound exists

+∞
n=0 |a nε (x ε − c ε ) n | =: R ε of absolute values terms converges to a ρ-moderate net, then (2.20) holds for N = 0.This includes Example 11 of geometric hyperseries, Example 12 of a function with a flat point if both x, c are finite, and Example 14 of the exponential hyperseries at c = 0 if x is finite.(iii) In Example 15 of Dirac delta at c = 0, if |bx| ≤ 1 (therefore, x is an infinitesimal number) we have

. 22 )
Since ρ R ⊆ ρ R c by Rem.5.(ii), property (2.21) also shows that Def.19 does not depend on the representatives involved.Proof.It suffices to set K := R ∨ max n≤N a n , where R ∈ ρ R and N ∈ N come from (2.20).

.
This proves thatf (n) (c) n! (x − c) n n∈N is eventually ρ R-bounded in ρ R c and hence x ∈ ρ σ conv f (n) (c) n! c, c by Cor.25.

Theorem 39 .
Let U ⊆ ρ R be an open set and f , g ∈ ρ σ GC ω (U, ρ R).Then the setO := int {x ∈ U | f (x) = g(x)}is clopen in the sharp topology.Proof.For simplicity, considering f − g, without loss of generality we can assume g = 0. We only have toshow that O is closed in U .Assume that c is in the closure of O in U , i.e. c ∈ U, ∀r ∈ ρ R >0 ∃c ∈ B r (c) ∩ O. (4.4)We have to prove that c ∈ O.We first note that for each c ∈ O, we have B p (c) ⊆ O for some p ∈ ρ R >0 and hencef (x) = 0 ∀x ∈ B p (c).(4.5)Now, fix n ∈ N in order to prove that f (n) (c) = 0. From (4.4), for all r ∈ ρ R >0 we can find cr ∈ B r (c) ∩ O such that f (n) (c r ) = 0 from (4.5).From sharp continuity of f (n) , we have f (n) (c) = lim r→0 + f (n) (c r ) = 0. Since f ∈ ρ σ GC ω (U, ρ R) and c ∈ U , we can hence find σ > 0 such that f (x) = ρ n∈ σ N f (n) (c)n! (x − c) n = 0 for all x ∈ B σ (c), i.e. c ∈ O.
because the notion of equality ∼ ρ in the two quotient sets is the same and because if (x ε ) is ρ-moderate and (x ε ) ∼ ρ (y ε ), then also(y ε ) is ρ-moderate.(iii) Condition (2.2) of being weakly ρ-moderate represents a constrain on what coefficients a n we can consider in a hyperseries.For example, if (a n ) n∈N is a sequence of real numbers satisfying |a n | ≤ p(n), where p ∈ R[x] is a polynomial, then p(n) ≤ ρ −nQ ε for all ε sufficiently small and for all n we always have a formal HPS.Condition Def.8.(iv) follows because derivatives δ (k) (x) ∈ ρ R are always moderate.It remains to prove Def. 8.(iii) for all we can assume that both (2.2) and (2.3) hold with