The Colombeau theory of generalized functions

Transcription

1 The Colombeau theory of generalized functions Tạ Ngọc Trí Mathematics Master thesis specialized in Analysis 1 supervised by Prof. Dr. Tom H. Koornwinder KdV Institute, Faculty of Science University of Amsterdam The Netherlands Key words: Schwartz distributions, tempered distributions, Fourier transform, convolution, multiplication of distributions, Colombeau generalized functions, generalized complex numbers, tempered generalized functions.

2 Abstract In this report we discuss the nonlinear theory of generalized functions proposed by J. F. Colombeau in the 1980 s in two research monographs. Some motivations coming from the famous linear theory of L. Schwartz will also be discussed. Finally, results by the author will be presented about the class of L 1 ( n )-functions as a subset of the class of generalized functions.

3 Contents Introduction 3 1 Impossibility and Degeneracy esults in Schwartz Distribution Theory The Schwartz distributions Test function spaces Schwartz distributions Differentiation Product of a distribution and a smooth function Convolution and Fourier transform Product of distributions and related problems The Colombeau theory of generalized functions Preliminaries and notations Definition of Colombeau generalized functions Properties of the differential algebra G( n ) Nonlinear properties of G( n ) Generalized complex numbers Point values of generalized functions Integrals of generalized functions Weak concepts of equality in G(Ω) The tempered generalized functions and their Fourier transform 49 3 L 1 () embedded in the Colombeau generalized functions 56 Discussion and Conclusions 64 Acknowledgments 65 eferences 66 2

4 Introduction Master Thesis Tạ Ngọc Trí 3 The theory of distributions, initiated by L. Schwartz, has opened the door for important developments in modern Mathematics, especially in Partial Differential Equations (see [Sch01], [Hör83]). For this theory, L. Schwartz won the Fields Medal in That prestigious prize is the recognition by the mathematical world for his extremely important work. Soon after the introduction of his own theory, L. Schwartz published a paper in which he showed an impossibility result (see [Sch54]) about the product of two arbitrary distributions. However, in some applications there is need for such a product. Various mathematicians looked for a way around the Schwartz impossibility result (see [Obe92], [Obe01], [os87]). They tried to find methods to define the product of two arbitrary distributions. Some of them partly solved that issue (see [Mik66], [Tys81]). But the need for a full solution remained. In the 1980 s such a theory was proposed by J. F. Colombeau. In two successive monographs ([Col84] and [Col85]) he proposed the theory of generalized functions. In this new theory the distributions are a subset, and we can multiply two arbitrary generalized functions. After the appearance of the Colombeau theory, some mathematicians have relied on it, and got some interesting results about the solutions of nonlinear partial differential equations occurring in nature (see [Obe92], [Obe01], [Gko01]). These results show the importance of the new theory. In this thesis, the author presents some aspects of the issues above. In the first chapter we discuss the Schwartz theory of distributions and the issues to occur. Another part of the thesis is devoted to the main contents of the Colombeau theory of generalized functions. Some questions which came up in the study of this theory are discussed and answered in the last chapter. This is the graduation thesis for the 2 year Master of Mathematics program at Korteweg de Vries Institute for Mathematics (KdV), University of Amsterdam (UvA), The Netherlands.

5 Chapter 1 Impossibility and Degeneracy esults in Schwartz Distribution Theory 1.1 The Schwartz distributions To solve the issue of non-existence of derivatives for many ordinary functions, L. Schwartz proposed the distributions in the 1940 s. This theory has contributed to the development of some important fields in Mathematics since then, such as the development of Partial Differential Equations. I would like to sketch some main ideas of this distribution theory as well as some issues which arise in the process of using it in Mathematics Test function spaces 2... α n n, where i x j, j := 1, 2,..., n and i = 1; We start with some terminologies that will be used later. A multi-index (or, to be precise, an n-multi-index) is an n-tuples α = (α 1, α 2,..., α n ) of nonnegative integers; its length(or order) is α = α 1 +α 2 + +α n. With each multiindex α is associated the differential operator α = α 1 1 α 2 j = x j, or D α = D α 1 1 D α n n where D j = their order is α = α 1 + α α n. Let be given a non-empty open set Ω n. A complex-valued function f defined on Ω is said to belong to C (Ω) if α f exists and is continuous for every multi-index α. The support of a continuous function f : Ω C, denoted by supp f, is the closure in Ω of the set {x Ω : f(x) 0}. We notice that supp f is a closed subset of Ω. If K is a compact set in n, then 4

6 Master Thesis Tạ Ngọc Trí 5 we denote by D K the set {f C ( n ) : supp f K}. A topological vector space over K, (K :=, or C) is a vector space X over K, equipped with a topology that is compatible with the vector space structure, i.e., such that the mapping (x, y) x + y and (λ, x) λx are continuous. In a topological vector space X, a subset E is bounded if for every neighborhood V of 0, there is a number s > 0 such that E tv, for all t > s. If 0 has a bounded neighborhood, then X is said to be locally bounded. A subset E of a topological vector space X is said to be absorbing if, for all x X, there is t = t(x) 0 such that x te. If for all α C, α 1, we have αe E, then E is called a balanced subset of X. A topological vector space X (with topology τ) is called a locally convex if there is a local base for τ whose members are convex. A locally convex space is called a Fréchet space if it is also induced by a complete metric d satisfying d(x + z, y + z) = d(x, y) (d is translation invariant). A topological vector space X is said to have the Heine Borel property if every closed and bounded subset of X is compact. And we can see that: Proposition C (Ω) is a Fréchet space with the Heine-Borel property, and D K is a closed subspace of C (Ω), whenever K Ω. Proof. Please refer to [ud85], pp in detail. Here, we only sketch some main points. Choose compact sets K j, j = 1, 2,..., such that K j lies in the interior of K j+1 (denoted by Int K j+1 ) and Ω = j=1k j. The family of seminorms p N for N = 1, 2,..., defined by setting p N (f) = max{ α f(x) : x K N, α N} has the properties: separating points of C (Ω) and generating a topology with a countable local base. So, a compatible translation-invariant metric d can be defined in terms of p N, N = 1, 2,... as follows: d(f, g) = N=1 2 N p N (f, g) 1 + p N (f, g).

7 6 The Colombeau theory of generalized functions It is not difficult to show that this metric is complete, so C (Ω) is a Fréchet space. For each x Ω, the functional F x : f f(x) is continuous in the topology induced by the family of p N for N = 1, 2,.... We also notice that D K = ker F x, x Ω\K so D K is a closed subspace of C (Ω) for arbitrary compact subset K of Ω. For the proof of the Heine-Borel property of C (Ω), please refer to [ud85] for details. Lastly, we remark that if a sequence (f n ) converges to f in the topology defined above on C (Ω), then the sequence (f n ) converges uniformly to f on compact sets in Ω. Note. We would like to comment that the concepts of an increasing sequence of compact sets in [Fri98], p.35; or, monotone increasing sequence of compact subsets in [Yos74], p.27, for K j, j = 1, 2,... mean K j Int K j+1, j = 1, 2,.... The following example shown by Prof. Tom Koornwinder to me will explain the importance of this condition. Consider Ω = ( 1, 1), K n = [ 1+ 1 n, 1 n ] [ 1 n, 1 1 ] {0}, n = 1, 2,.... n We can easily verify that K j K j+1, j = 1, 2,..., n=1k n = Ω. However, K n is not a subset of Int K m whenever n < m. We could take the sequence (f n ) in C (Ω) such that f n (x) = 0 on K n and f n ( 1 2n ) = 1, for n = 1, 2,.... We can verify that the sequence of f n converges to 0 in the topology on C (Ω), obtained by the K ns, but f n does not converge to 0 uniformly in the usual sense! We also notice that C (Ω) is not locally bounded (with the topology induced by the family of p N, N = 1, 2,...), since if it was, it would have finite dimension and this is absurd (See [ud85], Theorem 1.23), so it follows from Theorem 1.39 in [ud85] that C (Ω) is not normable. We also get the same result for D K whenever K has nonempty interior Definition The union of the spaces D K, as K ranges over all compact subsets of Ω is called the test function space, denoted by D(Ω).

8 Master Thesis Tạ Ngọc Trí 7 It is clear that D(Ω) is a vector space, equipped with the usual definitions of addition and scalar multiplication of complex-valued functions. It is also obvious that φ D(Ω) if and only if φ C (Ω) and supp φ is a compact subset of Ω. To construct a locally convex topology τ on D(Ω) in which all Cauchy sequences do converge, we take for τ the collection of all unions of sets of the form φ + W, with φ D(Ω) and W β, where β is the collection of all convex balanced sets W D(Ω) : D K W τ K, for every compact K of Ω, with the topology τ K which has been defined in D K. Please refer to [ud85], pp for details. Here, we only list some important properties of this topology Proposition τ is a topology in D(Ω), and β is a local base for τ τ makes D(Ω) into a locally convex topological vector space. We also remark that τ has the following further properties: τ K coincides with the subspace topology D K that inherits from D(Ω) D(Ω) has the Heine-Borel property If (φ j ) is a Cauchy sequence in D(Ω), then (φ j ) D K for some compact K Ω, and lim k,j sup{ α φ k (x) α φ j (x) : x Ω; α N} = 0, where N = 0, 1, 2,.... If φ j 0 as j in τ, then there is a compact K Ω which contains the support of every φ j, j = 1, 2,..., and α φ j 0 uniformly as j, for every multi-index α In D(Ω), every Cauchy sequence converges. We would like to mention one special kind of topological vector space called Montel space in the following definition. Please refer to [Tre67], p.356 for further details.

9 8 The Colombeau theory of generalized functions Definition A topological vector space X is called a Montel space if X is locally convex Hausdorff such that every absorbing, covex, balanced and closed subset of X is a neighborhood of zero in X, and X also has the Heine-Borel property. Montel spaces share a special property which we will use later: a Montel space is always reflexive. Also to end this section, we recall the following result (All the details are in [Tre67], p.357) Proposition C (Ω) and D(Ω) are Montel spaces! Schwartz distributions We start with the definition of Schwartz distributions in [Fri98], p.7 as follows Definition A linear form u : D(Ω) C is called a distribution (or a Schwartz distribution) if, for every compact K Ω, there is a real number c 0 and a nonnegative integer N such that u, φ c sup α φ, α N for all φ D(Ω) with supp φ K. The vector space of distributions on Ω is denoted by D (Ω). Here, the vector space operations on the space of distributions are the usual ones on linear forms. We also see that all ordinary functions we often meet such as continuous functions, or functions in L p (Ω), 1 p are distributions. For instance, if f is continuous on Ω, then f : D(Ω) C such that f, φ = f(x)φ(x) dx, φ D(Ω), is a distribution. We can easily verify this conclusion. Here, we only mention a newcomer: the Dirac distribution. We Ω consider δ : D( n ) C such that δ, φ = φ(0), for all φ D( n ). Obviously, we get δ, φ = φ(0) 1 sup φ(x), for all φ D( n ), such that supp φ K, K a compact subset of n. Therefore, δ is a distribution and

10 Master Thesis Tạ Ngọc Trí 9 we call it the Dirac distribution. In fact, this strange function appeared in Physics at the beginning of the 20th century, and it was one of motivations for the appearance of distributions! We can describe distributions in another way as follows (refer [Fri98], p.9) Proposition A linear form u on D(Ω) is a distribution if and only if lim j u, φ j = 0 for every sequence (φ j ) which converges to zero in D(Ω) as j. We will consider a corollary of Proposition : a locally integrable function f on Ω yields a distribution defined by φ f, φ := f(x)φ(x) dx. Ω Here, a measurable f such that f(x) dx < for all compact subsets K K Ω is called a locally integrable one. The set of all locally integrable functions on Ω is denoted by L 1 loc (Ω). Now we turn to the proof that the above linear form is a distribution. Indeed, if (φ j ) 0 as j in D(Ω), then there is a compact subset K Ω such that supp φ j K, j = 1, 2,.... So, we can define f(x)φ Ω j(x) dx = f(x)φ K j(x) dx. Moreover, from fφ j f sup K φ j, and the uniform convergence of (φ j ) on K, we can apply the dominated convergence Lebesgue theorem, and we get lim j f, φ j = Differentiation One of the motivations for distribution theory was to get a space which contains ordinary functions and in which every element has a derivative, which coincides with ordinary derivative if the element is a differentiable function. D (Ω) is such a space in view of the following result: Proposition If u D (Ω), then the following linear form, denoted by α u (α a multi-index) is also a distribution. α u, φ = ( 1) α u, α φ, φ D(Ω), The proof is straight forward, see, for instance [Yos74], p.49. The motivation for this definition of α u is that it coincides with ordinary differentiation if u is a differentiable function. Indeed, if f C 1 (Ω), the

11 10 The Colombeau theory of generalized functions space of functions which have continuous derivatives up to order 1 on Ω, then we can remark that Ω j(fφ) dx j = 0 for all j = 1, 2,..., n. It follows from the Fubini s rule that Ω j(fφ) dx 1 dx n = 0. Hence Ω jfφ dx = f Ω jφ dx. Or j f, g = f, j g. So, if we identify f C 1 (Ω) with the distribution f D (Ω), f, φ = fφ dx, the above formula shows the Ω motivation of differentiation we have mentioned Examples a. If H(x) = { 1 if x 0, 0 if x < 0 (called Heaviside function), then H = δ. Indeed, we consider Ω =, and we have H, φ = ( 1) 1 H, φ = φ(x) dx = φ(x) = φ(0) = δ, φ, for all φ D (). Therefore, H = δ. 0 b. If f(x) = log x, then we can verify that f is locally integrable on. Therefore, the corresponding distribution defined by f is f, φ = log x φ(x) dx. We will calculate f. This distribution is usually denoted by 1. We have x 0 (log x ), φ = = 0 log x φ(x) dx log x φ(x) dx ɛ = lim ɛ 0 +[ log x φ(x) dx + = lim ɛ 0 +{[φ(ɛ) φ( ɛ)] log ɛ + = lim ɛ 0 +[ ɛ φ(x) x dx + ɛ 0 log x φ(x) dx ɛ ɛ φ(x) x log x φ(x) dx] φ(x) x dx + dx}], ɛ φ(x) x dx}

12 Master Thesis Tạ Ngọc Trí 11 since [φ(ɛ) φ( ɛ)] log ɛ 0, as ɛ 0 +. In fact 1 x does not belong to L1 loc(). So, we cannot represent the distribution 1 x in integral form. However, in the above approach we can define the distribution 1 x D () as log x, and it belongs to D ()\L 1 loc (). The expression lim ɛ 0 +[ ɛ φ(x) x ɛ φ(x) x the integral dx}], denoted by p.v. φ(x) x φ(x) x dx. dx + dx, is called the principal value of Product of a distribution and a smooth function In distribution theory we can get the product of a distribution u D (Ω), and a smooth function f C (Ω). This product, denoted by fu, is defined as follows fu, φ = u, fφ, for all φ D(Ω). It is not difficult to verify that the right-hand side is a distribution, and it justifies the above definition. For instance, if δ D (), then xδ = 0, since xδ, φ = δ, xφ = (xφ)(0) = 0, φ D(). For the case of u = 1 x, we can prove that x 1 x = 1 in D (). Indeed, we have x 1 x, φ = 1 x, xφ = lim = ɛ 0 +[ ɛ φ(x) dx xφ(x) x = 1, φ, φ D(). dx + ɛ xφ(x) x dx}] Leibniz s rule extends to the product of f C (Ω) and u D (Ω) by the following proposition Theorem Let f C (Ω) and u D (Ω), and let α be an arbitrary multi-index, then α (fu) = β+γ=α α! β!γ! β f γ u,

13 12 The Colombeau theory of generalized functions where α! = α 1!α 2!... α n! if α = (α 1, α 2,..., α n ). Proof. Please refer to [Fri98], p.24 for details. Next, we will extend the concept of support of an ordinary function to the case of distributions. Firstly, we say that a distribution u D (Ω) vanishes on an open set V of Ω if u, φ = 0 for all φ D(Ω) with support contained in V. Secondly, we notice the following result, for instance, in [Yos74], p Theorem If {V j } j I is a family of open subsets of Ω and if u vanishes on each V j, j I, then u vanishes on V = j I V j. Finally, we can say that the support of a distribution u, denoted by supp u, is the complement of the set {x Ω : u vanishes on a neighborhood of x}. So, as in the case of continuous functions, supp u is a closed subset of Ω. For instance, supp δ = {0}. Indeed, if x 0, so we can select U, an open neighborhood of x such that 0 / U, then δ, φ = φ(0) = 0. So, we have supp δ {0}. On the other hand, for every neighborhood V of 0, there is a function φ D( n ) such that supp φ V, φ(0) 0. So, with such φ we have δ, φ = φ(0) 0. Therefore, {0} supp δ. In summary, we have supp δ = {0}. We also notice that if f is continuous, or locally integrable on Ω, then supp f in the sense of distributions coincides with supp f in usual sense. For f L 1 loc (Ω), supp f means that x / supp f there is a neighborhood U of x in Ω on which f = 0 almost everywhere Convolution and Fourier transform For 1 p <, let L p ( n ) = {f defined and measurable on n : f(x) p dx < }, n where the integral is in Lebesgue sense and almost everywhere equal functions are identified with each other. When equipped with the norm f Lp := ( n f(x) p dx) 1 p, the space Lp ( n ), 1 p < becomes a separable Banach space (efer to [Ada75] in detail). If f, g L 1 ( n ), then the convolution of f and g, denoted by f g, is

14 Master Thesis Tạ Ngọc Trí 13 defined as follows: (f g)(x) := f(y)g(x y) dy. n It can be shown that (f g)(x) exists almost everywhere, f g L 1 ( n ), and f g L1 f L1 g L1. This operation makes L 1 ( n ) into a Banach algebra, but without unit (See [Arv98]). In fact we can also define the convolution of f and g if f L 1 ( n ) and g L p ( n ), 1 p (see [Ada75] for the definition of L ( n )). There are some interesting results related to that case, see again [Ada75] for details. Here, we will consider the extension of this convolution to distributions Definition If u, v D ( n ), we call the convolution of u and v, denoted by u v, the following linear form u v, φ = u(y), v(x), φ(x + y), where φ D( n ), whenever u, ψ with ψ(y) := v(x), φ(x + y), is welldefined and the resulting linear form u v is in D ( n ). For instance, we can prove that if at least one of u and v has compact support, the above definition will be satisfied and u v = v u. Moreover, in [Fri98] there are some other cases discussed where u v is well-defined emarks a) u δ = δ u = u, for all u D ( n ). Indeed, δ has compact support and u(y), δ(x), φ(x + y) = u(y), φ(y) = u, φ b) The above definition of convolution is still valid if f, g L 1 ( n ). Indeed, for arbitrary φ D( n ), set h(y) = g(x)φ(y + x) dx, n then h L 1 ( n ) by Moreover, we get h(y) g(x)φ(y + x) dx = g(t y) φ(t) dt n n sup φ(t) g(t y) dt = c g L1, t supp φ n

15 14 The Colombeau theory of generalized functions for y n. So, since f(y), g(x), φ(y + x) = f(y), h(y) = f(y)h(y) dy, n and f(y)h(y) c g L1 f(y), we get the existence of f(y), g(x), φ(y+x), so f g exists. We also have (by Fubini) f g, φ = f(y)g(x)φ(y + x) dx dy n n = ( f(y)g(t y) dy)φ(t) dt. n n = f(y)g(t y) dy, φ(t) n So, (f g)(t) = n f(y)φ(t y) dy as usual! Next, we will get an interesting result related to the convolution of u D ( n ) and ρ D( n ) (See, for instance, [Fri98], p.53) as follows Theorem If u D ( n ) and ρ D( n ), then (ρ u)(x) = u(y), ρ(x y), x n, and this function is a member of C ( n ). In literature ρ u is called a regularization of u, and later we will see that some mathematicians such as Mikusinki, Itano, Fisher,...have used regularizations of u, v D ( n ) to define the multiplication uv (efer to [Col84] p.31 for details). If f L 1 ( n ), the Fourier transform f (sometimes f, or Ff) of f is defined by f(ξ) = f(x)e ix.ξ dx, ξ n, where x.ξ is the usual inner n product on n and x.ξ = n j=1 x jξ j. We will extend this concept to distributions, or rather to a special subspace of D ( n ) called the space of tempered distribution Definition We denote by S( n ) the totality of functions f C ( n ), such that p α,β (f) := sup x n x β D α f(x) <,

16 Master Thesis Tạ Ngọc Trí 15 for all multi-indices α, β. Such functions are called rapidly decreasing ones. We can show that f D( n ), then f S( n ); or e x 2 of S( n ). is also a member The family of {p α,β : α, β are multi-indices} is the one of separating seminorms defined on S( n ), and it induces a locally convex topology. Furthermore, this topology is metrizable, and the induced distance is complete. So, S( n ) is a Fréchet space. We also have: φ j 0 in S( n ) α, β; p α,β (φ j ) 0 We also see that the embedding D( n ) S( n ) continuous, and D( n ) is dense in S( n ) for the above topology defined above in S( n ). One important result in S( n ) we need later is in the following lemma Lemma The Fourier transform F : S( n ) S( n ) is a continuous map. Now, we will define tempered distributions Definition S ( n ) is the subspace of D ( n ) consisting of distributions which extended to be continuous linear forms on S( n ). A sequence (u j ) 1 j in S ( n ) is said to converge to u S ( n ) if u j, φ u, φ for all φ S( n ) as j. The members of S ( n ) are called tempered distributions. We can show that u S ( n ) if and only if φ j 0 in S( n ) as j, implies that u, φ j 0 as j 0. Based on that remark, one can easily verify that L 1 ( n ) S ( n ). We also get that all distributions of D ( n ) with compact supports are tempered distributions, too. For instance, the Dirac function δ S ( n ). Indeed, if u D ( n ) and the compact set K in n is supp u, we fix ψ D( n ) such that ψ = 1 on some open set containing K. We define ũ, f := u, ψf, f S( n ) If f j 0 in S( n ) as j, then all D α f j 0 uniformly on n as j. Hence, all D α (ψf j ) 0 uniformly on n as j. It follows that

17 16 The Colombeau theory of generalized functions ũ is a continuous linear form on S( n ). Since ũ, φ = u, φ for φ D( n ), ũ is an extension of u. That concludes our arguments. In fact one can prove that every f L p ( n ), 1 p ; is a tempered distribution. Furthermore, if P is a polynomial, f S( n ), and u is a tempered distribution, then the distributions D α u, P u,and fu are also tempered distributions. Please refer to [ud85] for details. Now we will define the Fourier transform of a tempered distribution Definition The Fourier transform of u S ( n ) is the distribution û S ( n ) defined by û, φ := u, φ, φ S( n ). We remark that it follows from Lemma that if φ j 0 as j 0 in S( n ), then φ j 0 as j 0 in S( n ), and it verifies the above setting. There arises a consistency question that ought to be settled. If f L 1 ( n ), then f also a member of S ( n ), denoted by u f. So, there are two definitions of the Fourier transform of f, namely f(ξ) = f(x)e ix.ξ dx, ξ n, and n û f. The question is if they agree, i.e., whether the distribution û f corresponds to the function f. The answer is affirmative, and we can show this as follows û f, φ = u f, φ = f φ dx = fφ dx = f, φ, n for all φ S( n ) with the traditional Fourier transform and the Fourier transform of f as a tempered distribution in S ( n ) (since L 1 ( n ) S ( n )) Examples a) δ S ( n ), so we can define δ as follows δ, φ = δ, φ = φ(0) = e ix.0 φ(x) dx = φ(x) dx = 1, φ n n Hence, we have δ = 1. b) From the above example, we get 1 S ( n ), so we will consider 1, and we get 1, φ = 1, φ = φ(ξ) dξ = (2π) n 1 [ (2π) n e iξ.0 φ(ξ) dξ] n n, = (2π) n φ(0) = (2π) n δ, φ n

18 Master Thesis Tạ Ngọc Trí 17 where we have used the inverse Fourier transform φ(x) = 1 (2π) n e iξ.x φ(ξ) dξ. n Therefore, we have 1 = (2π) n δ. c) Next, we will show an interesting result related to the Fourier transform of a distribution with compact support. That is, if u is a distribution with compact support, then û(ξ) = u(x), e ix.ξ, and this û is in C ( n ). Even, if we extend the variable ξ to the complex domain, then we also have a very nice result called the Paley-Wiener theorem as follows Theorem a) If u is a distribution in S ( n ) with compact support, then û(ξ) = u(x), e ix.ξ, ξ C n, is an analytic function on C n. The function û(ξ), ξ C n is often said to be the Fourier-Laplace transform of u b) If u is a distribution in D ( n ) with compact support and supp u {x : x a}, where a is a positive real number, then there are constants c, N 0 such that û(ξ) c(1 + ξ N )e a Im ξ, ξ C n c) If u D( n ) and supp u {x : x a}, then there are constants c m 0, m = 0, 1,..., such that û(ξ) c m (1 + ξ ) m e a Im ξ, ξ C n, m = 0, 1,.... Proof. Please refer to [Fri98] in detail for a) and b). Here, we sketch some main points of the proof for c) (see the same reference as above) If u D( n ), then û(ξ) = u(x)e ix.ξ dx. n Also, since u D( n ), one can perform repeated partial integration in the above identity and get ξ α û(ξ) = ( i) α n α u(x)e ix.ξ dx, ξ C n,

19 18 The Colombeau theory of generalized functions for all multi-indices α, α 0. So one has, if supp u {x : x a}, then ξ α û(ξ) V a sup α u sup{e x Im ξ : x supp u}, x a where V a is the measure of the ball {x : x a}. Hence, ξ α û(ξ) V a sup α u e a. Im ξ, ξ C n, x a for all α, α 0, and these inequalities clearly imply c). Next, we would like to present a theorem called the structure theorem which shows the relationship among distributions and ordinary functions, and we will apply this result in Chapter 2. We can see the details of proof for this result in [Sch66], or [Fri98] Theorem The restriction of a distribution u D ( n ) to a bounded open set Ω n is a derivative of finite order of a continuous function. 1.2 Product of distributions and related problems In we have defined the product of f C (Ω) and u D (Ω). In practice as well as in mathematics view, we would like to define the product of two arbitrary distributions, say on m. Clearly, we cannot use the method of 1.1.4, because fφ is not a reasonable test function if f D ( m ) and φ D( m ). To overcome this, some suggestions came up in literature. However, we will see later that for such ways these still do not lead to the definition of the product of two arbitrary distributions. We are also interested in some ways to embed D ( m ) in an algebra, so that we can get the product. But some limitations arise, and we will end this section with such a result called Schwartz impossibility result. Next, we will discuss the first way of defining the product of two arbitrary distributions called method of regularization and passage to the limit (See [Mik66], or [Col84]).

20 Master Thesis Tạ Ngọc Trí Definition A δ-sequence is a sequence (δ n ), n = 1, 2,... of elements of D( m ) such that a) supp δ n {x m : x ɛ n } with lim n ɛ n = 0 b) m δ n (x) dx = 1. We remark that if S, T D ( m ), then from Theorem , the convolution S δ n and T δ n are C -functions on m, and they are regularizations of S and T. So, we can get (S δ n )(T δ n ) in the usual sense, and this motivates the definition of the distribution product below Definition We say that S and T are multiplicable with product S.T if, for any δ-sequence (δ n ), n = 1, 2,..., there exists lim n (S δ n )(T δ n ) in D ( m ) and its limit is independent of the choice of δ-sequence. Further motivation for the above definition is that lim n δ n = δ in D ( m ), so lim n (S δ n ) = S and lim n (T δ n ) = T in D ( m ). This definition was proposed by Hirata-Ogata, Mikusinki, Itano, Fisher,... (See [Col84]). Some products of distributions can indeed be realized in this way. For instance, we have Example In D (), we get in the sense of Definition x.δ = 1 2 δ Proof. We will sketch some main points. Denote by δ n (x) the function δ n ( x), n = 1, 2,..., one gets ( 1 x δ n).(δ δ n ), φ = for all φ D(). = ( 1 x δ n).δ n, φ = ( 1 x δ n), δ n.φ = 1 x, δ n δ n φ

21 20 The Colombeau theory of generalized functions If we expand φ(x) = φ(0) + xφ (0) + x 2 ψ(x), then ( 1 x δ n).(δ δ n ), φ = φ(0) 1 x, δ n δ n +φ (0) 1 x, δ n (xδ n ) + 1 x, δ n (x 2 ψ)δ n One can prove that the last term in the right hand-side of the above identity tends to 0 as n (See [Ita76] or [Mik66]...) for details). The first term tends to 0 as n because δ n δ n is an even function. Here, we will show that the second term tends to 1 2 φ (0). Indeed, we have φ (0) 1 x, δ n (xδ n ) = where α n = δ n (xδ n ), and it follows that 1 x α n(x) dx, α n = δ n [( x)δ n ] = x(δ n δ n ) + (xδ n ) δ n, since it is obvious that if ψ 1 and ψ 2 are in L 1 (), then x(ψ 1 ψ 2 ) = (xψ 1 ) ψ 2 + ψ 1 (xψ 2 ). Therefore, α n α n = x(δ n δ n ), and 1 x, α n = x, α n + α n x, α n α n = x, α n α n, since α n + α n is even. It follows that 1 x, α n = x, x(δ n δ n ) = 1 2 since δ n D(), n = 1, 2,... and δ n(x) dx = 1. So, we have for all φ D(), or in D (). Therefore, we have (δ n δ n )(x) dx = 1 2 lim ( 1 n x δ n)(δ δ n ), φ = 1 2 φ (0) = 1 2 δ, φ, lim n ( 1 x δ n)(δ δ n ) = 1 2 δ in the sense of Definition However, we have: 1 x.δ = 1 2 δ in D (),

22 Master Thesis Tạ Ngọc Trí Proposition In D (), there does not exist δ 2 in the sense of Definition Note. From this result we can see that this way of defining the product of distributions does not meet what we need! Proof. Conversely, we assume that there exists δ 2 D () in the sense of Definition Taking an arbitrary δ-sequence (δ n ), n = 1, 2,..., we always have the existence of lim n δ2 n, ψ, for all ψ D(). We take ψ D() such that ψ 1 in a neighborhood of 0. Then, δ 2 n, ψ = δ 2 nψ dx = δ 2 n dx. Because of the existence of lim n δn, 2 ψ, we get (δ n ), n = 1, 2,... bounded in L 2 (). It is well-known that in L 2 () the closed unit ball is weakly compact, so there is a subsequence (δ nk ), k = 1, 2,... of (δ n ), n = 1, 2,... which weakly converges to g in L 2 () (we can see this from the result of Problem 18 in [Hal67] that the weak topology of closed unit ball in L 2 () is metrizable). Hence, we have for all ψ L 2 () that g, ψ = lim k δ nk, ψ, so for ψ D(), we have lim k δ nk, ψ = g, ψ. It follows that δ = g L 2 (), and this is a contradiction. We conclude that (δn), 2 n = 1, 2,... never converges in D (). These arguments above are based on Prof. Tom Koornwinder s remarks on Colombeau theory. They clarify the arguments in [Obe92], p.24. There are some other examples in which there do not exist products of distributions like δ 2 above. Please refer to [Obe92] for details. Now we will show another way of defining the product of distributions based on the Fourier transform. We notice that, if u D ( m ) with compact support, then u (ξ) = u(x), e ix.ξ. Denoting by u its converse Fourier transform, we also have u (x) = 1 (2π) m u(ξ), eiξ.x. Also denote by M( m ) all the pairs (u, v) D ( m ) D ( m ) such that for every x m, there is a neighborhood of x, denoted by Ω x, such that

23 22 The Colombeau theory of generalized functions 1. (ωu) (Ψv) is integrable over m for all ω, Ψ D(Ω x ), 2. m (ωu) (Ψv) dx = m (ωv) (Ψu) dx for all ω, Ψ D(Ω x ), 3. m (ωu) (Ψv) dx depends continuously on ω D(Ω x ) for all Ψ D(Ω x ) For such a pair of u, v above, we can define the product of u and v as follows: Definition If (u, v) M( m ), the product of u and v in D (Ω x ), denoted by uv, is defined locally on Ω x, as follows: For any ω D(Ω x ), let Ψ D(Ω x ) with Ψ(x) = 1 on supp ω. Then uv, ω = (ωu) (Ψv) dx. m We remark that with the first condition above, the right-hand side exists. Moreover, it follows from the second that it is independent on the choice of Ψ. Indeed, If Ψ 1, Ψ 2 both play the role of Ψ, we have (ωu) (Ψ 1 v) dx = m (ωψ 2 u) (Ψ 1 v) dx = n (ωv) (Ψ 1 Ψ 2 u) dx m = (Ψ 1 ωu) (Ψ 2 v) dx = m (ωu) (Ψ 2 v) dx. m Therefore, Definition is fully defined. The questions arise what δ 2 and 1 δ are in the sense of this definition, and x what is the relationship between the product in the sense of this definition and the product by regularization and passage to the limit. It can be shown in three following results Proposition In D (), there does not exist δ 2 in the sense of Definition Proof. We take x = 0, ω and Ψ D(Ω 0 ) such that ω(0) = Ψ(0) = 1, and Ψ = 1 on supp ω, where Ω 0 is a neighborhood of 0. Assume that δ 2 M(), so (ωδ) (Ψδ) is integrable in. However, we have (ωδ) = 1, and (Ψδ) = 1 1. It follows that 2π 2π is integrable in. That is an absurdity. So, δ2 / M(). And this concludes our proposition.

24 Master Thesis Tạ Ngọc Trí Proposition In D (), there does not exist 1 δ in the sense of Definition 1.2.5, either. x Proof. We will use the argument of contradiction as follows. If there is 1 x δ, then we take Ω 0, a neighborhood of 0, and ω, Ψ D(Ω 0 ) such that ω(0) = Ψ(0) = 1, and Ψ = 1 on supp ω. We notice that if u D () arbitrarily, then one has (u, δ) M() = (ωu) (Ψδ) dx <, or ωu dx <, since (Ψδ) = 1 2π. Hence, based on the property of the Fourier transform, one has ωu is continuous on. Choosing ω = 1 on a neighborhood of 0, the restriction of u to this neighborhood would be a continuous function. However, obviously this is wrong in our case of u = 1 x. The results of Example and Proposition suggest that conditions in Definition are stricter than the ones in Definition We can see this more clearly by the following result of Tysk in [Tys81], or in [Col84] Proposition If u, v D ( m ) and there exists uv in the sense of Definition 1.2.5, then the product uv also exists in the sense of Definition for all δ-sequences such that δ n (x) 0; and both products are equal. Proof. Please see the references above. We have seen how mathematicians have partly solved the problem of distribution multiplication. Although such those methods sometimes helped scientists to solve some specific problems in nature as well as in theory (See [Obe92]). However, it is clear that our issue has not been fully solved up to now. To look back our issue, we also see that it is easy to arrive at a wrong conclusion if we are not careful with the distribution multiplication, even in what we have constructed. For instance, we have 1 x.x = 1, and x.δ = 0 in D (). If we assume that the ideal operation of multiplication in D ()

25 24 The Colombeau theory of generalized functions which extends the one in D() is associative, we get an absurd conclusion as follows: δ = ( 1 x.x).δ = 1.(x.δ) = 0(!). x Another example is related to the Heaviside function H(x). It is clear that H n = H, n = 2, 3,..., so H = H 2 = H 3. A naive argument gives H = 2H.H = 3H 2.H = 3H.H, hence H = 2H.H = 0 which contradicts that H = δ in D (). L. Schwartz in his theory of distribution (see [Sch54]) cautioned his readers by stating an impossibility result about defining distribution multiplication. This result pointed out some further basic difficulties in trying to construct in easy ways nonlinear extensions of the distributions (See [os87], p.27). This will be the topic of the next subsection Schwartz impossibility result Let A be an algebra containing the algebra C 0 () of all continuous functions on as a subalgebra. Let us assume that the function 1 C 0 () is the unit element in A. Further let us assume that there exists a linear map : A A extending the derivative of continuously differentiable functions and satisfying Leibniz s rule (ab) = a.b + a. b, then 2 ( x ) = 0. We remark from the conclusion of this theorem that we cannot hope to construct an algebra A containing D () such that Leibniz s rule is satisfied. Proof. One gets It follows that, (x x ) = x. x + x. ( x ) = x + x. ( x ). 2 (x x ) = 2. ( x ) + x. 2 ( x ). On the other hand, in C 1 (), and that means in A : (x x ) = 2 x. Therefore, 2 (x x ) = 2. x. It follows that x. 2 ( x ) = 0. Now, we will use the result In A, if xa = 0, then a = 0, so we conclude 2 ( x ) = 0. We only need to verify this last result: We can see that x(log x 1) and x 2 (log x 1) C 1 () by giving 0 as the value of these functions at 0. Using Leibniz s rule in A, we get {x(log x 1)x} = {x(log x 1)}.x + x(log x 1).

26 Master Thesis Tạ Ngọc Trí 25 So, 2 {x(log x 1)x} = 2 {x(log x 1)}.x + 2. {x(log x 1)}. Thus, 2 {x(log x 1)}.x = 2 {x 2 (log x 1)} 2 {x(log x 1)}. But, since coincides with the usual derivation operator on C 1 -functions, and x 2 (log x 1) C 1 (), one gets Therefore, in A we have So, {x 2 (log x 1)} = 2x(log x 1) + x. 2 {x 2 (log x 1)]} = 2. {x(log x 1)} {x(log x 1)}.x = 1. To simplify the notation we set y = 2 {x(log x 1)}, then y.x = 1; thus x.a = 0 = y.(x.a) = 0 = 1.a = 0 = a = 0. This concludes the proof. If there is an algebra A containing D () in which Leibniz s rule is satisfied, then δ = 0, since 2 ( x ) = 2δ in D (). That is an absurdity. This also explains to us why this result called Schwartz impossibility result.

27 Chapter 2 The Colombeau theory of generalized functions In the previous chapter we have seen the need to construct a new theory of generalized functions which should solve some issues raised there. In the 1980 s J. F. Colombeau proposed such a theory, and we will concerned with it in this chapter. 2.1 Preliminaries and notations For q = 1, 2,... and N is the set of all nonnegative integers we denote { } A q = φ D( n ) : φ(t) dt = 1 and t α φ(t) dt = 0 for 1 α q, n n where t = (t 1,..., t n ) n, α = (α 1,..., α n ) N n, and t α = (t 1 ) α 1... (t n ) α n. We can remark that A 1 A 2... A q A q+1... from the defenition of A q s. Moreover, we also get emark A q =. q=1 Proof. Indeed, if we assume that there is φ q=1 A q, then φ is an analytic function in C n. We notice that conditions of φ imply that φ(0) = 1 and α φ(0) = 0, α 1. So, φ(x) = 1, x n. On the other hand, since 26

28 Master Thesis Tạ Ngọc Trí 27 φ A q, q = 1, 2,..., then φ D( n ), and for each m = 1, 2,..., there is C m 0 such that φ(ξ) C m (1 + ξ ) m e a Im ξ, where ξ C n, and a is a positive number such that supp φ {x : x a} ( the Paley- Wiener theorem (Theorem )). In particular, when ξ = x n and ξ = x, we will get φ(x) 0. This is in contradiction to φ(x) = 1, x n. Therefore, we have q=1 A q =. We can also shorten the arguments for the proof above if we use the property of the Fourier transform in S( n ). Indeed, if φ q=1 A q, then φ S( n ), and so does φ, which yields a contradiction with φ(x) = 1, x n. Therefore, we also get q=1 A q =. We also have a very important following result: emark For each q = 1, 2,..., we have A q. Proof. The following proof comes from Prof. Tom Koornwinder s comments on books on Colombeau theory. This shows more clearly and correctly all the arguments for the above results in [Gko01], p.3. It is clear that we only need to verify in case of n = 1. Consider the following continuous linear forms on D() L 0 (ϕ) = ϕ(x) dx n L j (ϕ) = x j ϕ(x) dx, 1 j q n Clearly, (L j ), 0 j q are linearly independent, so for each ϕ we can consider a linear form defined on the finite dimensional linear subspace generated by L 0, L 1,..., L q. Now, we use the Hahn-Banach theorem for extension to a continuous linear functional on D (). Applying the result of Proposition , we have D() is a Montel space and D () = D(), here D () and D () are considered with the strong dual topology. Therefore, there are ψ k D(), k = 0, 1,..., q such that L j (ψ k ) = δ jk for j, k = 0, 1,..., q. Setting ϕ = ψ 0, we have L j (ϕ) = δ j0, or ϕ(x) dx = 1, and x j ϕ(x) dx = 0, 1 j q. n n

29 28 The Colombeau theory of generalized functions This implies that ϕ A q for each q = 1, 2,.... We notice that there is at least one another proof for this remark, but it focuses on technical calculations. Please see [Col85] for details. We denote by φ ɛ (t) the function 1 ɛ n φ(t ɛ ), where φ D(n ) and ɛ > 0. It is not difficult to verify that φ A q if and only if φ ɛ A q. We also denote T x the translation operator, that is the mapping φ φ( x). We also write φ ɛ,x (t) = (T x φ ɛ )(t). 2.2 Definition of Colombeau generalized functions First, we denote by E[ n ] the set of all the functions : A 1 n C, where (φ, x) (φ, x), which are C -functions in x for each fixed φ. It is easy to see that E[ n ] is an algebra with the pointwise operators Definition We say that an element E[ n ] is moderate if for every compact set K of n and every derivation operator α ( α is accepted even for order zero and in that case it is the identity operator), there is an N N such that for all φ A N, we have ( α )(φ ɛ, x) = O(ɛ N ) as ɛ 0, uniformly on K. We denote by E M [ n ] the set of all moderate elements of E[ n ]. Note that N = N(α, K). If we have N = N(α, K) for some N, then we may replace N by any N such that N > N. We remark that if 1 and 2 are elements of E[ n ], then we have α 1 1 α 2 2 and α 2 2 α 1 1 are both O(ɛ N 1 ) for some N 1. So, applying the Leibniz s rule and noticing that the sum of finitely many elements of orders O(ɛ N ) is also of such order, we get that E M [ n ] is a subalgebra of E[ n ]. Denote by { } Γ := β : N + such that if q < r then β(q) β(r) and lim β(q) =, q and we will show the concept of null functions as follows

30 Master Thesis Tạ Ngọc Trí Definition We say that an element of E[ n ] is null, if for every compact set K of n and every derivation operator α ( α is accepted even for order zero and in that case it is the identity operator), there is an N N and β Γ such that for all q N and all φ A q, we have ( α )(φ ɛ, x) = O(ɛ β(q) N ) as ɛ 0, uniformly on K. We denote by I the set of all null elements of E[ n ]. It is obvious that I E M [ n ]. It follows also from the definitions of E M [ n ] and I, and from Leibniz s rule that: if 1, 2 E M [ n ], and at least one of them belongs to I, then 1. 2 I. Therefore, I is an ideal in E M [ n ]. Now we can use the previous definitions in order to define the generalized functions on n Definition The algebra of generalized functions of Colombeau, denoted by G( n )(or G), is the quotient algebra E M [ n ]/I. We remark that f is a generalized function in G( n ) iff f = f + I, where f E M [ n ] is a representative of f. We also say that f = ḡ in G( n ) iff f g I, where f, g are representatives of f, ḡ respectively. From these remarks we see that G( n ) is an associative, commutative algebra. It is obvious that α E M [ n ] E M [ n ] and α I I, α. Therefore, we can define α : G( n ) G( n ) f α f, where α f = α f + I It follows that α is linear, and satisfies Leibniz s rule of product derivatives. Next, we shall show that all classes of ordinary functions such as C ( n ), C 0 ( n ) (continuous functions on n ), and the Schwartz distributions are still elements of G( n ). 2.3 Properties of the differential algebra G( n ) We will successively study the embeddings of C ( n ), C 0 ( n ) and D () into G( n ). In each case we will give the representatives in G( n ) of the corresponding functions or distributions.

31 30 The Colombeau theory of generalized functions Theorem An embedding C ( n ) G( n ) is defined by the mapping f C ( n ) f + I G( n ), where f(φ, x) = f(x), φ A 1 and x n. Proof. First, we show that the above mapping is well defined. Indeed, since f C ( n ), f(φ, ) = f( ) C ( n ). On the other hand, α f(φɛ, x) = α f(x), φ A 1, x n, and for f C ( n ), there is sup K α f(φɛ, x) = sup α f(x) = c < for an arbitrary compact K. K So, we have for each multi-index α, sup α f(φɛ, x) c c K ɛ, φ A 0 1 and 0 < ɛ < 1. Therefore, f E M [ n ], and it follows that f + I G( n ). We also see that if f C ( n ) and its image f + I = I, then it follows from the definition of I that for each compact K n we have f(x) = O(ɛ) as ɛ 0 uniformly on K. So, f = 0 on K, and because K is arbitrary, we get f = 0. Therefore, the above mapping is injective, and we have the embedding as a conclusion. To conclude this subsection, we remark from the result of the above theorem that in the C ( n )-case the derivative operators α on G( n ) coincide with the usual partial derivatives of functions when we consider them in C ( n ) Theorem An embedding C 0 ( n ) G( n ) is defined by the mapping f C 0 ( n ) f + I G( n ), where f(φ, x) = f(x + y)φ(y) dy = n f(y)φ(y x) dy, φ A 1, x n. n Proof. We shall show that f E M [ n ]. Indeed, since φ A 1, we have f(φ, ) C ( n ) and α f(φ, x) = ( 1) α n f(y) α φ(y x) dy, where α is the order of α

32 Master Thesis Tạ Ngọc Trí 31 Furthermore, one has f(φ ɛ, x) = f(y)φ ɛ (y x) dy n = 1 f(y)φ( y x ) dy = f(x + ɛt)φ(t) dt ɛ n ɛ n n It follows that α f(φɛ, x) = ( 1) α ɛ α +n f(y)( α φ)( y x ɛ n ) dy = ( 1) α ɛ α n f(x+ɛt) α φ(t) dt We notice that for arbitrary compact K and multi-index α, we have sup f(x+ɛt) α φ(t) dt sup f(x+δy) α φ(t) dt = c <, K n x K, y supp φ, δ [0,1] n and does not depend on ɛ, 0 < ɛ < 1 for all φ A 1. Therefore, we have f E M [ n ]. On the other hand, we can show that if f C 0 ( n ) and f I, then f = 0. Indeed, we notice that: Lemma For all f C 0 ( n ), we have lim ɛ 0 f(φɛ, x) = f(x). Proof. We can write lim ɛ 0 f(φɛ, x) = lim ɛ 0 n f(x + ɛt)φ(t)dt. Now, we notice that φ A 1, f is continuous. So, we can apply the dominated convergence Lebesgue theorem we get the conclusion above. If f I, then for all compact K, there exists N N such that for all φ A N, we have f(φ ɛ, x) = O(ɛ) as ɛ 0, uniformly on K. Hence, apply the result of Lemma above, we have f(x) = 0 on K. It follows that f = 0, and this concludes the proof of the above theorem. We notice that C 0 ( n ) is included in E M [ n ] as a linear subspace, not a subalgebra. It follows that C 0 ( n ) is not a subalgebra of G( n ), either. In fact, in general we have f(x + ɛt)φ(t) dt. n g(x + ɛt)φ(t) dt n f(x + ɛt)g(x + ɛt)φ(t) dt. n We also see another issue to arise: if f C 0 ( n ), then we can apply both Theorem and Theorem 2.3.2, so what happens? Nothing at all!

33 32 The Colombeau theory of generalized functions because we can show that f f I, so both f and f are representatives of f C ( n ). For convenience, we will show this in case of n = 1. Indeed, we have ( f f)(φ, x) = f(x) f(x + y)φ(y) dy. Therefore, one gets ( f f)(φ ɛ, x) = f(x) f(x + ɛt)φ(t) dt = [f(x + ɛt) f(x)]φ(t) dt. Since f C (), we can apply Taylor s formula up to order q to f at the point x, and we get f(x + ɛt) f(x) = q (ɛt) k f (k) (x) + ɛ q+1 t q+1 k! (k + 1)! f (q+1) (x + θɛt), k=1 where 0 < θ < 1. Hence, for arbitrary compact K, and for all q N, φ A q we have ( f f)(φ ɛ, x) = O(ɛ q+1 ) as ɛ 0, uniformly on K. This fits with Definition for α = 0, N = 0 and β(q) = q +1. It is similar to estimate α ( f f)(φ ɛ, x), so f f I. Finally, we talk about the embedding of D ( n ) into G( n ) Theorem An embedding D ( n ) G( n ) is defined by the mapping u D ( n ) ū + I G( n ), where ū(φ, x) = u(y), φ(y x), φ A 1 and x n. Proof. Set φ (t) = φ( t) for φ D( n ), then ū(φ, x) = (u φ )(x) (Theorem ). So, we have ū(φ, x) C ( n ). Also it follows from the above formula that α ū(φ, x) = ( 1) α u(y), ( α x φ)(y x). To prove that ū(φ, x) E M [ n ] and the above mapping is injective, we can refer to two solutions: the first one based on Theorem in [Col84], pp , and the other one in [os87], pp , which is based on the compact support of φ. To conclude we notice that this theorem is an extension of Theorem in case of continuous functions.

34 Master Thesis Tạ Ngọc Trí 33 It follows that the δ-dirac distribution has the form f δ + I G( n ), where f δ (φ, x) = δ(y), φ(y x) = φ( x). Also, we have the representative of δ 2 in G( n ) is φ 2 ( x) + I. To end this subsection we would like to make some historical notes. In fact before Schwartz introduced distributions as they are now, he had tried an approach starting from the convolution operator. He defined an operator T mapping D( n ) into the space E( n ) of C -functions with arbitrary support such that for continuous f, the corresponding operator actually maps φ D( n ) to the convolution of f and φ. However, later in the process of defining the product of a C -function with such an operator T, as well as defining the Fourier transform, he saw that this method was not very convenient. After that he turned to the definition as we know nowadays. However, it is the original realization of a distribution as a convolution operator which was extended in Colombeau s theory (with D( n ) being replaced by A 1 ). Please refer to [Sch01] for further details. The following subsection will show the more extended results on the representatives of elements in G( n ). 2.4 Nonlinear properties of G( n ) We notice that not only multiplication makes sense in G( n ), but also the more general nonlinear operations can do. To show that, firstly we consider a class of slowly increasing functions at infinity in the following definition: Definition A function f : n C is said to be slowly increasing at infinity if there are c > 0, N N such that f(x) c(1 + x ) N, x n. The set of all C ( n )-functions which together with all their derivatives are slowly decreasing at infinity is denoted by O M ( n ). Some special properties of functions in this class make them play an important role when we study the Fourier transform theory in Schwartz distribution theory (See [Fri98], pp , for example). Polynomials, or e ix are examples of slowly increasing functions at infinity, but e x is not. The next result, see [Col85], pp for details, will show us the issue we want to concern in this section.