On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field

Abstract Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.


Introduction
Let N be a non-Archimedean-ordered field extension of R that is real closed and complete in the order topology and whose Hahn group S N is Archimedean, i.e. (isomorphic to) a subgroup of R. Recall that S N is the set of equivalence classes under the relation ∼ defined on N * := N \{0} as follows: For x, y ∈ N * , we say that x is of the same order as y and write x ∼ y if there exist n, m ∈ N such that n|x| > |y| and m|y| > |x|, where | · | denotes the ordinary absolute value on N : |x| = max {x, −x}. S N is naturally endowed with an addition via [x] + [y] = [x · y] and an order via [x] < [y] if |y| |x| (which means n|y| < |x| for all n ∈ N), both of which are readily checked to be well defined. It follows that (S N , +, <) is an ordered group, often referred to as the Hahn group or skeleton group, whose neutral element is [1], the class of 1.
The theorem of Hahn [3] provides a complete classification of non-Archimedean-ordered field extensions of R in terms of their skeleton groups. In fact, invoking the axiom of choice, it is shown that the elements of our field N can be written as (generalized) formal power series (also called Hahn series) over its skeleton group S N with real coefficients, and the set of appearing exponents forms a well-ordered subset of S N . That is, for all x ∈ N , we have that x = q∈SN a q d q ; with a q ∈ R for all q, d a positive infinitely small element of N , and the support of x, given by supp(x) := {q ∈ S N : a q = 0}, forming a well-ordered subset of S N .
We define for x = 0 in N , λ(x) = min(supp(x)), which exists since supp(x) is well ordered. Moreover, we set λ(0) = ∞. Given a non-zero x = q∈supp(x) a q d q , then x > 0 if and only if a λ(x) > 0.
The smallest such field N is the Levi-Civita field R, first introduced in [5, 6]. In this case, S R = Q, and for any element x ∈ R, supp(x) is a left-finite subset of Q, i.e. below any rational bound r there are only finitely many exponents in the Hahn representation of x. The Levi-Civita field R is of particular interest because of its practical usefulness. Since the supports of the elements of R are left-finite, it is possible to represent these numbers on a computer. Having infinitely small numbers allows for many computational applications; one such application is the computation of derivatives of real functions representable on a computer [13,15], where both the accuracy of formula manipulators and the speed of classical numerical methods are achieved. For a review of the Levi-Civita field R, see [11] and references therein.
In the wider context of valuation theory, it is interesting to note that the topology induced by the order on N is the same as the valuation topology τ v introduced via the non-Archimedean (ultrametric) valuation | · | v : N → R, given by It follows, therefore, that the field N is just a special case of the class of fields discussed in [9]. For a general overview of the algebraic properties of formal power series fields, we refer to the comprehensive overview by Ribenboim [8], and for an overview of the related valuation theory the book by Krull [4]. A thorough and complete treatment of ordered structures can also be found in [7]. A more comprehensive survey of all non-Archimedean fields can be found in [1].

Weak local uniform differentiability and review of recent results
Because of the total disconnectedness of the field N in the order topology, the standard theorems of real calculus like the intermediate value theorem, the inverse function theorem, the mean value theorem, the implicit function theorem and Taylor's theorem require stronger smoothness criteria of the functions involved in order for the theorems to hold. In this section, we will present one such criterion: the so-called weak local uniform differentiability, we will review recent work based on that smoothness criterion and then present new results.
In [2], we focus our attention on N -valued functions of one variable. We study the properties of weakly locally uniformly differentiable (WLUD) functions at a point x 0 ∈ N or on an open subset A of N . In particular, we show that WLUD functions are C 1 , they include all polynomial functions, and they are closed under addition, multiplication and composition. Then, we generalize the definition of weak local uniform differentiability to any order. In particular, we study the properties of WLUD 2 functions at a point x 0 ∈ N or on an open subset A of N ; and we show that WLUD 2 functions are C 2 , they include all polynomial functions, and they are closed under addition, multiplication and composition. Finally, we formulate and prove an inverse function theorem as well as a local intermediate value theorem and a local mean value theorem for these functions.
Here, we only recall the main definitions and results (without proofs) in [2] and refer the reader to that paper for the details.
We extend the WLUD concept to higher orders of differentiability and we define WLUD k as follows.

Theorem 3 (Local Mean Value Theorem). Let
In [12], we formulate and prove a Taylor theorem with remainder for WLUD k functions from N to N . Then, we extend the concept of WLUD to functions from N n to N m with m, n ∈ N and study the properties of those functions as we did for functions from N to N . Then, we formulate and prove the inverse function theorem for WLUD functions from N n to N n and the implicit function theorem for WLUD functions from N n to N m with m < n in N.
As in the real case, the proof of Taylor's theorem with remainder uses the mean value theorem. However, in the non-Archimedean setting, stronger conditions on the function are needed than in the real case for the formulation of the theorem.

Theorem 4 (Taylor's Theorem with Remainder). Let
Before we define weak local uniform differentiability for functions from N n to N m and then state the inverse function theorem and the implicit function theorem, we introduce the following notation. Notation 1. Let A ⊂ N n be open, let x 0 ∈ A be given, and let f : A → N m be such that all the first-order partial derivatives of f at x 0 exist. Then, Df (x 0 ) denotes the linear map from N n to N m defined by the m × n Jacobian matrix of f at x 0 : ⎛

Definition 3 (WLUD). Let
Moreover, we say that f is WLUD on A if f is WLUD at every point in A.
We show in [12] that if f is WLUD at x 0 (respectively on A) then f is C 1 at x 0 (respectively, on A). Thus, the class of WLUD functions at a point x 0 (respectively, on an open set A) is a subset of the class of C 1 functions at x 0 (respectively, on A). However, this is still large enough to include all polynomial functions. We also show in [12] that if f , g are WLUD at x 0 (respectively, on A) and if α ∈ N then f + αg and f · g are WLUD at x 0 (respectively, on A). Moreover, we show that if f : A → N m is WLUD at x 0 ∈ A (respectively, on A) and if g : (1) f | Ω is one-to-one; (

2) f (Ω) is open and
As in the real case, the inverse function theorem is used to prove the implicit function theorem. But before we state the implicit function theorem, we introduce the following notation.

New results
This paper is a continuation of the work done in [2,12]. In the following section, we will generalize in Definition 5 and Definition 6 the concepts of WLUD k and WLUD ∞ to functions from N n to N ; and we will formulate (in Theorem 8 and Theorem 9 and their proofs) conditions under which a WLUD ∞ N -valued function at a point x 0 ∈ N or a WLUD ∞ N -valued function at a point x 0 ∈ N n will be analytic at that point. A ⊆ N be open, let x 0 ∈ A, and let

Theorem 7. Let
Then, there exists a neighbourhood U of x 0 in A such that, for any x, y ∈ U , we have that That is, the Taylor series ; and hence f is Proof. First, we note that lim sup Thus, the Taylor series R := 1 lim sup Then, λ 0 ∈ R and λ 0 < ∞. Then, for all x ∈ N satisfying λ(x − x 0 ) > λ 0 , we have that ; that is, it converges with respect to both the ultrametric absolute value | · | v and the ordinary absolute value | · |. For all k ∈ N, we have that ( Let δ > 0 in N be such that λ(δ) > max{λ 0 , t, 0}; this is possible since max{λ 0 , t, 0} < ∞. It follows from (2) that λ(δ) > λ(δ k ) and hence 0 < δ δ k for all k > N. Thus, Moreover, for all First, we show that Let x ∈ U be given. Taking the limit in (3) as k → ∞, we get: from which we obtain Since the convergence of the Taylor series above is in the order (valuation) topology, we will show that the derivatives of f at x to any order are obtained by differentiating the power series in Equation (4) term by term. That is, for all j ∈ N, First, note that since λ(l(l − 1) . . . (l − j + 1)) = 0, it follows that ∞ l=j l(l − 1) . . .
Using induction on j, it suffices to show that Let h ∈ N be such that x + h ∈ U . We will show that Thus, Now, let y ∈ U be given. Then Since convergence in the order topology (valuation topology) entails absolute convergence, we can interchange the order of the summations in the last equality [10,14]. We get: where we made use of Equation (5)

Moreover, we say that f is WLUD on A if f is WLUD at every point in A.
Using Definition 2 and Definition 4, the natural way to define k times weak local uniform differentiability (WLUD k ) at a point x 0 or on an open subset A of N n is as follows.
Definition 5. Let A ⊂ N n be open, let f : A → N , and let x 0 ∈ A be given. Then, we say that f is WLUD k at x 0 if f is k-times differentiable in a neighbourhood Ω of x 0 in A and if for every > 0 in N there exists δ > 0 in N such that B δ (x 0 ) ⊂ Ω, and for all ξ, η ∈ B δ (x 0 ) we have that Moreover, we say that f is WLUD k on A if f is WLUD k at every point in A. Definition 6. Let A ⊂ N n be open, let f : A → N , and let x 0 ∈ A be given. Then, we say that f is WLUD ∞ at x 0 if f is WLUD k at x 0 for every k ∈ N. Moreover, we say that f is WLUD ∞ on A if f is WLUD ∞ at every point in A. Now, we are ready to state and prove the analogue of Theorem 8 for functions of n variables. A ⊆ N n be open, let x 0 ∈ A, and let

Theorem 8. Let
Then, there exists a neighbourhood U of x 0 in A such that, for any η ∈ U, we have that