Basic properties of ultrafunctions
aa r X i v : . [ m a t h . F A ] F e b Basic properties of ultrafunctions
Vieri Benci ∗ Lorenzo Luperi Baglini † October 17, 2018
Dedicated to Bernard Ruf in occasion of his 60th birthday.
Abstract
Ultrafunctions are a particular class of functions defined on a non-Archimedean field R ∗ ⊃ R . They provide generalized solutions to func-tional equations which do not have any solutions among the real functionsor the distributions. In this paper we analyze sistematically some basicproperties of the spaces of ultrafunctions. Mathematics subject classification : 26E30, 26E35, 46F30.
Keywords . Ultrafunctions, Delta function, distributions, Non ArchimedeanMathematics, Non Standard Analysis.
Contents Λ -theory 3 ∗ Dipartimento di Matematica, Universit`a degli Studi di Pisa, Via F. Buonarroti 1/c, 56127Pisa, ITALY and Department of Mathematics, College of Science, King Saud University,Riyadh, 11451, SAUDI ARABIA. e-mail: [email protected] † Dipartimento di Matematica, Universit`a degli Studi di Pisa, Via F. Buonarroti 1/c, 56127Pisa, ITALY, e-mail: [email protected] Operations with ultrafunctions 18
In some recent papers the notion of ultrafunction has been introduced ([1], [2]).Ultrafunctions are a particular class of functions defined on a non-Archimedeanfield R ∗ ⊃ R . We recall that a non-Archimedean field is an ordered field whichcontain infinite and infinitesimal numbers.To any continuous function f : R N → R we associate in a canonical wayan ultrafunction e f : ( R ∗ ) N → R ∗ which extends f ; more exactly, to any func-tional vector space V (Ω) ⊆ L (Ω) ∩ C (Ω) , we associate a space of ultrafunctions e V (Ω) . The ultrafunctions are much more than the functions and among themwe can find solutions of functional equations which do not have any solutionsamong the real functions or the distributions.A typical example of this situation is analyzed in [2] where a simple Physicalmodel is studied. In this problem there is a material point interacting witha field and, as it usually happens, the energy is infinite. Therefore the needto use infinite numbers arises naturally. Other situations in which infinite andinfinitesimal numbers appear in a natural way are studied in [5] and in [6].In this paper we analyze systematically some basic properties of the spacesof ultrafunctions e V (Ω). In particular we will show that: • to any measurable function f we can associate an unique ultrafunction e f such that f ( x ) = e f ( x ) if f is continuous in a neighborhood of x ; • to every distribution T we can associate an ultrafunction e T ( x ) such that ∀ ϕ ∈ D , h T, ϕ i = R ∗ e T ( x ) e ϕ ( x ) dx where R ∗ is a suitable extension of theintegral to the ultrafunctions; • the vector space of ultrafunctions e V (Ω) is hyperfinite, namely it sharesmany properties of finite vector spaces (see Sec. 2.4); • the vector space of ultafunctions e V (Ω) has an hyperfinite basis { δ a ( x ) } a ∈ Σ where δ a is the ”Dirac ultrafunction in a ” (see Def. 18) and Σ ⊂ ( R ∗ ) N is a suitable set; • any ultrafunction u can be represented as follows: u ( x ) = X q ∈ Σ u ( q ) σ q ( x ) , where { σ a ( x ) } a ∈ Σ is the dual basis of { δ a ( x ) } a ∈ Σ ;2 any operator F : V (Ω) → D ′ (Ω) , can be extended to an operator e F : e V (Ω) → e V (Ω) ;the extension of the derivative and the Fourier transform will be analyzedin some detail.The techniques on which the notion of ultrafunction is based are related toNon Archimedean Mathematics (NAM) and to Nonstandard Analysis (NSA).The first section of this paper is devoted to a relatively elementary presentationof the basic notions of NAM and NSA inspired by [3] and [4]. Some technicalitieshave been avoided by presenting the matter in an axiomatic way. Of course, it isnecessary to prove the consistency of the axioms. This is done in the appendix;however in the appendix we have assumed the reader to be familiar with NSA. Let Ω be a subset of R N : then • C (Ω) denotes the set of continuous functions defined on Ω ⊂ R N ; • C (Ω) denotes the set of continuous functions in C (Ω) having compactsupport in Ω; • C k (Ω) denotes the set of functions defined on Ω ⊂ R N which have contin-uous derivatives up to the order k ; • D (Ω) denotes the set of the infinitely differentiable functions with compactsupport defined on Ω ⊂ R N ; D ′ (Ω) denotes the topological dual of D (Ω),namely the set of distributions on Ω; • H ,p (Ω) is the usual Sobolev space defined as the set of functions in L p (Ω)such that ∇ u ∈ L p (Ω) N ; • H (Ω) = H , (Ω) • for any ξ ∈ (cid:0) R N (cid:1) ∗ , ρ ∈ R ∗ , we set B ρ ( ξ ) = n x ∈ (cid:0) R N (cid:1) ∗ : | x − ξ | < ρ o ; • supp ( f ) = { x ∈ R N : f ( x ) = 0 } ; • mon ( x ) = { y ∈ R N : x ∼ y } ; • gal ( x ) = { y ∈ R N : x ∼ f y } . Λ -theory In this section we present the basic notions of Non Archimedean Mathematicsand of Nonstandard Analysis following a method inspired by [3] (see also [1]and [2]). 3 .1 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 fieldcontains (a copy of) the rational numbers. Its elements will be called numbers. Definition 1
Let K be an ordered field. Let ξ ∈ K . We say that: • ξ is infinitesimal if, for all positive n ∈ N , | ξ | < n ; • ξ is finite if there exists n ∈ N such as | ξ | < n ; • ξ is infinite if, for all n ∈ N , | ξ | > n (equivalently, if ξ is not finite). Definition 2
An ordered field K is called Non-Archimedean if it contains aninfinitesimal ξ = 0 . It’s easily seen that all infinitesimal are finite, that the inverse of an infinitenumber is a nonzero infinitesimal number, and that the inverse of a nonzeroinfinitesimal number is infinite.
Definition 3
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 nonzeroinfinitesimal numbers and infinite numbers in any superreal field. Infinitesimalnumbers can be used to formalize a new notion of ”closeness”: Definition 4
We say that two numbers ξ, ζ ∈ K are infinitely close if ξ − ζ isinfinitesimal. In this case, we write ξ ∼ ζ . Clearly, the relation ” ∼ ” of infinite closeness is an equivalence relation. Theorem 5 If K is a superreal field, every finite number ξ ∈ K is infinitelyclose 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 6
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: mon ( ξ ) = { ζ ∈ K : ξ ∼ ζ } , and the galaxy of ξ is the set of all numbers that are finitely close to it: gal ( ξ ) = { ζ ∈ K : ξ − ζ is finite } By definition, it follows that the set of infinitesimal numbers is mon (0) andthat the set of finite numbers is gal (0).4 .2 The Λ -limit In this section we will introduce a superreal field K and we will analyze its mainproperties by mean of the Λ-theory (see also [1], [2]). U will denote our ”mathematical universe”. For our applications a goodchoice of U is given by the superstructure on R : U = ∞ [ n =0 U n where U n is defined by induction as follows: U = R ; U n +1 = U n ∪ P ( U n ) . Here P ( E ) denotes the power set of E. Identifying the couples with the Ku-ratowski pairs and the functions and the relations with their graphs, it followsthat U contains almost every usual mathematical object. Given the universe U ,we denote by F the family of finite subsets of U . Clearly ( F , ⊂ ) is a directedset and, as usual, a function ϕ : F → E will be called net (with values in E ).We present axiomatically the notion of Λ-limit: Axioms of the Λ -limit • ( Λ- Existence Axiom.
There is a superreal field K ⊃ R such that everynet ϕ : F → R has a unique limit L ∈ K ( called the ”Λ-limit” of ϕ. ) The Λ- limit of ϕ will be denoted as L = lim λ ↑ U ϕ ( λ ) . Moreover we assume that every ξ ∈ K is the Λ- limit of some real function ϕ : F → R . • (Λ-2) Real numbers axiom . If ϕ ( λ ) is eventually constant , namely ∃ λ ∈ F , r ∈ R such that ∀ λ ⊃ λ , ϕ ( λ ) = r, then lim λ ↑ U ϕ ( λ ) = r. • (Λ-3) Sum and product Axiom . For all ϕ, ψ : F → R : lim λ ↑ U ϕ ( λ ) + lim λ ↑ U ψ ( λ ) = lim λ ↑ U ( ϕ ( λ ) + ψ ( λ )) ;lim λ ↑ U ϕ ( λ ) · lim λ ↑ U ψ ( λ ) = lim λ ↑ U ( ϕ ( λ ) · ψ ( λ )) . Theorem 7
The set of axioms { ( Λ -1) , ( Λ -2),( Λ -3) } is consistent. U (a net ϕ : F → U is called bounded if there exists n such that ∀ λ ∈ F , ϕ ( λ ) ∈ U n ). To this aim, consider a net ϕ : F → U n . (1)We will define lim λ ↑ U ϕ ( λ ) by induction on n . For n = 0 , lim λ ↑ U ϕ ( λ ) is defined bythe axioms ( Λ- (Λ-2),(Λ-3); so by induction we may assume that the limit isdefined for n − λ ↑ U ϕ ( λ ) = (cid:26) lim λ ↑ U ψ ( λ ) | ψ : F → U n − and ∀ λ ∈ F , ψ ( λ ) ∈ ϕ ( λ ) (cid:27) . Definition 8
A mathematical entity (number, set, function or relation) whichis the Λ -limit of a net is called internal . Definition 9
The natural extension of a set E ⊂ R is given by E ∗ := lim λ ↑ U c E ( λ ) = (cid:26) lim λ ↑ U ψ ( λ ) | ψ ( λ ) ∈ E (cid:27) where c E ( λ ) is the net identically equal to E . This definition, combined with axiom (Λ-1), entails that K = R ∗ . In this context a function f can be identified with its graph; then the naturalextension of a function is well defined. Moreover we have the following result: Theorem 10
The natural extension of a function f : E → F is a function f ∗ : E ∗ → F ∗ and for every net ϕ : F ∩ P ( E ) → E, and every function f : E → F , we havethat lim λ ↑ U f ( ϕ ( λ )) = f ∗ (cid:18) lim λ ↑ U ϕ ( λ ) (cid:19) . When dealing with functions, sometimes the ” ∗ ” will be omitted if the do-main of the function is clear from the context. For example, if η ∈ R ∗ is aninfinitesimal, then clearly e η denotes exp ∗ ( η ) . The following theorem is a fundamental tool in using the Λ-limit:6 heorem 11 (Leibnitz Principle)
Let R be a relation in U n for some n ≥ and let ϕ , ψ : F → U n . If ∀ λ ∈ F , ϕ ( λ ) R ψ ( λ ) then (cid:18) lim λ ↑ U ϕ ( λ ) (cid:19) R ∗ (cid:18) lim λ ↑ U ψ ( λ ) (cid:19) . When R is ∈ or = we will not use the symbol ∗ to denote their extensions,since their meaning is unaltered in R ∗ . Definition 12
An internal set is called hyperfinite if it is the Λ -limit of a net ϕ : F → F . Definition 13
Given any set E ∈ U , the hyperfinite extension of E is definedas follows: E ◦ := lim λ ↑ U ( E ∩ λ ) . All the internal finite sets are hyperfinite, but there are hyperfinite sets whichare not finite. For example the set R ◦ := lim λ ↑ U ( R ∩ λ )is not finite. The hyperfinite sets are very important since they inherit manyproperties of finite sets via Leibnitz principle. For example, R ◦ has the maxi-mum and the minimum and every internal function f : R ◦ → R ∗ has the maximum and the minimum as well.Also, it is possible to add the elements of an hyperfinite set of numbers orvectors as follows: let A := lim λ ↑ U A λ be an hyperfinite set; then the hyperfinite sum is defined in the following way: X a ∈ A a = lim λ ↑ U X a ∈ A λ a. In particular, if A λ = (cid:8) a ( λ ) , ..., a β ( λ ) ( λ ) (cid:9) with β ( λ ) ∈ N , then setting β = lim λ ↑ U β ( λ ) ∈ N ∗ we use the notation β X j =1 a j = lim λ ↑ U β ( λ ) X j =1 a j ( λ ) . .5 Qualified sets When we have a net ϕ : Q → U n , where Q ⊂ F , we can define the Λ-limit of ϕ by posing lim λ ∈ Q ϕ ( λ ) = lim λ ↑ U e ϕ ( λ )where e ϕ ( λ ) = (cid:26) ϕ ( λ ) for λ ∈ Q ∅ for λ / ∈ Q As one can expect, if two nets ϕ, ψ are equal on a ”large” or a ”qualified” subsetof F then they share the same Λ-limit. The notion of ”qualified” subset of F can be precisely defined as follows: Definition 14
We say that a set Q ⊂ F is qualified if for every bounded net ϕ we have that lim λ ↑ U ϕ ( λ ) = lim λ ∈ Q ϕ ( λ ) . By the above definition, we have that the Λ-limit of a net ϕ depends onlyon the values that ϕ takes on a qualified set (it is in this sense that we couldimagine Q to be ”large”). It is easy to see that (nontrivial) qualified sets exist.For example by (Λ-2) we deduce that, for every λ ∈ F , the set Q ( λ ) := { λ ∈ F | λ ⊆ λ } is qualified. In this paper, we will use the notion of qualified set via the followingTheorem: Theorem 15
Let R be a relation in U n for some n ≥ and let ϕ , ψ : F → U n .Then the following statements are equivalent: • there exists a qualified set Q such that ∀ λ ∈ Q, ϕ ( λ ) R ψ ( λ ); • we have (cid:18) lim λ ↑ U ϕ ( λ ) (cid:19) R ∗ (cid:18) lim λ ↑ U ψ ( λ ) (cid:19) . Proof : It is an immediate consequence of Theorem 11 and the definition ofqualified set. (cid:3)
In this section, we will introduce the notion of ultrafunction and we will analyzeits first properties. 8 .1 Definition of Ultrafunctions
Let Ω be a set in R N , and let V (Ω) be a (real or complex) vector space suchthat D (Ω) ⊆ V (Ω) ⊆ L (Ω) ∩ C (Ω) . Definition 16
Given the function space V (Ω) we set e V (Ω) := lim λ ↑ U V λ (Ω) = Span ∗ ( V (Ω) ◦ ) , where V λ (Ω) = Span ( V (Ω) ∩ λ ) . e V (Ω) will be called the space of ultrafunctions generated by V (Ω) . So, given any vector space of functions V (Ω), the space of ultrafunctiongenerated by V (Ω) is a vector space of hyperfinite dimension that includes V (Ω),and the ultrafunctions are Λ-limits of functions in V λ . Hence the ultrafunctionsare particular internal functions u : ( R ∗ ) N → C ∗ . Observe that, by definition, the dimension of e V (Ω) (that we denote by β ) isequal to the internal cardinality of any of its bases, and the following formulaholds: β = lim λ ↑ U dim( V λ (Ω)) . Since e V (Ω) ⊂ (cid:2) L ( R ) (cid:3) ∗ , it can be equipped with the following scalar product( u, v ) = Z ∗ u ( x ) v ( x ) dx, where R ∗ is the natural extension of the Lebesgue integral considered as a func-tional Z : L (Ω) → C . Notice that the Euclidean structure of e V (Ω) is the Λ-limit of the Euclideanstructure of every V λ given by the usual L scalar product. The norm of anultrafunction will be given by k u k = (cid:18)Z ∗ | u ( x ) | dx (cid:19) . Remark 17
Notice that the natural extension f ∗ of a function f is an ultra-function if and only if f ∈ V (Ω) . Proof.
Let f ∈ V (Ω) , and Q ( f ) = { λ ∈ F | f ∈ λ } . Since, for every λ ∈ Q ( f ) , f ∈ V λ (Ω) and, as we observed in section 2.3, Q ( f ) is a qualified set, itfollows by Theorem 15 that f ∗ ∈ e V (Ω) . Conversely, if f / ∈ V (Ω) then by Leibnitz Principle it follows that f ∗ / ∈ V ∗ (Ω)and, since e V (Ω) ⊂ V ∗ (Ω), this entails the thesis. (cid:3) .2 Delta, Sigma and Theta Basis In this section we introduce three particular kinds of bases for V (Ω) and westudy their main properties. We start by defining the Delta ultrafunctions : Definition 18
Given a number q ∈ Ω ∗ , we denote by δ q ( x ) an ultrafunction in e V (Ω) such that ∀ v ∈ e V (Ω) , Z ∗ v ( x ) δ q ( x ) dx = v ( q ) . (2) δ q ( x ) is called Delta (or the Dirac) ultrafunction concentrated in q . Let us see the main properties of the Delta ultrafunctions:
Theorem 19
We have the following properties:1. For every q ∈ Ω ∗ there exists an unique Delta ultrafunction concentratedin q ;
2. for every a, b ∈ Ω ∗ δ a ( b ) = δ b ( a ); k δ q k = δ q ( q ) . Proof.
1. Let { e j } βj =1 be an orthonormal real basis of e V (Ω) , and set δ q ( x ) = β X j =1 e j ( q ) e j ( x ) . Let us prove that δ q ( x ) actually satisfies (2). Let v ( x ) = P βj =1 v j e j ( x ) beany ultrafunction. Then Z ∗ v ( x ) δ q ( x ) dx = Z ∗ β X j =1 v j e j ( x ) β X k =1 e k ( q ) e k ( x ) ! dx == β X j =1 β X k =1 v j e k ( q ) Z ∗ e j ( x ) e k ( x ) dx == β X j =1 β X k =1 v j e k ( q ) δ j,q = β X j =1 v k e k ( q ) = v ( q ) . So δ q ( x ) is a Delta ultrafunction centered in q .It is unique: if f q ( x ) is another Delta ultrafunction centered in q then forevery y ∈ Ω ∗ we have: δ q ( y ) − f q ( y ) = Z ∗ ( δ q ( x ) − f q ( x )) δ y ( x ) dx = δ y ( q ) − δ y ( q ) = 0and hence δ q ( y ) = f q ( y ) for every y ∈ Ω ∗ . δ a ( b ) = R ∗ δ a ( x ) δ b ( x ) dx = δ b ( a ) . k δ q k = R ∗ δ q ( x ) δ q ( x ) = δ q ( q ). (cid:3) Definition 20
A Delta-basis { δ a ( x ) } a ∈ Σ (Σ ⊂ Ω ∗ ) is a basis for e V (Ω) whoseelements 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 ∈ Ω ∗ the equation Z ∗ δ a ( x ) σ b ( x ) dx = δ ab (3) holds. The set Σ ⊂ Ω ∗ is called set of independent points. The existence of a Delta-basis is an immediate consequence of the followingfact:
Remark 21
The set { δ a ( x ) | a ∈ Ω ∗ } generates all e V (Ω) . In fact, let G (Ω) bethe vectorial space generated by the set { δ a ( x ) | a ∈ Ω ∗ } and suppose that G (Ω) is properly included in e V (Ω) . Then the orthogonal G (Ω) ⊥ of G (Ω) in e V (Ω) contains a function f = 0 . But, since f ∈ G (Ω) ⊥ , for every a ∈ Ω ∗ we have f ( a ) = Z ∗ f ( x ) δ a ( x ) dx = 0 , so f ↿ Ω ∗ = 0 and this is absurd. Thus the set { δ a ( x ) | a ∈ Ω ∗ } generates e V (Ω) , hence it contains a basis. Let us see some properties of Delta- and Sigma-bases:
Theorem 22
A Delta-basis { δ q ( x ) } q ∈ Σ and its dual basis { σ q ( x ) } q ∈ Σ satisfythe following properties:1. if u ∈ e V (Ω) , then u ( x ) = X q ∈ Σ (cid:18)Z ∗ σ q ( ξ ) u ( ξ ) dξ (cid:19) δ q ( x );
2. if u ∈ e V (Ω) , then u ( x ) = X q ∈ Σ u ( q ) σ q ( x ); (4)
3. if two ultrafunctions u and v coincide on a set of independent points thenthey are equal;4. if Σ is a set of independent points and a, b ∈ Σ then σ a ( b ) = δ ab ;
5. for any q ∈ Ω ∗ , σ q ( x ) is well defined. roof.
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 u ( x ) = X q ∈ Σ (cid:18)Z δ q ( ξ ) u ( ξ ) dξ (cid:19) σ q ( x ) = X q ∈ Σ u ( q ) σ q ( x ) .
3. It follows directly from 2.4. If follows directly by equation (3)5. Given any point q ∈ Ω ∗ 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 toprove 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 σ ′ q ( x ) = X a ∈ Σ σ ′ q ( a ) σ a ( x ) . Then, by (4), it follows that σ ′ q ( x ) = σ q ( x ) . (cid:3) Let Σ be a set of independent points, and let L Σ : e V (Ω) → e V (Ω) be thelinear operator such that L Σ σ a ( x ) = δ a ( x )for every a ∈ Σ . Proposition 23 L Σ is selfadjoint, positive and Z ∗ L Σ u ( x ) v ( x ) dx = X a ∈ Σ u ( a ) v ( a ) . Proof.
Since u ( x ) = P a ∈ Σ u ( a ) σ a ( x ) and v ( x ) = P a ∈ Σ v ( a ) σ a ( x ) , then Z ∗ L Σ u ( x ) v ( x ) dx = Z ∗ L Σ X a ∈ Σ u ( a ) σ a ( x ) ! X b ∈ Σ v ( b ) σ b ( x ) ! dx == X a ∈ Σ X b ∈ Σ u ( a ) v ( b ) Z ∗ δ a ( x ) σ b ( x ) dx = X a ∈ Σ u ( a ) v ( a ) . Hence, clearly, L Σ is selfadjoint and positive. (cid:3) From now on, we consider the set Σ fixed once for all and we simply denotethe operator L Σ by L. Since L is a positive selfadjoint operator, A = L / is awell defined positive selfadjoint operator. For every a ∈ Σ we set θ a ( x ) = Aσ a ( x ) . Theorem 24
The following properties hold:1. { θ a ( x ) } a ∈ Σ is an orthonormal basis; . for every a, b ∈ Σ , θ a ( b ) = θ b ( a ) ;3. for every ultrafunction u we have u ( x ) = X a ∈ Σ u ( a ) σ a ( x ) = X a ∈ Σ u ( a ) θ a ( x ) = X a ∈ Σ u ( a ) δ a ( x ) , where we have set, for every a ∈ Σ ,u ( a ) := ( A − u )( a ) = Z ∗ θ a ( ξ ) u ( ξ ) dξ ; u ( a ) = ( A − u )( a ) = ( L − u )( a ) = Z ∗ σ a ( ξ ) u ( ξ ) dξ ;
4. for every ultrafunctions u, v we have Z ∗ u ( x ) v ( x ) dx = X a ∈ Σ u ( a ) v ( a ) = X a ∈ Σ u ( a ) v ( a );
5. for every ultrafunction u we have Z ∗ u ( x ) dx = X a ∈ Σ u ( a ) . Proof: { θ a ( x ) } a ∈ Σ is a basis since it is the image of the basis { σ a ( x ) } a ∈ Σ respect to the invertible linear application L. It is orthonormal: for every a, b ∈ Σwe have Z ∗ θ a ( x ) θ b ( x ) dx = Z ∗ Aσ a ( x ) Aσ b ( x ) dx = Z ∗ Lσ a ( x ) σ b ( x ) == σ b ( a ) = δ ab.
2) We have θ a ( b ) = Z ∗ θ a ( x ) δ b ( x ) dx = Z ∗ θ a ( x ) Aθ b ( x ) dx == Z ∗ Aθ a ( x ) θ b ( x ) dx = Z ∗ δ a ( x ) θ b ( x ) dx = θ b ( a ) .
3) The equality u ( x ) = X a ∈ Σ u ( a ) σ a ( x )has been proved in Theorem 22, (4); the equality u ( x ) = X a ∈ Σ u ( a ) θ a ( x ) , u ( a ) = R ∗ θ a ( ξ ) u ( ξ ) dξ , follows since { θ a ( x ) } a ∈ Σ is an orthonormal basis.And ( A − u )( a ) = Z ∗ δ a ( ξ ) A − u ( ξ ) dξ == Z ∗ A − δ a ( ξ ) u ( ξ ) dξ = Z ∗ θ a ( ξ ) u ( ξ ) dξ since A (and, so, A − ) is selfadjoint.The equality u ( x ) = X a ∈ Σ u ( a ) δ a ( x ) , where u ( a ) = R ∗ σ a ( ξ ) u ( ξ ) dξ , follows by point (1) in Theorem 22. And u ( a ) = Z ∗ σ a ( ξ ) u ( ξ ) dξ = Z ∗ L − δ a ( ξ ) u ( ξ ) dξ == Z ∗ δ a ( ξ ) L − u ( ξ ) dξ = ( L − u )( a ) .
4) We have that R ∗ u ( x ) v ( x ) dx = P a ∈ Σ u ( a ) v ( a ) since { θ a ( x ) } a ∈ Σ is anorthonormal basis: Z ∗ u ( x ) v ( x ) dx = Z ∗ X a ∈ Σ u ( a ) θ a ( x ) ! X b ∈ Σ v ( b ) θ b ( x ) dx ! == X a ∈ Σ X b ∈ Σ u ( a ) v ( b ) Z ∗ θ a ( x ) θ b ( x ) dx = X a ∈ Σ u ( a ) v ( a );the equality R ∗ u ( x ) v ( x ) dx = P a ∈ Σ u ( a ) v ( a ) follows by expressing u ( x ) inthe Delta basis and v ( x ) in the Sigma basis: Z ∗ u ( x ) v ( x ) dx = Z ∗ X a ∈ Σ u ( a ) δ a ( x ) ! X b ∈ Σ v ( b ) σ b ( x ) ! dx == X a ∈ Σ X b ∈ Σ v ( b ) u ( a ) Z ∗ δ a ( x ) σ b ( x ) dx = X a ∈ Σ u ( a ) v ( a ) .
5) This follows by expressing u ( x ) in the Delta basis: Z ∗ u ( x ) dx = Z ∗ X a ∈ Σ u ( a ) δ a ( x ) dx = X a ∈ Σ u ( a ) Z ∗ δ a ( x ) dx = X a ∈ Σ u ( a ) . (cid:3) .3 Canonical extension of a function Let V ′ (Ω) denote the dual of V (Ω) and let M denote the set of measurablefunctions in R N . If T ∈ V ′ (Ω) and if there is a function f ∈ M such that ∀ v ∈ V (Ω) , h T, v i = Z f ( x ) v ( x ) dx then T and f will be identified, and with some abuse of notation we shall write T = f ∈ V ′ (Ω) ∩ M . With this identification, V ′ (Ω) ∩ M ⊂ L . Definition 25 If T ∈ [ V ′ (Ω)] ∗ , there exists a unique ultrafunction f T ( x ) suchthat ∀ v ∈ e V (Ω) , h T, v i = Z ∗ e T ( x ) v ( x ) dx. In particular, if u ∈ [ V ′ (Ω) ∩ M ] ∗ , e u will denote the unique ultrafunction suchthat ∀ v ∈ e V (Ω) , Z ∗ u ( x ) v ( x ) dx = Z ∗ e u ( x ) v ( x ) dx. Notice that V ′ (Ω) ∩ M is a space of distributions which contains the deltameasures, so to every Delta distribution δ q is associated an ultrafunction which,by definition, is the Delta ultrafunction centered in q , as expected. Definition 26 If f ∈ V ′ (Ω) ∩ M , g ( f ∗ ) is called the canonical extension of f . Inthe following, since f and f ∗ can be identified, we will write e f instead of g ( f ∗ ) . Thus any function f : R N → R can be extended to the function f ∗ : ( R ∗ ) N → R ∗ which is called the natural extension of f and if f ∈ V ′ (Ω) ∩ M , we have alsothe canonical extension of f given by e f : ( R ∗ ) N → R ∗ If f / ∈ V (Ω) , by Remark 17, e f = f ∗ , thus f ∗ / ∈ e V (Ω) . Example: if Ω = ( − , , then | x | − / ∈ V ( − , ′ ∩ M ; the ultrafunction ^ | x | − / is different from (cid:0) | x | − / (cid:1) ∗ since the latter is not defined for x = 0 , while (cid:18) ^ | x | − / (cid:19) x =0 = Z ∗ | x | − / δ ( x ) dx. heorem 27 If T ∈ [ V (Ω) ′ ] ∗ , then e T ( x ) = X q ∈ Σ h T, δ q i σ q ( x ) == X q ∈ Σ h T, θ q i θ q ( x ) == X q ∈ Σ h T, σ q i δ q ( x ) . In particular, if f ∈ [ V ′ (Ω) ∩ M ] ∗ e f ( x ) = X q ∈ Σ (cid:20)Z f ∗ ( ξ ) δ q ( ξ ) dξ (cid:21) σ q ( x ) = (5)= X q ∈ Σ (cid:20)Z f ∗ ( ξ ) θ q ( ξ ) dξ (cid:21) θ q ( x ) = (6)= X q ∈ Σ (cid:20)Z f ∗ ( ξ ) σ q ( ξ ) dξ (cid:21) δ q ( x ) . (7) Proof.
It is sufficient to prove that ∀ v ∈ V (Ω) , Z X q ∈ Σ h T, δ q i σ q ( x ) v ( x ) dx = h T, v i . We have that
Z X q ∈ Σ h T, δ q i σ q ( x ) v ( x ) dx = X q ∈ Σ h T, δ q i Z σ q ( x ) v ( x ) dx == * T, X q ∈ Σ (cid:18)Z σ q ( x ) v ( x ) dx (cid:19) δ q + = h T, v i . The other equalities can be proved similarly. (cid:3)
In this section we will show that the space of ultrafunctions is reacher than thespace of distribution, in the sense that any distribution can be represented byan ultrafunction and that the converse is not true.
Definition 28
Let D ⊂ e V (Ω) be a vector space. We say that two ultrafunctions u and v are D -equivalent if ∀ ϕ ∈ D, Z ∗ ( u ( x ) − v ( x )) ϕ ( x ) dx = 0 . We say that two ultrafunctions u and v are distributionally equivalent if theyare D (Ω) -equivalent. heorem 29 Given T ∈ D ′ , there exists an ultrafunction u such that ∀ ϕ ∈ D (Ω) , Z ∗ u ( x ) ϕ ∗ ( x ) dx = h T, ϕ i . (8) Proof:
Let { e j ( x ) } j ∈ J be an orthonormal basis of the hyperfinite space e V (Ω) ∩ D (Ω) ∗ and take u ( x ) = X j ∈ J h T ∗ , e j i e j ( x ) . Now take ϕ ∈ D . Since ϕ ∗ ∈ e V (Ω) ∩ D (Ω) ∗ , we have that ϕ ∗ ( x ) = X j ∈ J (cid:18)Z ∗ ϕ ∗ ( ξ ) e j ( ξ ) dξ (cid:19) e j ( x ) . Thus Z ∗ u ( x ) ϕ ∗ ( x ) dx = Z ∗ X j ∈ J h T ∗ , e j i e j ( x ) ϕ ∗ ( x ) dx = X j ∈ J (cid:28) T ∗ , e j Z ∗ e j ( x ) ϕ ∗ ( x ) dx (cid:29) == * T ∗ , X j ∈ J (cid:18)Z ∗ e j ( x ) ϕ ∗ ( x ) dx (cid:19) e j + = h T ∗ , ϕ ∗ i = h T, ϕ i . (cid:3) The following proposition shows that the ultrafunction u associated to thedistribution T by (8) is not unique: Proposition 30
Take T ∈ D ′ (Ω) and let V T = { u ∈ e V (Ω) : ∀ ϕ ∈ D (Ω) , Z ∗ u ( x ) ϕ ∗ ( x ) dx = h T, ϕ i} , let u ∈ V T and let v be any ultrafunction. Then1. v ∈ V T if and only if u and v are D (Ω) -equivalent;2. V T is infinite. Proof:
1) If v ∈ V T then ∀ ϕ ∈ D (Ω) , R ∗ ( u ( x ) − v ( x )) ϕ ∗ ( x ) dx = h T, ϕ i −h
T, ϕ i = 0, so u and v are D (Ω)-equivalent; conversely, if u and v are D -equivalent then ∀ ϕ ∈ D (Ω) , R ∗ u ( x ) ϕ ∗ ( x ) dx = R ∗ v ( x ) ϕ ∗ ( x ) dx. Since R ∗ u ( x ) ϕ ∗ ( x ) dx = h T, ϕ i then v ∈ V T .
2) Let v = 0 be any ultrafunction in the orthogonal (in e V (Ω)) of e V (Ω) ∩D (Ω) ∗ . Then u and u + v are D (Ω)-equivalent, since R ∗ ( u ( x ) + v ( x )) ϕ ∗ ( x ) dx = R ∗ u ( x ) ϕ ∗ ( x ) dx + R ∗ v ( x ) ϕ ∗ ( x ) dx = R ∗ u ( x ) ϕ ∗ ( x ) dx + 0. Since the orthogonalof e V (Ω) ∩ D (Ω) ∗ is infinite, we obtain the thesis. (cid:3) emark 31 There is a natural way to associate a unique ultrafunction to adistribution (see also [1]). In order to do this it is sufficient to split e V (Ω) intwo orthogonal component: e V (Ω) ∩ D (Ω) ∗ and (cid:16) e V (Ω) ∩ D (Ω) ∗ (cid:17) ⊥ . As we haveseen in the proof of the above theorem every ultrafunction in V T can be spittedin two components, u + v where v ∈ (cid:16) e V (Ω) ∩ D (Ω) ∗ (cid:17) ⊥ and u ∈ e V (Ω) ∩ D (Ω) ∗ is univocally determined. Then, we have an injective map i : D ′ (Ω) → e V (Ω) given by i ( T ) = u where u ∈ V T ∩ D (Ω) ∗ . Remark 32
The space of ultrafunctions is richer than the space of distribu-tions; for example consider the function u ( x ) := f ( x ) min (cid:0) x − , α (cid:1) where α > is an infinite number and f ( x ) is a function with compact supportsuch that f (0) = 1 . Since u ∈ [ V ′ (Ω) ∩ M ] ∗ , e u is well defined (see def. 25). Onthe other hand, e u does not correspond to any distribution since Z ∗ e u ( x ) ϕ ∗ ( x ) dx = Z ∗ f ∗ ( x ) min (cid:0) x − , α (cid:1) ϕ ∗ ( x ) dx is infinite when ϕ ( x ) ≥ and ϕ (0) > . In [1] Section 6, it is presented an ellipticproblem which has a solution in the space of ultrafunctions, but no solution inthe space of distributions.
Definition 33
Given the operator F : V (Ω) → D ′ (Ω) , the map e F : e V (Ω) → e V (Ω) defined by e F ( u ) = ^ F ∗ ( u ) (9) is called canonical extension of F (” ∼ ” is defined by 25). By the definition of e F , we have that ∀ v ∈ e V (Ω) , Z ∗ e F ( u ( x )) v ( x ) dx = Z ∗ F ∗ ( u ( x )) v ( x ) dx. (10)18omparing Definition 33 with Theorem 27 we have that e F ( u ( x )) = X q ∈ Σ h F ∗ ( u ) , δ q i σ q ( x ) == X q ∈ Σ h F ∗ ( u ) , θ q i θ q ( x ) == X q ∈ Σ h F ∗ ( u ) , σ q i δ q ( x ) . In particular, if F : V (Ω) → V ′ (Ω) ∩ M ∗ : e F ( u ( x )) = X q ∈ Σ (cid:20)Z F ∗ ( u ( ξ )) δ q ( ξ ) dξ (cid:21) σ q ( x ) = (11)= X q ∈ Σ (cid:20)Z F ∗ ( u ( ξ )) θ q ( ξ ) dξ (cid:21) θ q ( x ) == X q ∈ Σ (cid:20)Z F ∗ ( u ( ξ )) σ q ( ξ ) dξ (cid:21) δ q ( x ) . A good generating space to define the derivative of an ultrafunction is the fol-lowing one: V (Ω) = H , (Ω) ∩ C (Ω) ⊆ L (Ω) ∩ C (Ω) . In order to simplify the exposition, we will assume that Ω ⊆ R . The gener-alization of the notions exposed in this section is immediate.Let u ∈ f V (Ω) be a ultrafunction. Since V (Ω) ∗ ⊂ H (Ω) ∗ , we have that thederivative dudx = ∂u = u ′ is in L (Ω) ⊂ [ V ′ G ∩ M ] ∗ . Then we can apply Definition33:
Definition 34
We set Du = e ∂u = f ∂u. The operator D : f V (Ω) → f V (Ω) is called (generalized) derivative of the ultrafunction u. By (11) we have the following representation of the derivative: ∀ u ∈ f V (Ω) , Du ( x ) = X q ∈ Σ (cid:20)Z ∗ u ′ ( ξ ) δ q ( ξ ) dξ (cid:21) σ q ( x ) . If u ′ ∈ f V (Ω) ⊂ (cid:2) V (Ω) (cid:3) ∗ , we have that Du ( x ) = X q ∈ Σ u ′ ( q ) σ q ( x ) = u ′ ( x ) .
19n particular, if u ∈ H , (Ω) ∩ C (Ω) , Du = u ′ and so D extends the operator ddx : H , (Ω) ∩ C (Ω) → V (Ω) to the operator D : f V (Ω) → f V (Ω) . In this section we will investigate the extension of the one-dimensional Fouriertransform. A good space to work with the Fourier transform is the space V F ( R ) = H ( R ) ∩ L ( R , | x | ) . It is easy to see that the space V F ( R ) can be characterized as follows: V F ( R ) = (cid:8) u ∈ H ( R ) : ˆ u ∈ H ( R ) (cid:9) . In fact, if ˆ u ∈ H ( R ) , then R |∇ u ( ξ ) | dξ < + ∞ and hence R | u ( x ) | | x | dx < + ∞ . Then V F ( R ) ⊂ L ( R , | x | ) , so V F ( R ) ⊂ H ( R ) ∩ L ( R , | x | ) which is aHilbert space equipped with the norm k u k V F ( R ) = Z | u ( x ) | | x | dx + Z | ˆ u ( ξ ) | | ξ | dξ. Moreover Z | u ( x ) | dx = Z | u ( x ) | (1 + | x | ) 11 + | x | dx ≤≤ (cid:18)Z | u ( x ) | (1 + | x | ) dx (cid:19) (cid:18)Z | x | ) dx (cid:19) ≤≤ const. (cid:16) k u k L ( R ) + k u k L ( R , | x | ) (cid:17) . Thus, V F ( R ) ⊂ L ( R ) . Recalling that the functions in H ( R ) are continuous,we have that V F ( R ) ⊂ C ( R ) ∩ H ( R ) ∩ L ( R ) ∩ L ( R , | x | ) . We use the following definitions of Fourier transform: if u ∈ f V F ( R ), we set F ( u )( k ) = b u ( k ) = 1 √ π Z ∗ u ( x ) e − ikx dx ; (12) F − ( u )( x ) = 1 √ π Z ∗ b u ( k ) e ikx dx. (13)Now, in order to deal with the Fourier transform in an easier way, we needa new axiom whose consistency is easy to be verified (see Appendix): Axiom 35 ( FTA ) (Fourier transform axiom) If u ∈ f V F ( R ) then F ∗ ( u ) ∈ f V F ( R ) and ¯ u ∈ f V F ( R ) (here ¯ u is the complex conjugate of u ) .
20t is immediate to see that, by this axiom, for every ultrafunction, u we have F ∗ ( u ) = e F ( u )and hence, since there is no risk of ambiguity, we will simply write F ( u ) . It is well known that in the theory of tempered distributions we have that: F ( δ a ) = e − iak √ π ; F (cid:18) e iax √ π (cid:19) = δ a . In the theory of ultrafunctions an analogous result holds:
Proposition 36
We have that:1. F (cid:16) g e iax √ π (cid:17) = δ a ( k ); F ( δ a ( x )) = ^ e − iak √ π ; π R ∗ ] e − iax g e ikx dx = δ a ( k ) . Proof.
1. For every v ∈ V F , Z ∗ F g e iax √ π ! v ( k ) dk = Z ∗ (cid:18) π Z ∗ ] e − iak e ixk dx (cid:19) v ( k ) dk == 12 π Z ∗ Z ∗ ] e − iak e ixk v ( k ) dkdx == 1 √ π Z ∗ ] e − iak F − ( v ( k )) dx = v ( a ) . Hence, 1 holds.2 - We have F ( δ a ( x )) = Z ∗ δ a ( x ) e − ikx dx = Z ∗ δ a ( x ) ] e − ikx dx = ^ e − ika . π Z ∗ g e iax ] e − ikx dx = 12 π Z ∗ g e iax e − ikx dx = F g e iax √ π ! = δ a ( k ) . (cid:3) By our definitions we have that: g e ikx = X q ∈ Σ (cid:20)Z ∗ e ikξ δ q ( ξ ) dξ (cid:21) σ q ( x ); g e ixk = X q ∈ Σ (cid:20)Z ∗ e ixξ δ q ( ξ ) dξ (cid:21) σ q ( k ) . g e ikx = g e ixk or not. The following Corollaryanswers this question. Corollary 37
We have that: g e ikx = g e ixk . Proof.
By the previous proposition, we have that ] e − ikx = √ π F ( δ k ( x )) = Z ∗ δ k ( x ) e − ixk dk = Z ∗ δ x ( k ) e − ixk dx = ] e − ixk . Replacing x with − x we get the result. (cid:3) Since F : V F ( R ) → V F ( R ) is an isomorphism, it follows that, for any Delta-basis { δ a } a ∈ Σ , the set ( g e iax √ π ) a ∈ Σ = { F ( δ − a ) } a ∈ Σ is a basis and we get the following result: Theorem 38 If u ∈ V F ( R ) , then u ( x ) = 1 √ π X k ∈ Σ b u ( k ) g e ikx . where we have set (see Theorem 24) b u ( k ) = Z ∗ b u ( ξ ) σ k ( ξ ) dξ. Proof.
Since n g e ikx √ π o k ∈ Σ is a basis, any u ∈ V F ( R ) has the following repre-sentation: u ( x ) = 1 √ π X k ∈ Σ u k g e ikx . Let us compute the u k ’s: we have Z δ k ( x ) σ b ( x ) dx = Z δ k ( x ) σ b ( x ) dx = δ kb and so Z b δ k ( x ) c σ b ( x ) dx = δ kb and by Proposition 36, Z ] e − ikx √ π c σ b ( x ) dx = δ kb . { c σ k ( x ) } k ∈ Σ is the dual basis of n ^ e − ikx √ π o k ∈ Σ , namely { c σ k ( − x ) } k ∈ Σ is thedual basis of n g e ikx √ π o k ∈ Σ . Hence, since bb v ( x ) = v ( − x ) , we have: u k = Z u ( ξ ) c σ k ( − ξ ) dξ = Z u ( ξ ) cc σ k ( − ξ ) dξ == Z b u ( ξ ) σ k ( ξ ) dξ = Z b u ( ξ ) σ k ( ξ ) dξ = b u ( k ) . (cid:3) In this section we prove that the axiomatic construction of ultrafunctions iscoherent. We assume that the reader knows the key concepts in NonstandardAnalysis (see e.g. [7]).The following result has already been proved in [1]. Here we give an alter-native proof of this result based on Nonstandard Analysis:
Theorem 39
The set of axioms { (Λ - , (Λ - , (Λ - } is consistent. Proof.
Let U , V be mathematical universes and let h U , V , ⋆ i be a nonstan-dard extension of U that is | U | + -saturated. We denote by F the set of finitesubsets of U and, for every λ ∈ F , we pose F λ = { S ⊂ V | S is hyperfinite and λ ⋆ ⊂ S } . By saturation T λ ∈F F λ = ∅ . We take Λ ∈ T λ ∈F F λ . For any given net ϕ : F → U we define its Λ-limit aslim λ ↑ U ϕ ( λ ) = ϕ ⋆ (Λ)and we pose K = lim λ ↑ U R = (cid:26) lim λ ↑ U ϕ ( λ ) | ϕ : F → R (cid:27) . With these choices the Λ-limit satisfies the axioms (Λ-1) , (Λ-2),(Λ-3): theonly nontrivial fact is (Λ-2). Let ϕ be an eventually constant net, and let λ ∈ F , r ∈ R be such that ∀ λ ∈ { η ∈ F | λ ⊂ η } ϕ ( λ ) = r. By transfer it follows that ∀ λ ∈ { η ∈ F | λ ⊂ η } ⋆ = { η ∈ F ⋆ | λ ⋆ ⊂ η } wehave: ϕ ⋆ ( λ ) = r ⋆ . But r = r ⋆ and λ ⋆ ⊂ Λ by construction. So, since Λ ∈ F ⋆ , ϕ ⋆ (Λ) = r. (cid:3) ∗ to denote theextensions of objects in U in the sense of Λ-limit (not to be confused with theextensions obtained by applying the star map ⋆ : e.g., the field K = R ∗ is asubfield of R ⋆ ) . We observe that, given a set S in U , its hyperfinite extension (in the senseof the Λ-limit) is S ◦ = lim λ ↑ U ( S ∩ λ ) = S ⋆ ∩ Λand we use this observation to prove that, given a set of functions V (Ω), byposing e V (Ω) = Span ( V (Ω) ◦ ) = Span ( V (Ω) ⋆ ∩ Λ)we obtain the set of ultrafunctions generated by V (Ω).The only nontrivial fact to prove is that, for every function f ∈ V (Ω) , its natural extension f ∗ is an ultrafunction. First of all, we observe that, bydefinition, f ∗ = f ⋆ . Also, since f ∈ V (Ω) , by transfer it follows that f ⋆ ∈ V (Ω) ⋆ . And, by our choice of Λ , we also have that f ⋆ ∈ Λ since, by construction, { f } ⋆ = { f ⋆ } ⊂ Λ . It remains to prove the coherence of the axioms (Λ-1) , (Λ-2),(Λ-3) combinedwith F T A . Theorem 40
The set of axioms { (Λ - , (Λ - , (Λ - , F T A } is consistent. Proof.
The basic idea is to chose an hyperfinite set Λ ∈ T λ ∈F F λ ,where F λ is defined in Theorem 39 (which automatically ensures the satisfaction of(Λ-1) , (Λ-2),(Λ-3)), with one more particular property that will ensure the sat-isfaction of F T A.
We start by considering a generic hyperfinite set Λ ′ ∈ T λ ∈F F λ and we let B ′ = { e i ( x ) | i ∈ I } be any hyperfinite basis for Span ( V F ( R ) ⋆ ∩ Λ ′ ) . Now we pose B = { F j ( e i ( x )) : 0 ≤ j ≤ , i ∈ I } ∪ { F j ( e i ( x )) : 0 ≤ j ≤ , i ∈ I } , where F denotes the Fourier transform. Since F = id, we have that B isclosed by Fourier transform and complex conjugate. We now poseΛ = Λ ′ ∪ B and it is immediate to prove that, with this choice, F T A is ensured, because B is a set of generators for ^ V F ( R ) closed by Fourier transform and complexconjugate. (cid:3) eferences [1] Benci V., Ultrafunctions and generalized solutions,
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