Generalized Functions Beyond Distributions

Ultrafunctions are a particular class of functions defined on a Non Archimedean field R^{*}\supset R. They have been introduced and studied in some previous works ([1],[2],[3]). In this paper we introduce a modified notion of ultrafunction and we discuss sistematically the properties that this modification allows. In particular, we will concentrated on the definition and the properties of the operators of derivation and integration of ultrafunctions.


Introduction
In some recent papers the notion of ultrafunction has been introduced and studied ( [1], [2], [3]).Ultrafunctions are a particular class of functions defined on a Non Archimedean field R * ⊃ R. We recall that a Non Archimedean field is an ordered field which contains infinite and infinitesimal numbers.In general, as we showed in our previous works, when working with ultrafunctions we associate to any continuous function f : R N → R an ultrafunction f : (R * ) N → R * which extends f ; more exactly, to any vector space of functions V (Ω) ⊆ L2 (Ω) ∩ C(Ω) we associate a space of ultrafunctions V (Ω).The spaces of ultrafunctions are much larger than the corrispective spaces of functions, and have much more "compactness": these two properties ensure that in the spaces of ultrafunctions we can find solutions to functional equations which do not have any solutions among the real functions or the distributions.
In [3] we studied the basic properties of ultrafunctions.One property that is missing, in general, is the "locality": local changes to an ultrafunction (namely, changing the value of an ultrafunction in a neighborhood of a point) affects the ultrafunction globally (namely, they may force to change the values of the ultrafunction in all the points).This problem is related to the properties of a particular basis of the spaces of ultrafunctions, called "Delta basis" (see [2], [3]).The elements of a Delta basis are called Delta ultrafunctions and, in some precise sense, they are an analogue of the Delta distributions.More precisely, given a point a ∈ R * , the Delta ultrafunction centered in a (denoted by δ a (x)) is the unique ultrafunction such that, for every ultrafunction u(x), we have * u(x)δ a (x)dx = u(a). 1t would be useful for applications to have an orthonormal Delta basis, namely a Delta basis {δ a (x)} a∈Σ such that, for every a, b ∈ Σ, * δ a (x)δ b (x)dx = δ a,b ; unfortunately, this seems to be impossible.
The main aim of this paper is to show how to modify the constructions exposed in [3] (that will be recalled) to avoid such unwanted issues.We will show how to construct spaces of ultrafunctions that have "good local properties" and that have Delta bases {δ a (x)} a∈Σ that are "almost orthogonal" where, by saying that a Delta basis is "almost orthogonal", we mean the following: for every a, b ∈ Σ, if |a − b| is not infinitesimal 2 then * δ a (x)δ b (x)dx = 0.
We will also discuss a few other properties of ultrafunctions that were missing in the previous approach but that hold in this new context.
The techniques on which the notion of ultrafunction is based are related to Non Archimedean Mathematics (NAM) and to Nonstandard Analysis (NSA).In particular, the most important notion that we use is that of Λ-limit (see [1], [2], [3]).In this paper this notion will be considered known; however, for sake of completeness, we will recall its basic properties in the Appendix.

Notations
If X is a set then • P (X) denotes the power set of X and P f in (X) denotes the family of finite subsets of X; • F (X, Y ) denotes the set of all functions from X to Y and F R N = F R N , R .
Let Ω be a subset of R N : then • C (Ω) denotes the set of continuous functions defined on Ω ⊂ R N ; • C 0 (Ω) denotes the set of continuous functions in C (Ω) having compact support in Ω; • C k (Ω) denotes the set of functions defined on Ω ⊂ R N which have continuous derivatives up to the order k; • C k 0 (Ω) denotes the set of functions in C k (Ω) having compact support; • C 1 ♯ (R) denotes the set of functions f of class C 1 (Ω) except than on a discrete set Γ ⊂ R and such that, for any γ ∈ Γ, the limits lim x→γ ± f (x) exist and are finite; • D (Ω) denotes the set of the infinitely differentiable functions with compact support defined on Ω ⊂ R N ; D ′ (Ω) denotes the topological dual of D (Ω), namely the set of distributions on Ω; x ∈ X means "for almost every x ∈ X";

Definition of Ultrafunctions
In this section we introduce a few Desideratum that will be used to introduce ultrafunctions in a slightly different way with respect to what we did in [2], [3].
The space of ultrafunctions will be denoted by F (R).With some abuse of notation we will call ultrafunction also the restriction of u to any internal subset of R * .
In particular, we have that so, being a Λ-limit of finite dimensional vector spaces, the vector space of ultrafunctions has hyperfinite dimension.Moreover, given any vector space of functions W ⊂ F (R), we can define the space of ultrafunctions generated by W as follows: Let us observe that where for every λ ∈ X we pose The space of ultrafunctions F (R) is too large for applications.We want to have a smaller space U(R) ⊂ F (R) which satisfies suitable properties for applications.We list the main properties that we would like to obtain for U(R).

Desideratum 1. There is an infinite number
Desideratum 1 states that the ultrafunctions have an uniform compact support and are bounded in R * .From these conditions it follows that, if u(x) ∈ U(R), then u(x) ∈ L p (R) * for every p; in particular, u(x) is summable and it is in L 2 (R) * .So U(R) ⊂ L 2 (R) * , and this allows to give to U(R) the euclidean structure and the norm induced by * , where This request, which may seem strange at first sight, will allow to associate to every point a ∈ [−β, β] a delta (or Dirac) ultrafunction centered in a, namely an ultrafunction δ a (x) such that, for every ultrafunction u(x), we have * u(x)δ a (x)dx = u(a).
Desideratum 3 is introduced for a few different reasons.First of all, it is important to have the characteristic functions of intervals even if, due to Desideratum 2, we will have to pay attenction in choosing the right definition of characteristic functions; moreover, it is important to have the extensions of C 1 functions in U(R) (one could ask this property for continuous function but, as we will show later, this request seems difficult to obtain if we want also the other Desideratums that we are presenting here).Finally, we will show that from Desideratum 3 it follows that the delta functions have compact support concentrated around their center: in fact we will show that, ∀a ∈ gal(0), supp (δ a ) ⊂ mon(a).
However it would be nice to have the previous property in the following more general fashion:

Our next desideratum is the following:
Desideratum 5.There exists a linear map (•) : Desideratum 5 substantially states that it is possible to define the projection of an L 1 loc (Ω) * function on U(R).In particular, this is useful to associate canonically an ultrafunction to every function f ∈ L 1 loc (Ω) since, in general, it will be false that f * ∈ U(R) (but when f * ∈ U(R) by Desideratum 5 we have f * = f ).Desideratum 6.There exists a map D : U(R) → U(R) such that Desideratum 6 simply states that it is possible to define a derivative on U(R) which satisfies a few expected properties.
In the next sections we show how to construct a space that satisfies all the Desideratum that we presented.

Construction of a canonical space of ultrafunctions
We want to consider a special subset of ultrafunctions.Let β be an infinite number; we set where γ 0 = −β; γ ℓ = β and, for j = 0, 1, ..., ℓ − 1, we require that where η is an infinitesimal number.Moreover, it is useful to assume that R ⊆ Γ.For j = 0, 1, ..., ℓ − 1, we set For every a, b ∈ Γ we denote by χ [a,b] (x) the characteristic function of [a, b] defined in a slightly different way: For every j = 0, 1, ..., ℓ − 1, we set The set of functions will be referred to as the set of grid functions.
Definition 2. We denote by U(R) the space of ultrafunctions which can be represented as follows: where, ∀j ∈ J, v j (x) ∈ C 1 (R).We will refer to U(R) as the canonical space of ultrafunctions.
Proposition 4. U(R) is an hyperfinite dimensional vector space, and Proof.
is a set of generators for U(R), and its cardinality is * , it can be equipped with the following scalar product where * is the natural extension of the Lebesgue integral considered as a functional : The norm of a (canonical) ultrafunction will be given by .

Canonical ultrafunctions have a few interesting properties:
Proposition 5.The following properties hold: are well defined and we set 4. if u ∈ U(R) then for j = 0 the limit is well defined and for j = l the limit is well defined.
5. if, for every j = 0, ..., ℓ we set then, for k = j, V (I j ) and V (I k ) are orthogonal; 6. U(R) can be splitted in orthogonal spaces as follows: 2) It follows by (1). 3 and and these limits exist because u j−1 , u j are continuous on I j−1 , I j respectively.
4) The same as in 1).5) This is immediate since, if j = k, if u ∈ V (I j ) and v ∈ V (I k ) then the supports of u and v are disjoint.

Delta and Sigma basis
Following the approach presented in [3], in this section we introduce two particular bases for U(R) and we study their main properties.We start by defining the Delta ultrafunctions.In order to do this, it is useful to observe that the value of an ultrafunction u for γ j , j = 1, ..., ℓ − 1, can be defined as follows: where u(x + ), u(x − ) are defined by (2).The fact that this definition makes sense follows by points 3) and 4) in Proposition 5. Moreover we pose These observations are relevant in the following definition: δ q (x) is called the Delta (or Dirac) ultrafunction concentrated in q.
For every i = j, for every v ∈ V (I i ) we have If q = γ 0 we pose and if q = γ ℓ we pose The verification that these definitions are well posed is equal to the one carried out for q ∈ I j .
If q = γ j , j = 0, ℓ we set The Delta function in q is unique: if f q (x) is another Delta ultrafunction centered in q then for every y ∈ [−β, β] we have: Let us observe that, as the previous proof shows, in every point γ j of the grid Γ, with the exceptions of −β, β, it is possible to define three delta functions centered in γ j , namely δ − γ j (x), δ + γ j (x) and δ γ j (x), which satisfy the following properties: for every ∀v ∈ U(R), we have Moreover, it is immediate to prove that the conditions in (4) charatecterize uniquely the functions δ − γ j (x), δ + γ j (x) and δ γ j (x).So we will consider (4) as a definition for δ − γ j (x), δ + γ j (x) and δ γ j (x).
whose elements are Delta ultrafunctions.Its dual basis {σ a (x)} a∈Σ is called Sigma-basis.We recall that, by definition of dual basis, for every a, b ∈ Σ the equation *
Proof. 1) It is an immediate consequence of the definition of dual basis.
2) Since {δ q (x)} q∈Σ is the dual basis of {σ q (x)} q∈Σ we have that 3) It follows directly from the previous point.4) If follows directly by equation ( 5). 5) Given any point q ∈ (−β, β) clearly there is a Delta-basis {δ a (x)} a∈Σ with q ∈ Σ.Then σ q (x) can be defined by mean of the basis {δ a (x)} a∈Σ .We have to prove that, given another Delta basis {δ a (x)} a∈Σ ′ with q ∈ Σ ′ , the corresponding σ ′ q (x) is equal to σ q (x).Using (2), with u(x) = σ ′ q (x), we have that Then, by ( 4), it follows that σ ′ q (x) = σ q (x).6) As we proved in Theorem 8, if q ∈ I j then δ q is an element of V (I j ), so supp(δ q (x)) ⊂ I j .Now δ q ∈ V (I j ), so there is a corrispective function σ q ∈ V (I j ) which is the sigma function centered in q.If we extend this function to [−β, β] by posing σ q (x) = 0 for x / ∈ I j we obtain, by uniqueness, exactly the sigma function centered in q in U(R).And supp(σ q (x)) ⊂ I j .
9) It is a straightforward consequence of the points 6 and 7, since for every j ∈ J we have I j ∪ I j+1 ⊂ mon (q) .

Canonical extension of functions
We start by defining a map which will be very useful in the extension of functions.
Remark 13.Notice that, if u ∈ L 2 (R) * , then u = P V u where is the orthogonal projection.
The following theorem shows that u is well defined and unique.
In particular, if f ∈ L 1 loc (R), the function f * is well defined.From now on we will simplify the notation just writing f .Example 15.Take |x| −1/2 ∈ L 1 loc (−1, 1), then makes sense for every x ∈ R * ; in particular and it is easy to check that this is an infinite number.Notice that the ultra- since the latter is not defined for x = 0 (and they also differ for |x| > β).
Now we want to show some interesting relations between f and f * .More precisely we are interested in the following question.
Take f ∈ L 1 loc (R) and Ω ⊂ R ; which are the conditions that ensure the following: , f and f * are not defined pointwise and hence the above equality must be intended for almost every x.

Corollary 17.
Let Ω ⊂ R be an open set and let f, g ∈ L 1 loc (R); then Proof.This follows immediatly by applying the previous theorem to the function h(x) = f (x) − g(x), since the operation f → f is linear.
Proof.Let {δ a (x)} a∈Σ be a Delta basis, let y ∈ Ω * and let y ∈ I j .Since, by (11), for every q ∈ Σ with q / ∈ I j σ q (y) = 0, by (7) we deduce that Now let g j (x) be the function such that ) is an ultrafunction.By construction, we have that g j (y) = f (y) since, by (6), But, by definition, g j (y) = f * (y); hence we deduce that f * (y) = f (y).
Example 19.If f (x) = 1, then By Theorem 18 and the above example, we get: will be called the canonical extension of f (x).With some abuse of notation, f (x) will be called the "canonical extension of f (x)" even when f (x) ∈ L 1 loc (R).Example 22.For a fixed k ∈ R, the function e ikx defines a unique ultrafunction e ikx .Notice that e ikx is different from the natural extension of e ikx even if ∀x ∈ gal (0) , e ikx = e ikx .

Derivative
Definition 23.For every ultrafunction u ∈ U(R), the derivative Du(x) of u(x) is the ultrafunction defined by the following formula: where P U u ′ denotes the orthogonal projection of u ′ on U(R) w.r.t. the L 2 scalar product and, for every j = 1, ..., l − 1, Theorem 24.For every u, v ∈ U(R) the following equality holds: Proof.We have: Now let us compute the two terms of the sum separately; the first one: The second one: Example 1: By (10) we have that  9)

Definite integral
Since every ultrafunction is an internal function, the definite integral is well defined: This observation is important to prove the following theorem: A question that arises is: does it hold, for ultrafunctions, some kind of "rule of integration by parts for continuous functions", at least for the points in Γ?
The answer is no, as a simple computation shows.Nevertheless, we have the following: Proof.By (11), since u, v ∈ U(R)∩C 1 (R) * then Du = u ′ and Dv = v ′ .Moreover, since U(R) = l−1 j=0 I j , if for every j = 0, ..., l − 1 we denote by P j the orthogonal projection on I j we have Now, if m = n + 1, since u and v are continuous we have In the general case, and since u, v are continuous we have The previous proposition is, in general, false if at least one between u, v is not in C 1 (R) * .The reason is that, by definition, the derivative has the following expression: and the presence of ℓ−1 j=1 △u(γ j )δ γ j (x) is what makes (14) to be false.Just for sake of completeness, we now show how to obtain a relaxed version of ( 14) by considering a different possible notion of derivative on U(R).The relaxed version of ( 14) is the following: since the functions in U(R) are piecewise C 1 functions, does it hold, for ultrafunctions, an analogue of the rule of integration by parts for piecewise C 1 functions?Namely, is it true that, if u, v ∈ U(R) and γ n < γ m ∈ Γ, then With the operator D the answer is no.But there is a different linear operator that actually satisfies (15) : Definition 29.We denote by D 2 u(x) the linear operator such that, for every u ∈ U(R) , we have Since U(R) = l−1 j=0 I j , if we denote by P j the orthogonal projection on I j , we have Moreover we have that, if u(x) is continuous in γ j , γ j+1 , then This new linear operator is what we need to obtain the generalization to U(R) of the rule of integration by parts for piecewise continuous functions: Theorem 30.(Integration by parts for piecewise C 1 functions) For every u, v ∈ U(R) and γ n < γ m ∈ Γ we have In the general case we have In particular, since D 2 1 = 0, it is immediate to prove that the following holds: Corollary 31.(Fundamental Theorem of Calculus for piecewise continuous functions) For every u ∈ U(R) and γ n < γ m ∈ Γ we have Of course, the derivative D 2 has also many drawbacks, e.g. for every grid function g we have D 2 (g) = 0.So in the following we will only consider the derivative D.

Ultrafunctions and distributions
In this section we briefly explain how to associate an ultrafunction to every distribution T ∈ C −∞ (R), where Note that, by definition, if T ∈ C −∞ (R) then there exists a natural number k and a function f ∈ C 1 (R) such that: So it is natural to introduce the following definition: Definition 32.Given a distribution T ∈ C −∞ (R) , let k be the minimum natural number such that there exists f ∈ C 1 (R) with T = ∂ k f.We denote by T the ultrafunction T (x) = D k f * .
T will be called the ultrafunction associated with the distribution T .
Proof.Let us suppose that T = ∂ k f, where k, f are given as in Definition 32.Then, by (10), since ϕ * (β) = ϕ * (−β) = 0, we have that * In the forthcoming paper [4] we will show that, actually, it is possible to define an embedding of the whole space of distributions in a particular space of ultrafunctions; this definition will be used to construct a particular algebra, related to ultrafunctions, in which the distributions can be embedded.

APPENDIX -Λ-theory
In this section we present the basic notions of Non Archimedean Mathematics and of Nonstandard Analysis following a method inspired by [6] (see also [1] and [2]).

Non Archimedean Fields
Here, we recall the basic definitions and facts regarding Non Archimedean fields.In the following, K will denote an ordered field.We recall that such a field contains (a copy of) the rational numbers.Its elements will be called numbers.Definition 34.Let K be an ordered field.Let ξ ∈ K.We say that: It is easily seen that all infinitesimal are finite, that the inverse of an infinite number is a nonzero infinitesimal number, and that the inverse of a nonzero infinitesimal number is infinite.Definition 36.A superreal field is an ordered field K that properly extends R.
It is easy to show, due to the completeness of R, that there are nonzero infinitesimal numbers and infinite numbers in any superreal field.Infinitesimal numbers can be used to formalize a new notion of "closeness": Definition 37. We say that two numbers ξ, ζ ∈ K are infinitely close if ξ − ζ is infinitesimal.In this case, we write ξ ∼ ζ.
Clearly, the relation "∼" of infinite closeness is an equivalence relation.
Theorem 38.If K is a superreal field, every finite number ξ ∈ K is infinitely close to a unique real number r ∼ ξ, called the shadow or the standard part of ξ.
Given a finite number ξ, we denote its shadow as sh(ξ), and we put sh(ξ) = +∞ (sh(ξ) = −∞) if ξ ∈ K is a positive (negative) infinite number.Definition 39.Let K be a superreal field, and ξ ∈ K a number.The monad of ξ is the set of all numbers that are infinitely close to it: and the galaxy of ξ is the set of all numbers that are finitely close to it: By definition, it follows that the set of infinitesimal numbers is mon(0) and that the set of finite numbers is gal(0).

The Λ-limit
In this section we will introduce a superreal field K and we will analyze its main properties by mean of the Λ-theory (see also [1], [2]).
We set X = P f in (F(R, R)); we will refer to X as the "parameter space".Clearly (X, ⊂) is a directed set and, as usual, a function ϕ : X → E will be called net (with values in E).
We present axiomatically the notion of Λ-limit: Axioms of the Λ-limit • (Λ-1) Existence Axiom.There is a superreal field K ⊃ R such that every net ϕ : X → R has a unique limit L ∈ K (called the "Λ-limit" of ϕ.)The Λ-limit of ϕ will be denoted as Moreover we assume that every ξ ∈ K is the Λ-limit of some real function ϕ : X → R.
Theorem 40 is proved in [1] and in [3].The notion of Λ-limit can be extended to sets and functions in the following way: Definition 41.Let E λ , λ ∈ X, be a family of sets.We define A set which is a Λ-limit is called internal.In particular, if ∀λ ∈ X, E λ = E, we set lim λ↑Λ E λ = E * , namely A function which is a Λ-limit is called internal.In particular, if ∀λ ∈ X, Notice that, while the Λ-limit of a constant sequence of numbers gives this number itself, the Λ-limit of a constant sequence of sets is a larger set and the Λ-limit of a constant sequence of functions is an extension of this function.
In a similar way it is possible to extend operator and functionals.Finally, the Λ-limits satisfy the following important Theorem:

Hyperfinite sets and hyperfinite sums
Definition 44.An internal set is called hyperfinite if it is the Λ-limit of a net ϕ : X → X.
Definition 45.Given any set E ∈ U, the hyperfinite extension of E is defined as follows: All the internal finite sets are hyperfinite, but there are hyperfinite sets which are not finite.For example the set is not finite.The hyperfinite sets are very important since they inherit many properties of finite sets via Leibnitz principle.For example, R • has the maximum and the minimum and every internal function f : R • → R * has the maximum and the minimum as well.

f
* : E * → R * is called the natural extension of f.