Distributions

12 Key excerpts on "Distributions"

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  Mathematical Analysis tools for engineering

  • Franco Tomarelli(Author)
  • 2019(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Chapter 8 Distributions or Generalized Functions A distribution is a function in a generalized sense: this extension goes further beyond the notion of function defined almost everywhere in the Lebesgue sense, which was introduced for an L p function in subsection 5.2.2. A distribution can be computed at no value of the independent variable. Nevertheless a distribution keeps the most important features of a function. The key point is that it is possible to evaluate the action of a distribution on a compactly supported smooth function (test functions): such evaluation, called duality pairing , or pairing for short, mimics the action of devices for instantaneous measurements of physical quantities. The pairing of a distri-bution with a test function has similar formal properties as the integration of the product of an integrable function times a smooth function. On the other hand, the Distributions provide an handy quantitative model of several natural phenomena, like impulsive forces, concentrated sources or electrical dipoles, which can be satisfactorily described neither by classical functions, nor by functions defined almost everywhere. Moreover, the distri-butions have derivatives of every order in a suitable weak sense, that are called distributional derivatives or weak derivatives, to mark the di ↵ erence from classical derivatives, pointwise defined as limit of incremental quotients. limit. First informal examples of Distributions appeared as a natural description of physical phenomena notably in electrical circuits analysis, then they were given an e ↵ ective and rigorous theoretical framework by the work of Laurent Schwartz (1915–2002). 8.1 Distributions We recall the notation of multi-indexes, see Definition 2.2, for representing partial derivatives 334 8 Distributions or Generalized Functions D ↵ x = D ↵ 1 1 D ↵ 2 2 · · · D ↵ n n = ✓ @ @ x 1 ◆ ↵ 1 ✓ @ @ x 2 ◆ ↵ 2 · · · ✓ @ @ x n ◆ ↵ n , and begin by introducing some formal definitions.

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  Analytic Functions Integral Transforms Differential Equations

  • Filippo Gazzola, Franco Tomarelli, Maurizio Zanotti(Authors)
  • 2023(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Chapter 3 Distributions Distributions or generalized functions are mathematical objects which differ from the elementary functions in the sense that they cannot be computed pointwise for every value of the independent variable; nevertheless they keep many typical features of functions. The main issue is that one can evaluate the action of any distribution on compactly supported smooth functions (test functions): such evaluation, called duality pairing (pairing for short; dualit`a in Italian and crochet in French), provides an operative and quantitative description of natural phenomena and of devices for instantaneous measure- ments (snapshots of physical quantities). The pairing of a distribution with a test function has formal properties analogous to the integration of the prod- uct of an integrable function times a smooth function. Moreover Distributions have all derivatives in a suitable weak sense (distributional derivatives); this property allows to work with Distributions as solutions or as given source terms of differential equations: for instance we can study Maxwell’s system of equations for electromagnetic field when concentrated charges, dipoles and/or currents in thin conducting structures (wires or metal foils) are present. Historically, Distributions showed up first as a natural description of physi- cal phenomena notably in electrical circuits analysis, subsequently they were given an effective and rigorous theoretical framework by the work of Laurent Schwartz (1915–2002) in the Forties of the XXth Century . 3.1 Definitions and tools We use the following notation: R n is the n-dimensional Euclidean space; any point x ∈ R n is represented by its coordinates x = (x 1 , x 2 , · · · , x n ) with respect to the canonical basis {e 1 , e 2 , · · · , e n }; the Euclidean norm of x ∈ R n is |x| = (x 2 1 + x 2 2 + · · · + x 2 n ) 1/2 ∀x ∈ R n .

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  Operational Calculus and Related Topics

  • A. P. Prudnikov, K.A. Skórnik(Authors)
  • 2006(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 3 Generalized Functions 3.1 Introduction Distributions are a generalization of locally integrable functions on the real line, or more generally, a generalization of functions that are defined on an arbitrary open set in the Euclidean space. Distributions were introduced as a result of difficulties with solving some problems of mathematical physics, quantum mechanics, electrotechnics and so forth. In these domains there are many theoretical and practical problems where the notion of function is not sufficient in this or that sense. In 1926 the English physicist Paul Dirac introduced a new element of mathematical formalism into quantum mechanics, [Dir]. He named it the Delta function and denoted it by δ ( t ). Dirac assumed that the delta function is defined on the real line and fulfills the following conditions: δ ( t ) = 0 for t = 0 , + ∞ for t = 0 , (3.1.1) and + ∞ -∞ δ ( t ) dt = 1 . (3.1.2) In the theory of real functions the two conditions (3.1.1) and (3.1.2) are contradictory. No real function exists to fulfill both conditions at the same time. On the other hand, both conditions give incorrect but highly convincing evidence of a physical intuition: δ ( t ) represents an infinitely large growth of electric tension in the infinitely short time where a unit of electricity loads. Nevertheless, the existence of mathematical models for which the search for mathematical description would lead naturally to Dirac’s deltas should not provide an excuse to use the imprecise mathematical notion that hides under δ ( t ), treat it as a function and at the same time assume that it fulfills conditions (3.1.1) and (3.1.2). Despite all formal objections many important results were achieved with the help of Dirac’s delta. In the 1930s it became obvious that Dirac’s delta has a fixed place in theoretical physics. As a result scientists sought a new mathematical theory that would help define Dirac’s

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  Invitation to Partial Differential Equations

  • Mikhail Shubin, Maxim Braverman, Robert McOwen, Peter Topalov, Maxim Braverman, Robert McOwen, Peter Topalov(Authors)
  • 2020(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 4 Distributions 4.1. Motivation of the definition. Spaces of test functions In analysis and mathematical physics one often meets technical difficulties caused by the nondifferentiability of certain functions. The theory of dis-tributions (generalized functions) makes it possible to overcome these diffi-culties. The Dirac δ -function, which naturally appears in many situations (one of them was mentioned in Section 3.4), is an example of a distribution. In the theory of Distributions a number of notions and theorems of anal-ysis acquire simpler formulations and are free of unnatural and irrelevant restrictions. The origin of the notion of distribution or generalized function may be explained as follows. Let f ( x ) be a physical quantity (such as temperature, pressure, etc.) which is a function of x ∈ R n . If we measure this quantity at a point x 0 with the help of some device (thermometer, pressure gauge, etc.), then we actually measure a certain average of f ( x ) over a neighbor-hood of x 0 , that is, an integral of the form f ( x ) ϕ ( x ) dx , where ϕ ( x ) is a function characterizing the measuring gadget and smeared somehow over a neighborhood of x 0 . The idea is to completely abandon the function f ( x ) and consider instead a linear functional by assigning to each test function ϕ the number (4.1) f, ϕ = f ( x ) ϕ ( x ) dx. Considering arbitrary linear functionals (not necessarily of the form (4.1)) we come to the notion of a distribution or generalized function . 57 58 4. Distributions Let us introduce now the needed spaces of test functions. Let Ω be an open subset of R n . Let E (Ω) = C ∞ (Ω) be the space of smooth (infinitely differentiable) functions on Ω; let D (Ω) = C ∞ 0 (Ω) be the space of smooth functions with compact support in Ω, i.e., functions ϕ ∈ C ∞ (Ω) such that there exists a compact K ⊂ Ω with the property ϕ | Ω K = 0. In general, the support of ϕ ∈ C (Ω) (denoted by supp ϕ ) is the closure (in Ω) of the set { x ∈ Ω : ϕ ( x ) = 0 } .

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  A Friendly Approach to Functional Analysis

  0

  • Amol Sasane(Author)
  • 2017(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  Chapter 6

  A glimpse of distribution theory

  In this last chapter of the book, we will see a generalisation of functions

  and their ordinary calculus to “Distributions” or “generalised functions”. These Distributions will be “continuous linear functionals on the vector space of test functions D (R

  d

  )”:

  Why study Distributions? We list three main reasons:

  (1)To mathematically model the situation when one has an impulsive force (imagine a blow to an object which changes its momentum, but the force itself is supposed to act “impulsively”, that is the time interval when the force is applied is 0!). Similar situations arise in other instances in mathematics and the applied sciences.

  (2)To develop a calculus which captures more general situations than the classical case. For example, what is the derivative of |x| at x = 0? It will turn out that this is also useful to talk about weaker notions of solutions of Partial Differential Equations (PDEs).

  (3)To extend the Fourier transform theory to functions that may not be absolutely integrable. For example, what1 is the Fourier transform of the constant function 1?

  It turns out that the theory of Distributions solves all of these three problems in one go. This seems like a miracle, and naturally there is a price to pay. The price is that everything classical is now replaced by a weaker notion. Nevertheless this is useful, since it is often sufficient for what one wants to do. An example is that, as opposed to functions on R , which have a well-defined value at every point x ∈ R , we can no longer talk about the value of a distribution at a point of R .

  Let us elaborate on reason (2) above, in the context of PDEs. An example we met earlier in Exercise 3.17, page 143, is that of the wave equation (which describes the motion of a plucked guitar string), and we had checked that for a twice continuously differentiable f

  gives a solution with the initial condition described by f (and zero initial velocity). However, when we pluck a guitar string, the initial shape needn’t be C2

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  Applied Functional Analysis

  • D.H. Griffel(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Thus the class of Distributions contains objects which correspond to ordinary functions as well as singular Distributions which do not. In the next section we shall define operations on Distributions analogous to the operations of ordinary algebra and calculus applied to functions. This will justify calling Distributions ‘generalised functions’, and will allow us to use Distributions for most of the purposes for which ordinary functions are used.

  1.3 Operations on Distributions

  Definition 1.16 (Addition) If ƒ and g are Distributions and a and b are complex numbers, we define the distribution aƒ + bg to be the functional φ a 〈ƒ ,φ 〉 + b 〈g ,φ 〉 for all φ in .

   

  Strictly speaking we should verify that this defines a linear continuous functional on , but that is very easy and is left to the reader. The same applies to other definitions in this section.

   

  Definition 1.17 (Multiplication) If ƒ is a distribution and h is a smooth function (cf. Definition 1.2), we define the product of ƒ and h to be the distribution hƒ : φ 〈ƒ ,h φ 〉 for all φ in .

   

  Note that if φ is a test function and h is smooth, then h φ is a test function, and therefore 〈ƒ ,h φ 〉 is well-defined. If h is not smooth, then neither is h φ , hence h , and Definition 1.17 does not work. Thus we cannot define the product of a distribution with a function which is discontinuous or has a discontinuous derivative. Distributions cannot in general be multiplied. The difficulty can be seen in the following

   

  Example 1.18 The Heaviside function H is defined by

  and an almost identical function H 1 is defined by

  H and H 1 are locally integrable, and they generate the same regular distribution . There is no distinction between H 1 and H in generalised function theory, and this reflects the fact that from the point of view of the physicist the distinction between them is artificial: one could never distinguish between H 1 and H experimentally. If Definition 1.17 were extended to discontinuous functions, we would have 〈H 1 δ ,φ 〉 = 0, Now, any definition of the product of two Distributions would have to agree with Definition 1.17 in the case that one of the Distributions was regular. Since H 1 and H generate the same distribution

  H ,

  we would have two different values for 〈

  H δ

  ,φ 〉, and thus an inconsistent theory. This shows that it is impossible to define the product of δ

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  Foundations of Image Science

  • Harrison H. Barrett, Kyle J. Myers(Authors)
  • 2013(Publication Date)
  • Wiley-Interscience

    (Publisher)

  theory of Distributions, originally developed by Laurent Schwartz (1950). We give a brief summary of this theory in the next section, but then make use of all three approaches to delta functions in what follows. The hope is that this strategy will allow delta functions to be understood at several different levels of rigor. The reader who is content with a less rigorous development can jump to Sec. 2.2 without loss of continuity.

  2.1 THEORY OF Distributions

  2.1.1 Basic concepts

  In brief, a distribution is a linear, continuous functional that maps a function t(x) to a real or complex number. For simplicity, we assume initially that t(x) is a real, scalar-valued function of a single real variable x. We have seen in Chap. 1 that the general form of a bounded, continuous, linear functional on a Hilbert space is given by the Riesz representation theorem, (1.24) , as

  (2.1)

  where g(x), known as the kernel of the functional, must lie in the Hilbert space (Gohberg and Goldberg, 1981, p. 61). If the Hilbert space is 2 , g(x) must be square-integrable.

  If we give up on the requirement that the kernel lie in a Hilbert space, we can define a wider class of functional known as Distributions. When we do so, we shall always write the distribution in the Riesz form, like (2.1) , but then g(x) need not be a square-integrable function or even a function at all in the conventional sense. All we have to do is specify the mapping rule of the functional, and that, in turn, will give meaning to the kernel function, which we then term a generalized function.

  As a simple example, consider the function g(x) = 1/x. We might be tempted to define a functional using this kernel as

  (2.2)

  but this integral is not well defined (unless t(0) = 0) because of the singularity at x

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  Delta Functions

  Introduction to Generalised Functions

  • R F Hoskins(Author)
  • 2009(Publication Date)
  • Woodhead Publishing

    (Publisher)

  As a result we have only been able to define a restricted sub-class of the generalised functions introduced by Laurent Schwartz and called by him Distributions. We have only discussed what are properly described as Distributions of finite order. Thus, to begin with, we dealt with regular Distributions (those defined by ordinary functions and integration processes) and delta functions: these were all identified as linear functionals which are relatively bounded on D with respect to the uniform norm || f ||, and they are accordingly referred to as Distributions of order 0. Next we have Distributions like the first derivative δ ′ of the delta function, which are linear functionals relatively bounded on D with respect to the norm || f || (1), and not with respect to the uniform norm || f || itself: these are similarly described as Distributions of order 1. And so on, with Distributions of order p, where p ∈, being defined as just those linear functionals which are relatively bounded on D with respect to the norm || f || (p), and not with respect to any norm ||| f || (q) where q < p. This restriction is not of any immediate major importance: Distributions of finite order are generally the most widely applicable and in practice will be found to suffice for most practical purposes. But for completeness we need to give here some account of the full scope of the Schwartz theory, as well as of its limitations and of some of the subsequent modifications and extensions of the theory which have been proposed. 8.4.2 Schwartz Distributions Consider first the problem of defining general Schwartz Distributions - that is, to define the class of all continuous linear functionals on D, including “Distributions of infinite order”. Briefly, suppose we take the norm || f || (p) defined by equation (8.5), and the corresponding concept of p -convergence, and then allow p to tend to infinity

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  Theories of Generalised Functions

  Distributions, Ultradistributions and Other Generalised Functions

  • R F Hoskins, J S Pinto(Authors)
  • 2005(Publication Date)
  • Woodhead Publishing

    (Publisher)

  18 CHAPTER 1. INTRODUCTION TO Distributions and D^iyf{x) = D^+''f{x) . The general definition for Distributions in Μ* can be framed in the following way: the space Τ>{Έί^) is the linear space of all infinitely differentiable functions φ{χ), defined on IR* and taking real or complex values, which have compact support. Convergence in this space (which is equivalent to what we have called a;-uniform convergence over compact subsets in the case m = 1) is defined thus: the sequence (<^TI )n€N C D(1R*) is a null-sequence if and only if the following conditions hold: (1) there exists a ball Β{0,ε) = {χ e IR* : (ΣΓ =ι 4)^ < ^} such that each of the functions ψη{χ) vanishes identically outside Β{0,ε), (2) lim„_oo{supj.gB(o,e) |£>'VJ„(x)|} = 0 for every multi-index r of dimension m . Condition (2) ensures that each sequence (jD '^Vn)„gN converges uniformly to zero. A sequence {φη)ηζΚ converges to the limit ψ in the sense of the space 2>(IR*) if and only if {φ -ψη)ηζκ * null-sequence. A distribution on IR* is a linear functional μ on ^(ΠΙ*) which is continuous in the sense that lim <μ,ψη>= lim μ{ψη) = 0 η—>oo η—»oo for every null-sequence (¥'n)neN in ^(ΠΙ*). 1.2.2 Topological vector spaces The theory of topological vector spaces is a generalisation of the relatively familiar material associated with normed linear space theory and we begin by reviewing such basic topological concepts as are necessary. In ordineiry Euclidean space IR* the distance between two points χ and y is given by d{x,y)= Y ^ixk-ykf . (1.23) For any given point χ G IR* we can then define the open ball B{x, ε) of radius ε > 0 as the set of all points y whose distance from χ is less than ε: B(x,£) = {j/G ]R*:d(x,y)<5} . (1.24) A subset G of IR* is said to be open if and only if for each point χ G G there exists an open ball B(x, ε) which is contained in G. A subset F of IR* is said to be closed if and only if its complement F*^ = JliCF is open.

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  Distribution Theory

  With Applications in Engineering and Physics

  • Petre Teodorescu, Wilhelm W. Kecs, Antonela Toma(Authors)
  • 2013(Publication Date)
  • Wiley-VCH

    (Publisher)

  According to Theorem 1.5, f 2 S 0 (R). 1.3 Operations with Distributions 1.3.1 The Change of Variables in Distributions In geometry, mechanics and mathematical analysis the transformations of inde- pendent variables are frequently used [13], so as to simplify the calculations and interpretation of the results. These changes, applied to functions, lead to new functions. The methodology of these changes of variables can be extended from functions to Distributions. Let T W R n ! R n be an application defined by the relation x D h( u), hence x i D h i ( u 1 , . . . , u n ) , i D 1, n , (1.117) which represents a transformation from Cartesian coordinates ( x 1 , x 2 , . . . , x n ) 2 R n to the coordinates ( u 1 , u 2 , . . . , u n ) 2 R n . We see that the functions h i , i D 1, n, are of class C 1 (R n ) and that the punc- tual transformation is bijective. Therefore, the transformation (1.117) allows for the inverse punctual transform T 1 W R n ! R n , defined by the formula u D h 1 ( x ) , u i D h 1 i ( x 1 , . . . , x n ) . (1.118) The Jacobians of the transformations T and T 1 are @( x )/@( u) and @( u)/@( x ) for which we have @( x )/@( u) D (@( u)/@( x )) 1 , (@( u)/@( x )) ¤ 0. To see how to approach the definition of the change of variables, we shall con- sider the case of a locally integrable function which can be identified with a distri- bution of function type. Let f 2 L 1 loc (R n ) and ' 2 D(R n ). We have ( f ( h( u)), '( u)) D Z R n f ( h( u))'( u)du D Z R n f ( x )'( h 1 ( x )) ˇ ˇ ˇ ˇ @( u) @( x ) ˇ ˇ ˇ ˇ dx . (1.119) 32 1 Introduction to the Distribution Theory This equality can be transcribed as ( f ( x ), ψ( x )) D ( f ( h( u)), '( u)) , (1.120) where ψ( x ) 2 D(R n ), and has the expression ψ( x ) D '( h 1 ( x )) ˇ ˇ ˇ ˇ @( u) @( x ) ˇ ˇ ˇ ˇ .

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  Distributions and Fourier Transforms

  • William F. Donoghue(Author)
  • 1969(Publication Date)
  • Academic Press

    (Publisher)

  PROOF: We first note that the distribution T() is of order 0, for if it were of order N > 0, the terms T(n-k) would be of order N - k, and thus all terms in the equation except the first would be of order at most N - 1, whence T') of order N - 1, a contradiction. Since this is so, on any compact sub- 22. Distributions IN ONE DIMENSION 107 interval the terms T(n-k) are all functions of bounded variation for k 2 1, and therefore T') is the sum of a continuous function and one of bounded varia- tion on any such subinterval, in particular, T') is bounded and T(-') is Lipschitzian on such subintervals, and therefore continuous. Thus T() finally appears as a continuous function. This theorem, as well as the previous ones, shows that there is nothing essentially to be gained by the use of Distributions for the study of ordinary differential equations. The situation is quite different when the equations to be studied are partial differential equations, however. We should also remark that there exist theorems showing that a distribution Tin more than one vari- able is locally of the form P(D)S, where P (D) is a differential operator with constant coefficients, and f ( x ) is a continuous function. We do not give the proof, although it is not hard, since the theorem would serve only to show that the class of Distributions will arise naturally when we seek a class containing the continuous functions and closed under differentiation. We consider finally the division problem in one dimension. In general, given a distribution T and a Cm-function a(x), the division problem is the problem of finding a distribution S such that US = T. If the function a(x) never vanishes, an obvious solution is (l/a)T and this solution is unique. On the other hand, when a(x) does have zeros, a solution S, if it exists, cannot be unique, since any measure p supported by the zero set of a(x) satisfies the equation up = 0.

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  Lecture Notes on Applied Analysis

  • Roderick Wong(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  Chapter 6 DISTRIBUTION THEORY 6 . 1 . THE DIRAC DELTA FUNCTION In many books on applied mathematics and mathematical physics, one will encounter the function δ ( x ) defined by the equations δ ( x − a ) = 0 , x = a, (6 . 1 . 1) and ∞ −∞ φ ( x ) δ ( x − a ) dx = φ ( a ) , a ∈ R , (6 . 1 . 2) for any continuous function φ ( x ) on R ; see, e.g., [2, p.83], [4, p.152], [5, p.183] and [13, p.288]. However, mathematically, these two equations are inconsistent in the classical sense of a function and an integral, since the value of the integral of a function which is zero everywhere except for a fi-nite number of points should be zero. The function given in (6 . 1 . 1) − (6 . 1 . 2) is known as the Dirac delta function , and it was introduced by Dirac in his theory of quantum mechanics. There are now two mathematically mean-ingful approaches to help interpret the delta function δ ( x ). One approach is to use Functional Analysis, and to consider δ a := δ ( x − a ) as a continuous linear functional acting on a space of smooth functions; the action of δ a on a particular function φ ( x ) is given by the value φ ( a ). This approach led to the Schwartz’s theory of Distributions; see [15, p.141] and [18, p.77]. The other approach is to find a sequence of functions δ n ( x − a ) such that lim n →∞ ∞ −∞ δ n ( x − a ) φ ( x ) dx = φ ( a ) , a ∈ R . (6 . 1 . 3) This approach was introduced by Temple [21], and followed by Lighthill [13] and Jones [9]. The sequence δ n ( x ) satisfying (6.1.3) is called a delta sequence , and we write, symbolically, lim n →∞ δ n ( x − a ) = δ ( x − a ) , x ∈ R . (6 . 1 . 4) 247 248 LECTURE NOTES ON APPLIED ANALYSIS It seems that the second approach is more acceptable to physicists and applied mathematicians. But it was the first approach that has made a greater impact in analysis, and especially in the theory of partial differential equations; see, e.g., [8] and [20].

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