Stieltjes analytic functions and higher order linear differential equations

J. Math. Anal. Appl. 526 (2023) 127259 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa Regular Articles Stieltjes analytic functions and higher order linear differential equations Víctor Cora a, F. Javier Fernández a, F. Adrián F. Tojo a,b,∗ aDepartamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Santiago, Spain bCITMAga, 15782, Santiago de Compostela, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 19 January 2023 Available online 24 March 2023 Submitted by S. Hencl Keywords: Stieltjes derivative Analytic functions Higher order linear differential equations Exponential Stieltjes polynomials In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the g-analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients. © 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/). 1. Introduction In recent times, there has been a rise in popularity of the Stieltjes derivative. It turns out that Stieltjes differential equations model more precisely some systems and phenomena than classical ordinary differential equations do, as authors have shown in [7,9,10]. The reason behind it is that the Stieltjes derivative generalizes times scales [7], which is a theory that allows differential equations to have impulsive and stationary behavior. This is particularly interesting when modeling systems that suffer brusque changes or stay latent for some periods of time. In the literature related to Stieltjes differential equations we can find from foundational works regarding the Stieltjes derivative [14], to the study of the first order linear problem along with Picard and Peano type existence results [7]as well as existence results in another settings [9,10]. In [4], for the first time, the authors were able to Stieltjes-differentiate functions at every point of their domain. This allows to successfully define the notion of higher order Stieltjes derivatives and, thus, to consider higher order differential problems [4,6]. *Corresponding author. E-mail addresses: [email protected] (V. Cora), [email protected] (F.J. Fernández), [email protected] (F.A.F. Tojo). https://doi.org/10.1016/j.jmaa.2023.127259 0022-247X/© 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/). 2V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 Since we can differentiate functions an arbitrary number of times, we can define the space of C∞- differentiable (or Ck-differentiable) functions, as they do in [4]. A question that comes to mind is: what non trivial examples of C∞-differentiable functions can we give? In this work we focus on what it should be a smaller subset, that is, the space of Stieltjes-analytic functions. Note that these functions are not obvious to define. First, we have to clarify what a Stieltjes-monomial is, giving birth to a Stieljtes-polynomial theory, so that then we can define the Stieltjes-analytic functions as an infinite series of these monomials. This is no trivial task, as we have to pay attention to several technical difficulties related to the specific behavior of Stieltjes differentiation. We then apply this Stieltjes-analytic function theory to find solutions of differential equations, analogously to how it is used in the classical case. We show a way of translating any higher order linear Stieltjes differential equation with constant coefficients to a difference equation on the coefficients of a Stieltjes-analytic function, which always has a solution. We then prove that such Stieltjes-analytic function actually solves the original problem. In particular, we apply this method to the first order linear problem and define the exponential series. We show it is indeed a Stieltjes-analytic function, study its properties, interval of convergence and whether or not it has an analytic continuation to the whole real line. We then compare the exponential series to the exponential function given in [4,7], the solution of the first order linear problem that the literature considers, and show that they are equivalent. The structure of this work is as follows: In Section 2we present some preliminary concepts and definitions needed to prove the statements in the following sections. In Section 3we introduce the notion of g-monomial and g-polynomial. We study their properties and relations to monomials associated to other (special) derivators which we will later be able to calculate explicitly. In Section 4we finally introduce the Stieltjes-analytic functions. We study and compare them with the classical analytic functions. We show some examples of Stieltjes-analytic functions with interesting behaviors as well. Finally, in Section 5, we apply the Stieltjes-analytic function theory to differential equations and we show, as an application, a method that solves any higher order linear homogeneous Stieltjes differential equation with constant coefficients as well as some nonhomogeneous cases. We then define the exponential series. We study where it converges, what rules it follows when we change the center point, its relation to some specific exponential series and compare it to the exponential function defined in [4,7]. 2. Preliminaries In this Section we provide some basic notions related to the Stieltjes derivative which will be needed later on to introduce the Stieltjes-analytic functions. We recommend the reader to check the cited bibliography, in particular [4,7,11]. 2.1. Derivators Definition 2.1. We will call any g:R →Rleft-continuous non-decreasing function a derivator. To fix notation, we will reserve the letter gfor derivators. It is actually very common to find derivators in the literature defined only on an interval [a, b]⊂R. However, we will assume without loss of generality that every derivator is defined on the whole real line. To see this, take a derivator g:[a, b]→R, with a <b. We can frame gin Definition 2.1 by taking g(x)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩g(b)+x−b, x > b, g(x),x∈[a, b], g(a)+x−a, x < a. V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 3 Given a derivator g, we define the pseudometric dg(x, y)=|g(x)−g(y)|,forx, y ∈R. Definition 2.2. Given any x ∈Rand ε >0, we define the ball of center xand radius εas the set Bg(x, ε):={y∈R|dg(x, y)<ε}. Define τg:= {U⊂R|∀x∈U∃ε>0:Bg(x, ε)⊂U}. τgis a topology where the balls are open sets. We denote by τuthe usual topology. From now on, Fwill denote either Ror C. Definition 2.3. Given any X⊂R, we say a function f:X→Fis g-continuous at a point x ∈Xif ∀ε>0∃δ>0 such that y∈Bg(x, δ)∩X⇒|f(x)−f(y)|<ε. We say fis g-continuous on Xif it is g-continuous at every point x ∈X. Proposition 2.4 ([7, Proposition 3.2]). If f:X→Fis g-continuous on X, then 1. fleft-continuous at every point x ∈X; 2. if gis continuous at x ∈X, then fis continuous at x ∈X; 3. if gis constant on some [α, β], then fis constant on [α, β] ∩X. Denote by Cg(X, F)the set of g-continuous maps defined on Xthat take values on F. We denote by BC g(X, F)the subset of bounded g-continuous maps. Both these sets are vector spaces. Proposition 2.5. {Bg(a, r) | a, r∈Q}is a countable basis of τg. As a consequence, τgis second countable. Proof. Fix some x ∈Rand ε >0. We will prove that there are ax, rx∈Qsuch that x ∈Bg(ax, rx) ⊂ Bg(x, ε). Since gis left-continuous, there exists a sequence {an}n∈N⊂Qthat converges to xfrom the left, and is such that dg(an,x)=|g(an)−g(x)|→0. Hence, we can choose some n ∈Nsuch that for ax=an, dg(ax, x) <ε 2. Let rx∈Qbe such that dg(ax, x) <r x<ε 2. Thus, x ∈Bg(ax, rx). Now, for all y∈Bg(ax, rx), we have dg(x, y)≤dg(x, ax)+dg(ax,y)<ε. Then Bg(ax, rx) ⊂Bg(x, ε). For any U∈τg, we have U= x∈U Bg(x, εx), where εxis so that Bg(x, εx) ⊂U. Now, there are ax, rx∈Qsuch that x ∈Bg(ax, rx) ⊂Bg(x, εx) ⊂Ufor all x ∈U. Clearly, 4V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 U= x∈U Bg(ax,r x). Denote g(x+) = lim y→x+g(y) ∈R, for any x ∈R. Define Δg(x) := g(x+) −g(x). Δg(x) measures the jump of gat any given point x ∈R. Definition 2.6. Given a derivator g, define Dg={x∈R|Δg(x)>0} as the set of discontinuities of g. Define also Cg={x∈R|gis constant on (x−ε, x +ε)forsomeε>0}. Note that Cgis open in the usual topology. Hence, we can write Cg= n∈Λ (an,b n), where Λ ⊂N, (ak, bk) ∩(al, bl) =∅if k=land an, bn∈R ∪{+∞, −∞} for n ∈Λ. With this notation, we denote N− g:= {an}n∈Λ−Dg, N+ g:= {bn}n∈Λ−Dgand Ng:= N− g∪N+ g. Note that {(an, bn) | n ∈N} refers to the connected components of Cg. 2.2. g-derivative Definition 2.7 ([4, Definition 3.1]). Let a, b ∈Rbe such that a <b, a /∈N− gand b /∈Dg∪N+ g∪Cg. We define the Stieltjes derivative or g-derivative of a function f:[a, b] →Fat a point x ∈[a, b]as f g(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim y→x+ f(y)−f(x) g(y)−g(x),x∈Dg, lim y→x f(y)−f(x) g(y)−g(x),x/∈Dg∪Cg, lim y→b+ n f(y)−f(bn) g(y)−g(bn),x∈(an,b n)⊂Cg, (2.1) where (an, bn)is a connected component of Cg. Suppose xfalls in the second case, it could be that x ∈Ng, then we have to understand the corresponding limit as follows f g(x)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ lim y→x+ f(y)−f(x) g(y)−g(x),x∈N+ g, lim y→x− f(y)−f(x) g(y)−g(x),x/∈N− g. Let Ω ⊂Rbe an open set of the usual topology that satisfies ∀x∈Ω such that x∈(an,b n)⊂Cg,we have that bn∈Ω,(2.2) where (an, bn)is a connected component of Cg. Then, for f:Ω →Fand x ∈Ω, we define the Stieltjes derivative of fat xfollowing (2.1). V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 5 Observe that Definition 2.7 assumes functions have domains where the g-derivative is well-defined at all points. Note that we can always define the Stieltjes derivative of a function fat a point x ∈Ras long as f is defined in some neighborhood (x −ε, x +ε)of xand x /∈Cg, just following (2.1). Thus, we may say that some function fdefined on a neighborhood of a point x /∈Cgis g-differentiable at xwithout its domain satisfying the hypothesis of Definition 2.7. We strongly recommend the reader to explore the details of this definition as presented in [4, Definition 3.1], [4, Remarks 3.2 and 3.3] and [4, Definition 3.7]. There are examples of domains where the Stieltjes derivative is just not defined at all points. In fact, R can be one of those domains as we will see in Example 3.4. Remark 2.8. It follows from Definition 2.7 that, for x ∈Dg, f g(x)exists if and only if f(x+)exists and we have that f g(x)=f(x+)−f(x) Δg(x). Proposition 2.9. Fix a, b ∈Rsuch that a <b, a /∈N− gand b /∈Dg∪N+ g∪Cg. Given x ∈[a, b], denote x∗=xif x/∈Cg, bnif x∈(an,b n)⊂Cg. Then, given f1, f2:[a, b] →Fg-differentiable at x, we have that 1. λ1f1+λ2f2is g-differentiable at xfor any λ1, λ2∈Fand (λ1f1+λ2f2) g(x)=λ1(f1) g(x)+λ2(f2) g(x). 2. f1f2is g-differentiable at xand (f1f2) g(x)=(f1) g(x)f2(x∗)+f1(x∗)(f2) g(x)+(f1) g(x)(f2) g(x)Δg(x∗). If f1and f2are defined on a neighborhood of x /∈Cg, then (1) and (2) are satisfied (with x∗=x). Proofs of Proposition 2.9 can be found in [4, Proposition 3.9] and [11, Proposition 3.13]. The proof is reduced to computing the limit (2.1)in each case. Suppose f1and f2are g-continuous on [a, b] then, following Proposition 2.4, we obtain f(x)=f(x∗)forx∈[a, b], simplifying point 2 in Proposition 2.9. Note that Δgmay not be left-continuous and hence not g-continuous –cf. [5, Proposition 3.1]. In the classical case, if a function fis differentiable at a point x ∈R, then it is continuous at that point. As seen in [4, Remark 3.3], in the case of the Stieltjes derivative, the g-differentiability of a function only guarantees the g-continuity at points x /∈Ng∪Cg∪Dg. Definition 2.10. Fix a, b ∈Rsuch that a <b, a /∈N− gand b /∈Dg∪N+ g∪Cg. We say f:[a, b] →Fbelongs to C1 g([a, b], F)if the followings conditions are met 1. f∈Cg([a, b], F), 2. ∃f g(x)for all x ∈[a, b]and f g∈Cg([a, b], F). 6V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 Given k∈N, k>1, we say f:[a, b] →Fbelongs to Ck g([a, b], F)if, recursively, 1. f∈Cg([a, b], F), 2. ∃f g(x)for all x ∈[a, b]and f g∈Ck−1 g([a, b], F). We define C∞ g([a, b],F):=  k∈N Ck g([a, b],F). For k∈N∪{∞}, we also define BC k g([a, b],F):={f∈Ck g([a, b],F)|f(n) g∈BC g([a, b],F),∀n=0,...,k}. Analogously, we define the same sets for Ω ⊂Fan open set on the usual topology satisfying (2.2). Note that, thanks to Proposition 2.9, all of the above are vector spaces. 2.3. Lebesgue-Stieltjes integral Throughout the paper we will work with Lebesgue-Stieltjes integrals. The usual way of constructing Lebesgue–Stieltjes measures through a non-decreasing map is applying Caratheodory’s extension theorem. Here we present the theorem directly applied to derivators. For a full statement and proof see [1, Theorems 1.3.2 to 1.3.6]. For more details in the derivator’s case see [11, Example 1.46]. We will denote by B the Borel σ-algebra relative to τu, the usual topology of the real line. Theorem 2.11 (Caratheodory’s extension theorem). Let g:R →Rbe a derivator and μ∗ g:P(R) →[0, +∞]given by μ∗ g(A)=inf n∈N (g(bn)−g(an)) |A⊂ n∈N [an,b n),{[an,b n)}n∈N⊂C where C={[a, b)|a, b ∈R,a<b}. Then μ∗ gis an outer measure, the set Mg={A∈P(R)|μ∗ g(E)=μ∗ g(E∩A)+μ∗ g(E−A)),∀E∈P(R)} is a σ-algebra and the restriction μg=μ∗ g|Mgis a measure on Mg. In particular, C⊂Mgand B⊂Mg. If μis a measure on Bsuch that μ=μgon C, then μ=μgon B. Considering g=Id, we recover the classical construction of the Lebesgue measure. Furthermore, μg([a, b)) = g(b)−g(a), for all a, b ∈Rsuch that a <b, for more details see [11, Example 1.46]. For any x ∈R, we have μg({x})=Δg(x). V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 7 Definition 2.12. Let X∈Mgand consider the measure space (X, MgX, μg). Given any function f:X→F we say it is: 1. g-measurable, if f−1(U) ∈Mg, for all U∈B. 2. g-integrable or f∈L1 μg(X, F), if it is g-measurable and  X |f|dμg<∞. Proposition 2.13. Let X∈B. If f:X→Fis g-continuous on X, then f−1takes Borel sets onto Borel sets, in particular, fis g-measurable. Proof. Let U∈τube an open subset of F. Since fis g-continuous, we have that f−1(U)is an open set of τgintersected with X. From Proposition 2.5 we know that f−1(U)is a countable union of balls intersected with X. Balls are intervals and hence Borel sets, so we have that f−1(U) ∈B. Since Bis the smallest σ-algebra that contains τu, we have the result.  Note that if a function f:X→Fis such that f−1takes Borel sets onto Borel sets then it is g-measurable for any given derivator g. Theorem 2.14 (Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral). Let a, b ∈Rsuch that a <band w:[a, b] →F. Then the following concepts are equivalent: 1. The map wis g-absolutely continuous, that is, for each ε >0, there exists some δ>0such that, for any family {(an, bn)}m n=1 of pairwise disjoint open subintervals of [a, b], m  n=1 (g(bn)−g(an)) <δ⇒ m  n=1 |w(bn)−w(an)|<ε. We denote this as w∈ACg([a, b], F). 2. The map wsatisfies the following properties: (a) There exists w g(x)for all x ∈[a, b), except on a g-measurable set of null μg-measure. (b) w g∈L1 g([a, b), F). (c) For all x ∈[a, b], w(x)−w(a)=  [a,x) w gdμg. A more general result can be seen in [11, Theorem 2.71]. This same statement appears on [7, Theorem 5.1]. For a proof, see [14, Theorem 5.4]. Using the g-integrability of g-continuous functions, we obtain the following result. Proposition 2.15. Let a, b ∈Rbe such that a <band f∈BC g([a, b), F). Then fis g-integrable on [a, b) and the map F:x∈[a, b]→F(x)=  [a,x) fdμg is g-continuous and bounded on [a, b]. 8V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 Theorem 2.16 ([14, Theorem 2.4]). Let a, b ∈Rbe such that a <band f∈L1 μg([a, b), F). Consider the function F:[a, b] →Fgiven by F:x∈[a, b]→F(x)=  [a,x) fdμg. Then, there exists N⊂[a, b)such that μg(N) =0and F g(x) =f(x)for all x ∈[a, b) −N. Proposition 2.17. Let a, b ∈Rbe such that a <band f∈BC g([a, b], F). Then the map F:x∈[a, b]→F(x)=  [a,x) fdμg satisfies F g(x) =f(x)for all x ∈(a, b) −Cg. Besides, if a /∈N− gand b /∈Dg∪N+ g∪Cg, then F g(x) =f(x) for all x ∈[a, b]and F∈BC 1 g([a, b], F). Essentially, this result is proven in [4, Lemma 3.14]. See how Proposition 2.17 and Theorem 2.16 are so closely related, since μg(Cg) =μg(Ng) =0. Proposition 2.18 ([11, Proposition 1.52]). Let g1, g2:R →Rbe two derivators. Define g:R→Ras g(x)=g1(x)+g2(x),x∈R. Then gis also a derivator and μ∗ g(E)=μ∗ g1(E)+μ∗ g2(E),E∈P(R). In particular, we have that any subset g1and g2-measurable is g-measurable and μg(E)=μg1(E)+μg2(E),E∈Mg1∩Mg2. If we restrict to Borel sets, μg=μg1+μg2. This can help us compute integrals over μg. Let X∈Band f:X→Fa g1and g2-measurable map, then fis g-integrable if and only if is g1and g2-integrable and  X fdμg= X fd(μg1+μg2)= X fdμg1+ X fdμg2. 3. g-Monomials In this section we define the g-monomials and g-polynomials and present some interesting properties. We also compute the g-monomials explicitly in the case where gis either a continuous or a totally discontinuous derivator. Finally, in Theorem 3.22, we show that any g-monomial is in fact a combination of monomials of these two types of derivators, continuous or totally discontinuous. This relationship amongst monomials will have an impact as well on the exponential series, as we show in Section 5.2. 3.1. Basic notions We recall that, in the usual case, a function fis analytic on an open subset Ω, if ∀ x0∈Ω, there exists ε >0and {an}n∈N⊂Rsuch that V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 9 f(x)= n∈N an(x−x0)n for x ∈(x0−ε, x0+ε), where the convergence of the series is absolute and uniform. In this case, fis C∞-differentiable on Ωand also an=f(n)(x0) n! for all n ∈N, see [8, Chapter 1]. In a certain way, an analytic function is just an infinite sum of monomials or a polynomial of infinite degree. Whereas, in the usual case, polynomials represent the regular function prototype, for the case of a given derivator g, a polynomial needs not even be g-continuous. If we want to define the concept of a Stieltjes-analytic function, we have to look for a series of functions that are as regular as possible and that maintain their properties when we consider an infinite sum of them. In the classical case we have that  [x0,x) 1dμId =x−x0,2 [x0,x) s−x0dμId(s)=(x−x0)2,3 [x0,x) (s−x0)2dμId(s)=(x−x0)3, ..., which is a very specific instance of a Peano-Beaker series [2,16]. We can replicate this process and define the g-monomials as follows. Definition 3.1. Let g:R →Rbe a derivator and fix some x0∈R. We define gx0,0(x) = 1 for all x ∈R. Given any n ∈N, we define gx0,n :R →Rrecursively as gx0,n(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n [x0,x) gx0,n−1dμg,x≥x0, −n [x,x0) gx0,n−1dμg,x<x 0. We will call these functions g-monomials centered at x0, where x0is called the center of gx0,n. We will call the linear combinations of g-monomials centered at x0g-polynomials centered at x0. Remark 3.2. Generalizing the concept of monomial by repeatedly integrating the constant function 1 had already been introduced in the context of time scales, see [3, Section 1.6]. However, the framework that we present in this work is more general in the sense of given a time scale T, i.e., a nonempty closed subset of reals, we can recover, thanks to [14, Theorem 3.1], the Δ-derivative by use of the Stieltjes differentiation by considering the derivator g:x ∈R →g(x) =inf{s ∈T, s ≥x}. Thus, the monomials in [3, Section 1.6] coincide with the g-monomials for this particular choice of g. Remark 3.3. Note that gx0,n(x0) =0for all n ≥1. Let a, b ∈Rbe such that a <x 0<b. Since 1 ∈ BC g([a, b], R), Proposition 2.15 assures us that gx0,1is g-continuous and bounded on [a, b]. In particular, gx0,1is g-integrable and therefore gx0,2is well defined. By induction, we have that gx0,n is well defined, g-continuous and bounded on [a, b]for all n ∈N. Note that for x ∈[a, x0], gx0,n(x)=n [a,x) gx0,n−1dμg−n [a,x0) gx0,n−1dμg. 16 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 Proof. Suppose the above holds for some n ∈N. Given x ∈R, thanks to the Lemma 3.6, we have that gn+1(x) n+1 = x  x0 gndμg= x  x0 gn−1g1dμg=1 n x  x0 (gn) gg1dμg =1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg−n x  x0 gn−1Δgdμg⎞ ⎠=1 ngn(x)g1(x)−gn+1(x) n+1 , since Δg=0. Then gn+1(x)=gn(x)g1(x). The last result is trivial for n =1. Applying induction we get what we wanted.  Thus, any g-polynomial is just the composition of a classic polynomial with g. All g-polynomials are of the form p(g(x)) where p(x) =n k=0 akxk, x ∈R. 3.2.4. g-Monomials in the discontinuous case We will suppose now that gis a derivator such that gC=0(gB=g). The latter will allow us to compute the g-monomials explicitly. Proposition 3.16. Let g:R →Rbe a derivator such that gC=0. Fix some x0∈R. Given n ∈N, if x <x 0, we denote In x={σx:Dg∩[x, x0)→{0,...,n}|  y∈Dg∩[x,x0) σx(y)=n}. Then, gn(x) n!=(−1)n σx∈In x y∈Dg∩[x,x0) Δg(y)σx(y).(3.3) If x >x 0, define Jn x={σx:Dg∩[x0,x)→{0,1}|  y∈Dg∩[x0,x) σx(y)=n}. Then, gn(x) n!= σx∈Jn x y∈Dg∩[x0,x) Δg(y)σx(y).(3.4) Note that Jn xrepresents all possible subsets of nelements in Dg∩[x0, x). In particular, we have that, if |Dg∩[x0,x)|<n, gn(x) =0. Proof. Assume Dgis a set of isolated points. The general case is achieved combining Proposition 3.18 with the isolated points case. Since gC=0, we have that V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 17 g1(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩  s∈[x0,x) Δg(s),x>x 0, − s∈[x,x0) Δg(s),x≤x0, so the formula is true for n =1. Suppose that it also works for some arbitrary n ∈Nand proceed by induction. Take x ∈Rsuch that x <x 0. Since discontinuity points are a set of isolated points, there are only a finite number of them in [x, x0). Let {xk}mx k=1 =Dg∩[x, x0) ordered from highest to lowest, where mx=|Dg∩[x, x0)|∈N. Note that the right hand side of the equality (3.3)can be written as (−1)n j∈In∗ x Δg(x1)j1Δg(x2)j2···Δg(xmx)jmx, where In∗ x={j∈{0,...,n}mx| mx  k=1 jk=n}. Then, gn+1(x) (n+1)! =−1 n! [x,x0) gndμg=−1 n! mx  k=1 gn(xk)Δg(xk) =(−1)n+1 mx  k=1 ⎛ ⎝ j∈In∗ xk Δg(x1)j1Δg(x2)j2···Δg(xk)jk⎞ ⎠Δg(xk) (3.5) =(−1)n+1  j∈In+1∗ x Δg(x1)j1Δg(x2)j2···Δg(xmx)jmx,(3.6) as there is a one to one correspondence of addends in (3.5)and addends in (3.6). Note that mxk=k, for k∈{1, ..., mx}. Take x ∈Rsuch that x >x 0. Let {xk}mx k=1 =Dg∩[x0, x)be ordered from lowest to highest, where mx=|Dg∩[x0,x)|∈N. We will apply induction for n ∈N. The right side of the equality (3.4)can be written as  j∈Jn∗ x Δg(x1)j1Δg(x2)j2···Δg(xmx)jmx, where Jn∗ x={j∈{0,1}mx| mx  k=1 jk=n}. Then, 18 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 gn+1(x) (n+1)! =1 n! [x0,x) gndμg=1 n! mx  k=1 gn(xk)Δg(xk) = mx  k=1 ⎛ ⎝ j∈Jn∗ xk Δg(x1)j1Δg(x2)j2···Δg(xk−1)jk−1⎞ ⎠Δg(xk) = j∈Jn+1∗ x Δg(x1)j1Δg(x2)j2···Δg(xmx)jmx, as the same one to one correspondence holds again. Here, mxk=k−1, for k∈{1, ..., mx}. Note that, if we assume that there are mxdiscontinuities in [x0, x), for n ≥mx+1, gn(y) =0for y∈[x0, x].  3.3. Derivator approximation We will prove, in Theorem 3.22, that, in fact, the g-monomials are a combination of products of gC and gB-monomials. But, for that, it will be necessary for us to be able to approximate any derivator by derivators of which the set of discontinuity points is a set of isolated points. Let g:R →Rbe a derivator. Given m ∈N, we denote Dm g={x ∈R :Δg(x) ≥1 m}. Clearly, we have that Dg= m∈N Dm g. Note that, in fact, Dm gis a set of isolated points. One way of seeing this is that for all a, b ∈Rsuch that a <b, Dm g∩[a, b)is a finite set of points. This is easy to prove. Suppose it is not finite, then ∞≤  t∈Dm g∩[a,b) Δg(t)≤ t∈[a,b) Δg(t). However, we know that the previous sum is convergent and bounded by g(b) −g(a). Define gB,m :R →R given by gB,m(x)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  t∈Dm g∩[0,x) Δg(t),x>0, − t∈Dm g∩[x,0) Δg(t),x≤0. In particular, gB,m is left-continuous, nondecreasing and g-continuous. Given m ∈N, we define the gcontinuous derivator gm=gC+gB,m.(3.7) We have that, for any x ∈R, g(x)−gm(x)=gB(x)−gB,m(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  t∈[0,x) Δg(t)<1 m Δg(t),x>0, − t∈[x,0) Δg(t)<1 m Δg(t),x≤0, V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 19 and, again, g(x) −gm(x)is a g-continuous derivator. Note that, in particular, given R∈Rsuch that 0 <R, for all x ∈[−R, R], |g(x)−gm(x)|≤  t∈[−R,R) Δg(t)<1 m Δg(t). Proving that the above tends to zero when mtends to infinity, we would obtain that gm→guniformly on [−R, R]. Proposition 3.17. Let g:R →Rbe a derivator. For any a, b ∈Rsuch that a <b, we have lim m→∞  t∈[a,b) Δg(t)<1 m Δg(t)=0. Proof. If Dg∩[a, b)is finite, the above is trivial. Assume Dg∩[a, b)is infinite, let {tn}n∈N=Dg∩[a, b). Fix some ε >0, since the sum of the jumps is convergent, there exists n0∈Nsuch that ∞  n=n0 Δg(tn)<ε. Choose m0∈Nso that 1 m0≤Δg(tn)for n =1, ..., n0. Then, for m ≥m0, 0≤ t∈[a,b) Δg(t)<1 m Δg(t)≤ ∞  n=n0 Δg(tn)<ε.  Proposition 3.18. Let g:R →Rbe a derivator and fix some x0∈R. Fix R>0, for every n ∈N, gm n→gn uniformly on [x0−R, x0+R]. Proof. Note that we have just proved the case for n =1. Recall that g−gmis a derivator, so Proposition 2.18 applies. Suppose the result is true for some n ∈Nand denote A =[x0−R, x0+R]. For any x ∈A, gm n+1(x)−gn+1(x) n+1 = x  x0 gm ndμgm− x  x0 gndμg = x  x0 gm ndμgm− x  x0 gndμgm+ x  x0 gndμgm− x  x0 gndμg = x  x0 (gm n−gn)dμgm− x  x0 gnd(μg−μgm)≤ x  x0 |gm n−gn|dμgm+ x  x0 |gn|d(μg−μgm) ≤sup A |gm n−gn||gm 1(x)|+sup A |gn||g1(x)−gm 1(x)|. Since gm 1is bounded in Afor every m ∈Nand {gm 1}m∈Nconverges uniformly to g1, {gm 1}m∈Nis uniformly bounded. Choose some M>|gm 1(x)|for all x ∈Aand m ∈N. Then, sup Agm n+1 −gn+1≤(n+1)Msup A |gm n−gn|+sup A |gn|sup A |g1−gm 1|. 20 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 Hence, if C=sup A |gn|, sup Agm n+1 −gn+1≤(n+1)Msup A |gm n−gn|+Csup A |g1−gm 1|→0, when mtends to infinity.  We have now a way of approximating the monomials of any derivator from monomials of derivators with a finite number of discontinuities on bounded sets. We will prove that any monomial of any derivator gis in fact a combination of gCand gB-monomials. For that, a couple of results will be needed. Proposition 3.19. Let g:R →Rbe a derivator, X∈Mgand f∈L1 g(X, R). Then,  X fdμgB= t∈X∩Dg f(t)Δg(t). Proof. The proof is a direct consequence of [4, Lemma 2.3] since L1 g(X, R) ⊂L1 gB(X, R).  Lemma 3.20. Let g:R →Rbe a derivator such that Dgis a set of isolated points and fix some x0∈R. Then, for all n ∈N, m ∈Nand a, b ∈Rsuch that a <b, m [a,b) gC ngB m−1dμgB+n [a,b) gC n−1gB mdμgC=gB m(b)gC n(b)−gB m(a)gC n(a). Proof. Since the set of discontinuity points is a set of isolated points, there must be a finite amount of them in [a, b). Let {xi}k i=1 =Dg∩(a, b) ordered from lowest to highest and add x0=aand xk+1 =b. Thanks to Proposition 3.19, it follows that m [a,b) gC ngB m−1dμgB=m k  i=0 gC n(xi)gB m−1(xi)Δg(xi).(3.8) Let us calculate the second integral from the statement of the lemma. Note that gB mis constant on (xi−1, xi) for i=1,...,k+1, because it is gB-continuous. Since gCis continuous, singletons have null μgC-measure, hence, n [a,b) gC n−1gB mdμgC= k+1  i=1 n (xi−1,xi) gC n−1gB mdμgC= k+1  i=1 gB m(xi)n (xi−1,xi) gC n−1dμgC = k+1  i=1 gB m(xi)(gC n(xi)−gC n(xi−1)). We have that, for i =1, ..., k+1, gB m(xi)−gB m(a)=m [a,xi) gB m−1dμgB=m i−1  j=0 gB m−1(xj)Δg(xj). We can compute now the second integral, V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 21 n [a,b) gC n−1gB mdμgC= k+1  i=1 gB m(xi)(gC n(xi)−gC n(xi−1)) = k+1  i=1 ⎛ ⎝m i−1  j=0 gB m−1(xj)Δg(xj)+gB m(a)⎞ ⎠(gC n(xi)−gC n(xi−1)) =m k  j=0 gB m−1(xj)Δg(xj) k+1  i=j+1 (gC n(xi)−gC n(xi−1)) + gB m(a)(gC n(b)−gC n(a)) =m k  j=0 gB m−1(xj)Δg(xj)(gC n(b)−gC n(xj)) + gB m(a)(gC n(b)−gC n(a)). (3.9) Finally, adding (3.8)and(3.9) together, m k  j=0 gB m−1(xj)Δg(xj)gC n(b)+gB m(a)(gC n(b)−gC n(a)) =(gB m(b)−gB m(a))gC n(b)+gB m(a)(gC n(b)−gC n(a)) = gB m(b)gC n(b)−gB m(a)gC n(a). We obtain the following straightforward corollary. Corollary 3.21. Let g:R →Rbe a derivator where the set of discontinuity points is a set of isolated points and fix some x0∈R. For all n ∈N, m ∈Nand x ∈R, m x  x0 gC ngB m−1dμgB+n x  x0 gC n−1gB mdμgC=gB m(x)gC n(x). Finally, we have what is necessary to prove Theorem 3.22. Theorem 3.22. Let g:R →Rbe a derivator and fix some x0∈R. For all n ∈Nand x ∈R, gn(x)= n  k=0 n kgC k(x)gB n−k(x). Proof. We will prove it first for derivators such that the set of discontinuity points is a set of isolated points. For n =1, it is obvious that g1(x)=gC 1(x)+gB 1(x), for all x ∈R. Let us apply induction, suppose the result is true for some n ∈N. Thanks to Proposition 2.18, for any x ∈R, gn+1(x) n+1 = x  x0 gndμg= x  x0 n  k=0 n kgC kgB n−kdμg= n  k=0 n kx  x0 gC kgB n−kd(μgB+μgC) = n  k=0 n k⎛ ⎝ x  x0 gC kgB n−kdμgB+ x  x0 gC kgB n−kdμgC⎞ ⎠ 22 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 = n  k=1 n kx  x0 gC kgB n−kdμgB+ n−1  k=0 n kx  x0 gC kgB n−kdμgC(3.10) + x  x0 gB ndμgB+ x  x0 gC ndμgC = n  k=1 ⎛ ⎝n kx  x0 gC kgB n−kdμgB+n k−1x  x0 gC k−1gB n−k+1 dμgC⎞ ⎠ + x  x0 gB ndμgB+ x  x0 gC ndμgC. Fix k∈{1, ..., n}. Then, n kx  x0 gC kgB n−kdμgB+n k−1x  x0 gC k−1gB n−k+1 dμgC(3.11) equals 1 n+1n+1 k⎛ ⎝(n−k+1) x  x0 gC kgB n−kdμgB+k x  x0 gC k−1gB n−k+1 dμgC⎞ ⎠. Applying Corollary 3.21,(3.11)equals 1 n+1n+1 kgC k(x)gB n+1−k(x).(3.12) Splitting the sum (3.10)in terms like (3.11)and thanks to (3.12)we have that gn+1(x) n+1 = n  k=0 n k⎛ ⎝ x  x0 gC kgB n−kdμgB+ x  x0 gC kgB n−kdμgC⎞ ⎠ =1 n+1 n  k=1 n+1 kgC k(x)gB n+1−k(x)+ x  x0 gB ndμgB+ x  x0 gC ndμgC =1 n+1 n  k=1 n+1 kgC k(x)gB n+1−k(x)+gB n+1(x) n+1 +gC n+1(x) n+1 =1 n+1 n+1  k=0 n+1 kgC k(x)gB n+1−k(x). Hence, the result is true for derivators where the set of discontinuities is a set of isolated points. Now, let g:R →Rbe an arbitrary derivator. Take gmfor m ∈N, like in (3.7). We have that Dgm=Dm g, (gm)B=gB,m and then gmis derivator such that Dgmis a set of isolated points. Besides, thanks to Proposition 3.18, for all n ∈Nand x ∈R, gm n(x)→gn(x) V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 23 when mtends to infinity. Fix n ∈N, gm n(x)= n  k=0 n kgC k(x)gB,m n−k(x). Note that (gB)m=gB,m →gB, we can apply Proposition 3.18 so, gB,m k(x)→gB k(x) for all k∈Nand x ∈R. Then, when mtends to infinity, gm n(x)= n  k=0 n kgC k(x)gB,m n−k(x)→ n  k=0 n kgC k(x)gB n−k(x)=gn(x). 3.4. More calculus on g-monomials We will calculate now a new expression for the g-monomials using the integration by parts formula (Lemma 3.6) and iterative integrals. The purpose of this is to see how the g-monomials differ from being a power of g1. Define h1,k(x) k+1 = x  x0 gkΔgdμg,k≥0, hj+1,k(x) k+j+1 = x  x0 hj,k dμg,k≥0,j≥1. (3.13) We have the following result. Proposition 3.23. Let g:R →Rbe a derivator and n ∈N. For all x ∈R, gn(x)=gn−1(x)g1(x)− n−1  j=1 hj,n−1−j(x). Proof. The result is trivial for n =1. We proceed by induction. Suppose the above is true for some n ∈N. For all x ∈R, gn+1(x) n+1 = x  x0 gndμg= x  x0 gn−1g1dμg− n−1  j=1 x  x0 hj,n−1−jdμg. Applying Lemma 3.6, x  x0 gn−1g1dμg=1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg−n x  x0 gn−1Δgdμg⎞ ⎠ =1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg−h1,n−1⎞ ⎠. 24 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 By expression (3.13), gn+1(x) n+1 =1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg−h1,n−1⎞ ⎠− n−1  j=1 x  x0 hj,n−1−j(s)dμg =1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg−h1,n−1⎞ ⎠− n−1  j=1 hj+1,n−1−j(x) n =1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg−h1,n−1− n−1  j=1 hj+1,n−1−j(x)⎞ ⎠ =1 n⎛ ⎝gn(x)g1(x)− x  x0 gndμg− n  j=1 hj,n−j(x)⎞ ⎠. Hence, gn+1(x)=gn(x)g1(x)− n  j=1 hj,n−j(x). We now present a general formula obtained by applying the recursive expression that we have just computed. Proposition 3.24. Let g:R →Rbe a derivator and n ∈N. For all x ∈R, gn(x)=g1(x)n− n−1  k=1 g1(x)n−1−k k  j=1 hj,k−j(x). Proof. The result is trivial for n =1. We proceed by induction again. Suppose the above is true for some n ∈N. Applying Proposition 3.23, for all x ∈R, gn+1(x)=gn(x)g1(x)− n  j=1 hj,n−j(x) =⎛ ⎝g1(x)n− n−1  k=1 g1(x)n−1−k k  j=1 hj,k−j(x)⎞ ⎠g1(x)− n  j=1 hj,n−j(x) =g1(x)n+1 − n−1  k=1 g1(x)n−k k  j=1 hj,k−j(x)− n  j=1 hj,n−j(x) =g1(x)n+1 − n  k=1 g1(x)n−k k  j=1 hj,k−j(x). 4. Stieltjes-analytic functions In this section we introduce the Stieltjes-analytic functions. We study first some of the properties of series of g-monomials. We will try to replicate the classical analytic theory as far as possible. We will see several examples that will limit how far we can go. V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 25 Some studies have tried to develop a analytic function theory in the frame of time scales [13]. A comparison between this and our theory is particularly interesting since the Stieltjes derivative generalizes time scales [7]. 4.1. g-Monomial series We can start talking about Stieltjes-analytic functions now that we have the g-monomials already defined. Following the classical case, we want our functions to be an infinite sum of g-monomials, that is, f(x)= ∞  n=0 angn(x), for some {an}∞ n=0 ⊂Fand x ∈R. As we will see later, finding the set of convergence of a series of g-monomials can be more challenging than it seems. It is also interesting to ask whether there is any relationship between an analytic function in the usual sense defined by some coefficients at a fixed point, and the, a priori, Stieltjes-analytic function defined by the same coefficients using the g-monomials. We will prove, thanks to Theorem 3.22, that, under certain hypotheses, the map Series of g-monomials Stieltjes-analytic functions  Power series Analytic functions  f(x)= ∞  n=0 angn(x) f(x)= ∞  n=0 anxn, satisfies that f(x)= ∞  k=0  f(k)(gC 1(x)) k!gB k(x).(4.1) A result that we will often use is the following. Theorem 4.1 ([15, Theorem 8.3]). Given any sequence {aij}∞ i,j=0 ⊂F. If ∞  i=0 ∞  j=0 |aij|<∞, then ∞  i=0 ∞  j=0 aij = ∞  j=0 ∞  i=0 aij. For continuous derivators, since gn=gn 1, we can do the same analytic function theory as in the classical case. In this case, the convergence of a power series centered at x0either occurs only on x0or on a ball of positive radius. We can replicate the same result for continuous derivators, which are going to behave much better than those that have discontinuities. Recall that we continue with the notation gx0,n ≡gn. 4.1.1. Convergence In the general case, the convergence on the right hand side or the left hand side of x0does not imply convergence on the other. In fact, it will be a lot easier for series of g-monomials to converge at points at the 32 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 4.2. Stieltjes-analyticity Since asking for convergence on the left side of x0is a very strong hypothesis, recall Proposition 4.10, we decide to give this definition of Stieltjes-analytic function. Definition 4.11. Let g:R →Ra derivator. Given f:Ω →F, and Ω ⊂Ra (usual) open set. We say fis Stieltjes-analytic on Ω, if, for all y∈Ω, there exist δ>0and t ∈Rsuch that y∈(t, t +δ)and a sequence {an}∞ n=0 ⊂Fsatisfying f(x)= ∞  n=0 angt,n(x) for all x ∈(t, t +δ) ⊂Ω, and, moreover, the sum converges absolutely. With this definition, the function fin Example 4.9 is Stieltjes-analytic on Ω =(0, 1). For any y∈Ω, taking t =0the definition is satisfied. Note how this differs from the usual definition, for a function to be analytic at any given point it has to be written as a power series centered at that point. This is something we do not ask for here. It is easy to prove that the function fin Example 4.8 is Stieltjes-analytic on Ω =(−1, 1), since in that case gis a derivator whose set of discontinuities is a set of isolated points. Definition 4.11 is equivalent to the usual when g=Id. The space of Stieltjes-analytic functions defined on the same domain is a vector space. We will see how Stieltjes-analyticity behaves with integration and derivation. To do so, we will present some results that relate these concepts to the series of g-monomials. Proposition 4.12. Let {fn}n∈Nbe a sequence of functions defined on X⊂Rthat converges uniformly to some function f. If fnis g-continuous for all n ∈N, then fis g-continuous as well. The proof of this statement is practically identical to that of [7, Theorem 3.4]. As a consequence, we have that BC g(X, R)is a Banach space. It is easy to see now that any Stieltjes-analytic function is g-continuous in every point of their domain. In particular, any Stieltjes-analytic function is g-continuous. 4.2.1. Stieltjes-analyticity and integration Let us see how series of g-monomials behave with integration. Thanks to the Dominated Convergence Theorem, we will prove absolute convergence of the series of integrals on the same set where the g-monomial series converges. Proposition 4.13. Let g:R →Rbe a derivator and fix some x0∈R. Suppose that f(x)= ∞  n=0 angn(x) converges absolutely on [c1, c2], for some c1, c2∈Rsuch that x0∈(c1, c2). Then fis g-integrable in [c1, c2] and x  x0 fdμ= ∞  n=0 an n+1gn+1(x) for all x ∈[c1, c2]. Besides, the previous sum converges uniformly and absolutely on [c1, c2]. V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 33 Proof. Thanks to Proposition 4.12, fis g-continuous in [c1, c2]. Besides, |f(x)|≤max i=1,2 ∞  n=0 |an||gn(ci)|. Then f∈BC g([c1, c2]) and hence fis g-integrable. Applying Dominated Convergence Theorem, for all x ∈[c1, c2], x  x0 fdμ= lim m→∞ m  n=0 an x  x0 gndμg= lim m→∞ m  n=0 an n+1gn+1(x)= ∞  n=0 an n+1gn+1(x).(4.6) Denote has h(x)= ∞  n=0 |an||gn(x)|, for x ∈[c1, c2]. We have that his g-integrable since it is g-measurable and bounded. We can apply the Dominated Convergence Theorem to hagain so ∞  n=0  an n+1|gn+1(x)|= ∞  n=0 |an| [x0,x) |gn|dμg= [x0,x) hdμg, ∞  n=0  an n+1|gn+1(x)|= ∞  n=0 |an| [x,x0) |gn|dμg= [x,x0) hdμg. The series (4.6)is absolutely convergent for x ∈[c1, c2]. We get uniform convergence on [c1, c2]from Proposition 4.3. 4.2.2. Stieltjes-analyticity and differentiability Although integrability behaves well with series of g-monomials, we have to ask for more assumptions to ensure the convergence of the series of derivatives. The derivative will exist at points where |g1|is strictly less than the extremes of the convergence interval. Note that this is also required in the usual case. However, here it leads to more problems due to the constancy intervals of g–see Example 4.16. Proposition 4.14. Let g:R →Rbe a derivator and fix some x0∈R. Suppose that f(x)= ∞  n=0 angn(x) converges absolutely on [c1, c2], for some c1, c2∈Rsuch that x0∈(c1, c2). If c ∈(x0, c2)is such that |g1(c)|<|g1(c2)|, the series ∞  n=1 nangn−1(x) converges absolutely for x ∈[x0, c]. Besides, for any x ∈(x0, c) −Cg, f g(x)= ∞  n=1 nangn−1(x).(4.7) 34 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 If (x0, c)satisfies (2.2)then (4.7)holds for all x ∈(x0, c). Analogously, the same applies if c∈(c1,x 0)and |g1(c)|<|g1(c1)|. Proof. Take c ∈(x0, c2)such that |g1(c)|<|g1(c2)|. If we follow the proof of Proposition 4.6, we have that the series ∞  k=0 1 k! ∞  n=k |an|n! (n−k)! |gn−k(c)||gc,k(c2)|(4.8) converges. Therefore, since |g1(c)|<|g1(c2)|⇔|gc,1(c2)| =0 and the coefficient associated to |gc,1(c2)|in the series (4.8)is ∞  n=1 n|an||gn−1(c)|,(4.9) the sum (4.9)must converge. In particular, we have that the series ∞  n=1 nangn−1(x) converges absolutely on [x0, c]. Applying Proposition 4.13 to the previous sum, if x ∈[x0, c], f(x)−f(0) =  [x0,x) ∞  n=1 nangn−1(s)dμg(s). Thanks to Proposition 2.17, f g(x)= ∞  n=1 nangn−1(x) for all x ∈(x0, c) −Cg. If (2.2)holds with Ω =(x0, c), the g-derivative is well defined at all points, so equality (4.7)holds for all x ∈(x0, c). The same arguments are valid for points to the left of x0. Theorem 4.15. Let g:R →Rbe a derivator. If f:Ω →Fis a Stieltjes-analytic function that satisfies: (H)For all y∈Ω, there exist 0 <δand t ∈Rsuch that y∈(t, t +δ), gC(y) <g C(t +δ)and a sequence {an}∞ n=0 ⊂Fsatisfying f(x)= ∞  n=0 angt,n(x) for all x ∈(t, t +δ) ⊂Ω, and, moreover, the sum converges absolutely. Then Ωsatisfies condition (2.2), fis g-differentiable on Ωand f g:Ω→Fis again a Stieltjes-analytic function that satisfies (H). V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 35 Proof. Take y∈Ω. By (H), there are 0 <δ, t ∈Rand a sequence {an}∞ n=0 ⊂Fsuch that y∈(t, t +δ)and f(x)= ∞  n=0 angt,n(x) for all x ∈(t, t +δ] ⊂Ωand gC(y) <g C(t +δ)(we can assume that the series converges uniformly on (t, t +δ]choosing a suitable δ). By the continuity of gC, there exists δ∈(0, δ)such that y∈(t, t +δ)and gC(y) <g C(t +δ) <g C(t +δ). Applying Proposition 4.14, the series ∞  n=1 nangt,n−1(x) (4.10) converges absolutely on [t, t +δ]. Besides, we know that f g(x)exists at x ∈(t, t +δ) −Cgand equals the series (4.10). We can repeat the previous argument until we reach the following conclusion, for t=sup{x∈R:gC(x)<g C(t+δ)}, the series (4.10)converges absolutely on [t, t)and equals f gon (t, t) −Cg. Note that, if x ∈(t, t), then gC(x) <g C(t). Suppose x ∈(t, t) ∩Cgand x ∈(an, bn) ⊂Cg. We have that gC(x) =gC(bn) <g C(t) hence [x, bn] ⊂(t, t) ⊂Ωsince bn<t . Note that this proves that both (t, t)and Ωsatisfy hypothesis (2.2). Then, the series (4.10)equals f gon (t, t)by Definition 2.7. Choosing adequate tand δwe have that f g satisfies (H)at y. Since ywas arbitrarily chosen, we have that f gis Stieltjes-analytic.  Note that the hypotheses made in Theorem 4.15 are automatically fulfilled if gis continuous and strictly increasing. Take a, b ∈Rsuch that a <b, a /∈N− g, b /∈Dg∪N+ g∪Cgand [a, b] ⊂Ω, if fis a Stieltjes-analytic function that satisfies (1), applying Theorem 4.15 recursively we obtain that f∈C∞ g([a, b],F). In any case, we have f∈C∞ g(Ω,F). Example 4.16. Take g:R →Rgiven by g(x)=x, x ≤0, n, x ∈(n−1,n],n∈N. Fix x0=0. It is clear that gn(x) =xnfor x ∈(−∞, 0]. Now, g2(x) =0for x ∈(0, 1], g3(x) =0for x ∈(0, 2] and, in general, gn(x) =0for x ∈(0, n −1] for all n ≥2. In particular, any g-monomial series is finite for all x ∈(0, ∞), since for n ∈Ngreater than some fixed natural, gn(x) =0. Consider the series f(x)= ∞  n=0 n!gn(x). The series does not converge at points x <0, since the sequence {n!xn}∞ n=0 does not tend to 0when n tends to infinity. Even so, it converges absolutely at [0, ∞), since for every point x >0the sum is finite. In particular, we have that fsatisfies Definition 4.11 in (0, ∞). In fact, fhas g-derivatives of all orders that are Stieltjes-analytic functions too. Consider now fdefined only in (0, y), for some y>0. Take n ∈Nsuch 36 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 that n −1 <y≤n. Since gn+1(x) =0for x ∈(0, n], the coefficients {k!}∞ k=0 that define fare not unique anymore. We are free to choose the coefficients for k≥n +1. Note that we are in a similar situation to that of Example 3.4. Since fis defined only on (0, y), we can only differentiate at points in (0, n −1]. Its derivative can only be differentiated at points in (0, n −2], and so on. The coefficients will tell us what values f gtakes on (n −1, y], f(2) gon (n −2, y], etc. The last example proves that the uniqueness of the coefficients of a Stieltjes-analytic function is not ensured. Combining the arguments made in Theorem 4.15 and Proposition 4.14 we obtain the following result. Proposition 4.17. Let g:R →Rbe a derivator and fix some x0∈R. If f(x)= ∞  n=0 angn(x) converges absolutely on [c1, c2], for some c1, c2∈Rsuch that x0∈(c1, c2), gC 1(c2) >0and g1(c1) <0. If t2=sup{x∈R:gC 1(x)<g C 1(c2)} and t1=inf{x∈R:g1(c1)<g 1(x)}, the series ∞  n=k an n! (n−k)!gn−k(x) converges absolutely on (t1, t2), for all k∈N. Besides, (t1, t2)satisfies (2.2)and, for all x∈(t1,t 2), f(k) g(x)= ∞  n=k an n! (n−k)!gn−k(x). Proof. Note that x0∈(t1, t2). Besides, if x ∈(t1, t2), then gC 1(x) <g C 1(t2)and g1(t1) <g 1(x). That is because gC 1(t2) =gC 1(c2)and g1(t1) =g1(c1). Take x ∈(x0, t2). We have that x <c 2. Hence, from Proposition 4.6, the series ∞  k=0 1 k! ∞  n=k |an|n! (n−k)! |gn−k(x)||gx,k(c2)| converges. Now, since gC 1(x) <g C 1(c2), gx,k(c2) =0for all k∈N, so the series ∞  n=k |an|n! (n−k)! |gn−k(y)| converges for all k∈Nand y∈[x0, x]. Applying Proposition 4.13 and Proposition 2.17 recursively we have that f(k) g(y)= ∞  n=k an n! (n−k)!gn−k(y) (4.11) V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 37 for y∈(x0, x) −Cg. With xapproaching t2, the series (4.11)converges absolutely on [x0, t2)and equality (4.11)holds for all y∈(x0, t2). To the left of x0, the argument is the same. If x ∈(t1, x0)we only need that g1(c1) <g 1(x)to ensure gx,k(c1) =0for all k∈N. Note that, if the hypotheses are satisfied at both sides, we can g-differentiate at x0∈(t1, t2).  Note that if g=Id, the last result guarantees that, if (−R, R)is the convergence interval of a power series, it is the convergence interval of the series of its derivatives as well. We have achieved this without applying the Cauchy–Hadamard theorem [8, Lemma 1.1.6]. 4.2.3. Coefficients of a Stieltjes-analytic function We have seen that the relationship between a Stieltjes-analytic function and its coefficients can be more complex than it seems, see Example 4.16. Even so, that relation behaves like in the classical case if we ask for the right hypotheses. Theorem 4.18. Let fbe a Stieltjes-analytic function defined on Ωand fix y∈Ω. Assume fsatisfies (H), that is, there are δ>0, t ∈Rand a sequence {an}∞ n=0 ⊂Fsuch that y∈(t, t +δ), f(x)= ∞  n=0 angt,n(x) for all x ∈(t, t +δ) ⊂Ωand gC(y) <g C(t +δ)with the series converging absolutely. Suppose that t ∈Ω. Then f(n) g(t) n!=an for n ≥0. Proof. From Theorem 4.15, we have g-derivatives of fof all orders at t ∈Ω. From Proposition 4.17, exists t∈(y, t +δ)such that f(k) g(x)= ∞  n=k an n! (n−k)!gt,n−k(x) (4.12) for all x ∈(t, t)and k∈N. Take t ∈(t, t)to ensure uniform convergence of the series (4.12)on [t, t]. Then, lim x→t+f(k) g(x) = lim m→∞ m  n=k lim x→t+an n! (n−k)!gt,n−k(x) =akk!+ak+1(k+1)!Δg(t), for all k∈N. Now, if Δg(t) =0, f(k) gis continuous at tand f(k) g(t) = lim x→t+f(k) g(x)=akk!. Hence, f(k) g(t) k!=ak 38 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 for all k≥0. If Δg(t) =0, tis a discontinuity point of gand then, for k≥1, f(k) g(t) = lim x→t+ f(k−1) g(x)−f(k−1) g(t) g(x)−g(t)=akk!. Hence, f(k) g(t) k!=ak for all k≥0.  Remark 4.19. Note that, since fsatisfies the hypotheses of Theorem 4.15, f∈C∞ g(Ω, F)and fallows g-derivatives of all orders on Cg. If t ∈Cg, then Δg(t) =0and the argument we gave in Theorem 4.18 holds. 5. Differential equations and applications 5.1. Differential equations Assume from now on that ∞ /∈N+ g, just to guarantee intervals not bounded from above satisfy condition (2.2). Many of the things we say here are true without that assumption. The concept of Stieltjes-analytic function was born with the aim of solving differential equations, especially, linear differential equations. We now give a method to solve any higher order linear homogeneous Stieltjes differential equations with constant coefficients that works for some nonhomogeneous cases as well. Fix some x0∈R, consider the initial value problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ v(m) g(x)= m−1  k=0 λkv(k) g(x),λ k∈F, v(k) g(x0)=ck,c k∈F,k∈{0,...,m−1}. (5.1) We can study whether there is a Stieltjes-analytic solution to this problem. Suppose that a Stieltjes-analytic solution exists and we can center its g-monomial series on x0. Then, v(x)= ∞  n=0 angn(x) for some coefficients {an}n∈N⊂R. From problem (5.1)and v(k) g(x)= ∞  n=k an n! (n−k)!gn−k(x) we obtain the following difference equation by matching the coefficients: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ an+m(n+m)! = m−1  k=0 λkan+k(n+k)!,λ k∈F,n≥0, akk!=ck,c k∈F,k∈{0,...,m−1}. (5.2) V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 39 Theorem 5.1. Let λ0, λ1, ..., λm−1∈F. Given the mth-order linear difference equation with constant coefficients an+m+λm−1an+m−1+···+λ0an=0,n≥0.(5.3) If c0, c1, ..., cm−1are real numbers, there is a unique solution of (5.3), that satisfies a0=c0,a 1=c1, ..., a m−1=cm−1. This result can be found in [12, Theorem 4.3], along with the explicit solutions of (5.3). From Theorem 5.1, there is a unique sequence {an}∞ n=0 that solves problem (5.2). Consider then the g-monomial series defined by {an}∞ n=0. As we will prove in Section 5.3, that function is well defined and actually solves problem (5.1). In particular, we have that the problem (5.1)admits Stieltjes-analytic solutions. We will for now focus on the first order linear problem as it will help us later to solve higher order equations. Consider the problem v g(x)=λv(x), vg(x0)=1.(5.4) The associated difference equation would be an+1(n+1)!=λann!,n≥1, a0=1. We then obtain the sequence an=λn n!. In the classical case we obtain the exponential function when we consider the power series. The exponential is clearly a solution of (5.4)when g=Id. We will see that this actually translates to the general case. 5.2. The exponential series Due to how the literature understands the concept of exponential function associated with a derivator [4,7], we will talk about the exponential series instead of the exponential function. Later, we will see how the two concepts are related. In any case, we will prove that the exponential series in general solves the differential equation (5.4), as well as certain properties that are deduced from the series of g-monomials. Let g:R →Rbe a derivator, λ ∈F−{0}and fix some x0∈R. Consider the series ∞  n=0 λngn(x) n!. Take x >x 0. We have that ∞  n=0 |λ|ngn(x) n!≤ ∞  n=0 |λ|ng1(x)n n!≤e|λ|g1(x). Hence, the series converges absolutely on [x0, ∞). Thus, the previous series defines a Stieltjes-analytic function on the set (x0, ∞). Take now x <x 0such that x ∈Bg(x0, |λ|−1). Since gis left-continuous, the ball Bg(x0, |λ|−1)contains a neighborhood (x0−δ, x0]for some δ>0. We have that 40 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 ∞  n=0 |λ|n gn(x) n!≤ ∞  n=0 |λ|n|g1(x)|n<∞,(5.5) since |λg1(x)|<1. We have assured absolute convergence on Bg(x0, |λ|−1) ∪[x0, ∞) and, hence, on some neighborhood of x0. Definition 5.2. Given s ∈R, define the exponential series associated to gcentered at sas the function given by the series expg(λ;s)(x)= ∞  n=0 λngs,n(x) n! on those points where the sum converges absolutely. Choose some s ∈Bg(x0, |λ|−1)such that s <x 0. Take the exponential series associated to gcentered at s. Note first that the series ∞  n=0 |λ|n gs,n(x) n!(5.6) converges absolutely for x ∈Bg(s, |λ|−1) ∪[s, ∞), applying the same argument we did for x0. Thanks to the product formula for absolutely convergent series [15, Theorem 3.50], for some fixed sand all x ∈ Bg(s, |λ|−1) ∪[s, ∞), we have that ∞  n=0 |λ|n gn(s) n!∞  n=0 |λ|n gs,n(x) n!= ∞  n=0 n  k=0 |λ|k gk(s) k!|λ|n−k gs,n−k(x) (n−k)!  = ∞  n=0 |λ|n n  k=0  gk(s) k! gs,n−k(x) (n−k)! . Since, applying the change of center formula of Proposition 3.14, we have that ∞  n=0 |λ|n gn(x) n!≤ ∞  n=0 |λ|n n  k=0  gk(s) k! gs,n−k(x) (n−k)! , the series ∞  n=0 λngn(x) n! converges absolutely on Bg(s, |λ|−1) ∪[s, ∞), which is a bigger set than the one we calculated on (5.5). We could try to prove that expg(λ; x0)is defined in the whole real line taking balls of radius |λ|−1recursively. However, if there is a discontinuity with a jump bigger than |λ|−1this process is invalid. Take any x0∈R, we call Ωx0the maximal interval where ∞  n=0 λngn(x) n! converges absolutely. We have the following result. V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 41 Proposition 5.3. Let g:R →Rbe a derivator and fix some x0∈R. If Δg(x) <|λ|−1for all x <x 0, then Ωx0=R. Proof. Clearly, if Ωx0is not bounded from below, then Ωx0=R. Suppose otherwise and take t=infΩ x0∈R. Since lim x→t+g(x)−g(t)=Δg(t)<|λ|−1, there exists s >tsuch that |g(s)−g(t)|<|λ|−1. Since s ∈Ωx0, we can repeat the calculations made in (5.6) and deduce that ∞  n=0 λngn(x) n! converges absolutely on Bg(s, |λ|−1). Now, we can choose t<tin such a way that |g(s)−g(t)|<|λ|−1 and, hence, [t, s] ⊂Bg(s, |λ|−1) ⊂Ωx0, which contradicts that tis the infimum of Ωx0. Corollary 5.4. If Ωx0is bounded from below, then t =infΩ x0∈Ris such that Δg(t) ≥|λ|−1, which means Ωx0=(t, +∞). Besides, t=sup{s∈Dg∩(−∞,x 0)|Δg(s)≥| λ|−1}. Proposition 5.5. For all t ∈R, s ∈Ωtand x ∈Ωswe have that x ∈Ωtand expg(λ;t)(s)exp g(λ;s)(x)=exp g(λ;t)(x). Proof. Repeating the calculations of (5.6)we have that x ∈Ωt. Again, from the product formula for absolutely convergent series and Proposition 3.14, we have that expg(λ;t)(s)exp g(λ;s)(x)= ∞  n=0 λngt,n(s) n! ∞  n=0 λngs,n(x) n! = ∞  n=0 n  k=0 λn−kgt,n−k(s) (n−k)! λkgs,k(x) k!= ∞  n=0 λngt,n(x) n! =exp g(λ;t)(x). Applying the last result, it can be proven that expg(λ; t)is Stieltjes-analytic on Ωt. Translating Proposition 5.5 to the classical case, exp(s−t)exp(x−s)=exp(t−x). Proposition 5.6. Let g:R →Rbe a derivator and fix some x0∈R. The series ∞  n=0 λngn(x) n!(5.7) 48 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ bn+m= m−1  k=0 λkbn+k+rnn!,λ k∈F,n≥0, bk=ck,c k∈F,k∈{0,...,m−1}, (5.18) and solutions of problem (5.17). As we did in Lemma 5.16, we will bound again these sequences, as it is needed for the convergence of the g-monomial series. Lemma 5.19. If {bn}n∈Nsolves problem (5.18), then there exists a constant M>0such that |bn+m|≤ n  k=0 MkRn−k for all n ≥0, where Rn=|rn|n!for all n ≥1and R0>0is some positive number. Proof. Let {bn}n∈Nbe the solution to problem (5.18). Take C=max{1, m−1 k=0 |λk|, m−1 k=0 |bk|} and define M=C2≥C. Since M≥1, the powers of Mform a nondecreasing sequence. We have that, |bm|≤ m−1  k=0 |λk||bk|+|r0|≤C2+|r0|=M+|r0|. Define R0=M+|r0|. Let us bound bm+1 to show how the process continues. Note that |bk|≤M+|r0|=R0 for k=1, ..., m. So |bm+1|≤ m−1  k=0 |λk||bk+1|+|r1|≤MR0+|r1|=MR0+R1. Now, we have that |bk|≤MR0+R1for k=2, ..., m +1, hence, we can bound the following coefficient and so on. Let us apply induction, let n ∈Nbe such that |bi+m|≤ i  k=0 MkRi−k for all i ≤n. By definition of Mand Rk, |bi|≤n k=0 MkRn−kfor all i ≤n. Then |bn+1+m|≤ m−1  k=0 |λk||bk+1|+|rn+1|(n+1)!≤Mn  k=0 MkRn−k+Rn+1 = n+1  k=0 MkRn+1−k. Theorem 5.20. Let {an}n∈Nbe the solution of problem (5.17). Assume f(x)= ∞  n=0 rngn(x) (5.19) converges absolutely on [x0, c]for some c >x 0. Suppose x0/∈N− gand c /∈Dg∪N+ g∪Cg, so the g-derivative is well defined at all points of [x0, c], then v(x)= ∞  n=0 angn(x) (5.20) converges absolutely on [x0, c], is such that v∈Cm g([x0, c], F)and solves problem (5.16). V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 49 Proof. Let {an}n∈Nbe the solution of problem (5.17). Thanks to Lemma 5.19, there exists M>0such that |an+m|≤ 1 (n+m)! n  k=0 MkRn−k, for all n ≥0, where Rn=|rn|n!for all n ≥1and R0>0is some positive number. Denote ˇrn=Rn n!for all n ≥0. We are going to prove that the g-monomial series the sequence {an}n∈Ndefines converges absolutely on [x0, c). For any x ∈[x0, c], from the product formula for absolutely convergent series, the following series converges: ∞  n=0 Mn|gn(x)| n!∞  n=0 ˇrn|gn(x)|= ∞  n=0 n  k=0 Mk k!ˇrn−k|gk(s)||gn−k(x)|<∞,(5.21) note that ˇrnand |rn|define g-monomial series that converge at the same points. Then, for any x ∈[x0, c], we have that ∞  n=0 |an+m|(n+m)! n!|gn(x)|≤ ∞  n=0 n  k=0 MkRn−k n!|gn(x)|= ∞  n=0 n  k=0 Mkˇrn−k(n−k)! n!|gn(x)| = ∞  n=0 n  k=0 1 n kMk k!ˇrn−k|gn(x)|≤ ∞  n=0 n  k=0 Mk k!ˇrn−k|gn(x)|<∞, (5.22) applying Proposition 3.7 and equation (5.21). Applying now Propositions 4.13 and 2.17 mconsecutive times, we have that vk g(x)= ∞  n=0 an+k (n+k)! n!gn(x) for all x ∈[x0, c]and k∈{0, ..., m}, with those series converging absolutely on [x0, c]. Hence v(k) g(x0) = akk! =ckfor all k∈{0, ..., m −1}. Besides, for all x ∈[x0, c], m−1  k=0 λkv(k) g(x)+f(x)= m−1  k=0 λk ∞  n=0 an+k (n+k)! n!gn(x)+ ∞  n=0 rngn(x) = ∞  n=0 m−1  k=0 λkan+k (n+k)! n!+rngn(x)= ∞  n=0 an+m (n+m)! n!gn(x)=v(m) g(x). Thus, vsolves problem (5.16).  Remark 5.21. We can ensure convergence of the g-monomial series (5.20)at points to the left of x0if the series (5.19)converges at points to the left of x0as well. Suppose the g-monomial series (5.19)converges absolutely on [c1, c2]for some c1∈(−∞, x0)and c2∈(x0, +∞). From Theorem 5.20, the series (5.20) converges absolutely on [x0, c2]. We have two cases: 1. Δg(x) =0for all x ∈[c1, x0). Hence gn(x) =g1(x)nand |gn−k(x)||gk(x)|=|gn(x)|for all x ∈[c2, x0]. Thus, equation (5.21) implies the absolute convergence of the g-monomial series (5.22) for all x ∈[c2, c1]. Therefore, the series (5.20)converges absolutely and solves problem (5.16)on [c2, c1]. 50 V. Cora et al. / J. Math. Anal. Appl. 526 (2023) 127259 2. There exists some x1∈[c1, x0)such that Δg(x1) =0. That means, thanks to Proposition 4.10, that there exists some constant M>0such that |rn|n! ≤Mnfor all n ≥0. Applying Lemma 5.19 we have that there exists another constant C>0such that |an|n! ≤Cnfor all n ≥0. Hence, there exists some t <x 0such that both g-monomial series (5.20)and(5.19)converge absolutely on (t, +∞). Thus, vis well defined and solves problem (5.16)on (t, +∞). Declaration of competing interest The authors declare no competing interests. Acknowledgments The authors were partially supported by Xunta de Galicia, project ED431C 2019/02, and by the Agencia Estatal de Investigación (AEI) of Spain and by the European Community fund FEDER under grant MTM2016-75140-P. References [1] K.B. Athreya, S.N. Lahiri, Measure Theory and Probability Theory, Springer-Verlag, 2006. [2] M. Baake, U. Schlaegel, The Peano-baker series, Proc. Steklov Inst. Math. 275 (1) (2011) 155–159. [3] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, MA, 2012. [4] F.J. 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