DOCSLIB.ORG

Some Facts About Generalized Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Facts About Generalized Functions Appendix A Some Facts About Generalized Functions We recall here some basic facts and formulas from the theory of generalized func- tions. There are many mathematical textbooks on this subject. Physicists may find useful concise texts, for example Chaps. 2÷4 in [12], or Chaps. 2÷3 in [11]. Com- prehensive introduction to the subject can be found in [2]. The generalized functions that appear in field theory are of the so called Schwartz class, denoted as S(Rn) or S∗(Rn). The reason is that all such generalized functions have a Fourier transform. The generalized function of the Schwartz class1 (g. f.) is, by definition, a linear and continuous functional on the Schwartz space of functions, denoted as S(Rn). Elements of S(Rn) are called test functions. They are complex ∞ valued functions on Rn of the C class. Moreover, it is assumed that such functions, and all their derivatives, vanish in the limit |x|= (x1)2 + (x2)2 + ...+ (xn)2 → ∞, also when multiplied by any finite order polynomial in the variables x1,...,xn. Here x denotes arbitrary point in Rn and x1,...,xn are its Cartesian coordinates. The space S(Rn) is endowed with a topology, but we shall not describe it here. 2 Examples of test functions from the space S(R1) include e−ax and 1/ cosh(ax), where a > 0 is a real constant. On the other hand, (1 + x2)−1 is not test function from S(R1). The value of a generalized function F ∈ S∗(Rn) on a test function f ∈ S(Rn) is denoted in mathematical literature as F(x), f (x), but in physics the most popular is the misleading notation dn xF(x) f (x), e.g., dn x δ(x) f (x) in the case of the Dirac delta. One should keep in mind that the integral here is merely a symbol that replaces , from the mathematical notation—it is not the true integral. It may hap- pen however, that a generalized function is represented by an ordinary function F(x) such that the true integral dn xF(x) f (x) exists for all f ∈ S(Rn). Such F is called regular g.f. For example, the step function(x), x ∈ R1, is a regular generalized ∗( 1) ∞ ( ) ( ) = ∞ ( ) function from S R , because the integral −∞dx x f x 0 dx f x exists for every f ∈ S(R1). We show the integration range when we deal with the true integral. The Dirac delta δ(x) is the prominent example of non regular g.f. In the mathematical notation its definition has the form δ(x), f (x)= f (0). 1 Other names are also used: distribution for generalized function, and tempered distribution for generalized functions of the Schwartz class. H. Arod´z, L. Hadasz, Lectures on Classical and Quantum Theory of Fields, 343 DOI 10.1007/978-3-642-15624-3, C Springer-Verlag Berlin Heidelberg 2010 344 A Some Facts About Generalized Functions The derivative ∂i F of a g.f. F is defined as follows: n n d x ∂i F(x) f (x) =− d xF(x)∂i f (x) for all test functions f . One should remember that this is the definition, and not the formula of integration by parts. For example, d(x) ∞ df(x) ∞ df(x) dx f (x) =− dx (x) =− dx = f (0), dx −∞ dx 0 dx hence (x) = δ(x). Derivative of g.f. always exists and is a generalized function. ˜( ) = ( π)n/2 n ( ) ( ) The Fourier transform f k 2 Rn d x exp ikx f x of a test function f also is a test function from the space S(Rn). The g.f. F˜(x) ∈ S∗(Rn) such that for every f ∈ S(Rn) dn x F˜(x) f (x) = dnkF(k) f˜(k), is called the Fourier transform of the g.f. F. It exists for any F ∈ S∗(Rn). The oper- ation of taking the Fourier transform is continuous with respect to F. This property is used in order to facilitate computation of the Fourier transform of (x)—we first compute the Fourier transform of e−εx (x), where ε>0, and take the limit ε → 0+ at the end.2 Thus, ∞ −ε dx (˜ x) f (x) = lim dke k(k) f˜(k) ε→0+ −∞ ∞ ∞ ∞ 1 −ε i 1 = lim √ dk dx eikx k f (x) = lim √ dx f (x). ε→0+ 2π 0 −∞ ε→0+ 2π −∞ x + iε The r.h.s. of this formula defines the g.f. denoted as √i 1 . Therefore, 2π x+i0+ i 1 (˜ x) = √ . 2π x + i0+ 1 ε> Note that the g.f. x+iε is regular if 0. One can prove that 1 1 = P − iπδ(x), x + i0+ x 1 where the principal value distribution P x is defined as 2 The notation ε → 0+ means that ε = 0 is approached from the side ε>0. A Some Facts About Generalized Functions 345 1 −ε f (x) ∞ f (x) dx P f (x) = lim dx + dx x ε→0+ −∞ x ε x (it is not regular). The result for (˜ x) obtained above is often written in the form ∞ 1 dp eipx = iP + πδ(x). 0 x The form of a generalized function can be probed only with test functions. Because there is no test function with support consisting of just a single-point, it is not possible to tell what is the value of g.f. at a given point. One can however ( ) ∈ n check whether a g.f., say F x , is constant in a vicinity Vx0 of a point x0 R —it is sufficient to show that all the first derivatives of F(x) vanish in that vicinity, i.e., that ∂i F(x) f (x) = 0 ⊂ n δ( ) ∈ for every test function that has its support in Vx0 R . For example, for x S∗(R1) one may say that δ(x) = 0oneveryinterval(a, b) that does not contain 0, and that δ(x) = 0atx = 0, but not that δ(1) = 0. A consequence of the lack of definite value at a single point is that there is no general definition of product of generalized functions. We know the gener- ∗ n ∗ n alized functions F1(x) ∈ S (R ), F2(x) ∈ S (R ) if we know the values of n n n d xF1(x) f (x), d xF2(x) f (x) for every f ∈ S(R ). It is not possible to n infer from this what values should have d xF1(x)F2(x) f (x).Onlyinsome special cases, e.g., for certain regular generalized functions, such product can be defined. In particular, there is no problem with multiplication by an ordinary func- tion ψ(x), provided that ψ(x) f (x) ∈ S(Rn) for every f ∈ S(Rn). Then, the prod- uct ψ(x)F(x) is the g.f. defined by the formula dn x (ψ(x)F(x)) f (x) = dn xF(x)(ψ(x) f (x)). For example, if k is a fixed real number, eikxδ(x) is a generalized function, while xaδ(x) with non integer constant a > 0 is not (not all functions xa f (x) belong to S(R1) because of the problem with derivatives at x = 0). On the other hand, there is no difficulty with a product of generalized functions ∗ ∗ with different arguments. If F(x) ∈ S (Rn) and G(y) ∈ S (Rm), then we know dn xF(x) f (x) and dm yG(y)g(y) for all f ∈ S(Rn), g ∈ S(Rm). The general- ized function H(x, y) = F(x)G(y) ∈ S∗(Rn+m) is defined by its action on the test functions h(x, y) ∈ S(Rn+m) of the form h(x, y) = f (x)g(y), namely dn xdm yH(x, y)h(x, y) = dn xF(x) f (x) dm yG(y)g(y). 346 A Some Facts About Generalized Functions Such factorized test functions f (x)g(y) form a subset of S(Rn+m) that is sufficiently large to uniquely determine H(x, y) on the whole space S(Rn+m).An example: if x, y ∈ R1 are independent variables, then δ(x)δ(y) ∈ S∗(R2). Finally, let us consider the question whether ∞ 0 dx δ(x) =? dx δ(x) + dx δ(x), 0 −∞ or, in a more meaningful form, whether ? δ(x) = 1δ(x) = ((x) + (−x))δ(x) = (x)δ(x) + (−x)δ(x). The answer is that such a formula is wrong, because the products (x)δ(x), (−x)δ(x) are not defined. The way to correct the splitting consists in replacing (±x) with two smooth functions θ1(x), θ2(x) that obey the condition θ1(x) + θ2(x) = 1, and resemble (x), (−x), respectively. Moreover, these functions 1 1 should be such that θi (x) f (x) ∈ S(R ), i = 1, 2, for every f ∈ S(R ). Then we may safely write F(x) = θ1(x)F(x) + θ2(x)F(x) for any F(x) ∈ S∗(R1). Bibliography 1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards, Washington, DC (1964) 2. Vladimirov, V.S.: Methods of the Theory of Generalized Functions. CRC Press. Boca Raton, FL (2002) 3. Bogoljubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. Wiley- Interscience. New York, NY, London, Sydney, Toronto, ON (1980). 4. Jackiw, R., Manton, N.: Symmetries and conservation laws in gauge theories. Ann. Phys. 127, 257 (1980). 5. Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Co., Evanston, IL, Elmsford, NY (1961) 6.
Load more

Recommended publications
  • GENERALIZED FUNCTIONS LECTURES Contents 1. the Space
    GENERALIZED FUNCTIONS LECTURES Contents 1. The space of generalized functions on Rn 2 1.1. Motivation 2 1.2. Basic definitions 3 1.3. Derivatives of generalized functions 5 1.4. The support of generalized functions 6 1.5. Products and convolutions of generalized functions 7 1.6. Generalized functions on Rn 8 1.7. Generalized functions and differential operators 9 1.8. Regularization of generalized functions 9 n 2. Topological properties of Cc∞(R ) 11 2.1. Normed spaces 11 2.2. Topological vector spaces 12 2.3. Defining completeness 14 2.4. Fréchet spaces 15 2.5. Sequence spaces 17 2.6. Direct limits of Fréchet spaces 18 2.7. Topologies on the space of distributions 19 n 3. Geometric properties of C−∞(R ) 21 3.1. Sheaf of distributions 21 3.2. Filtration on spaces of distributions 23 3.3. Functions and distributions on a Cartesian product 24 4. p-adic numbers and `-spaces 26 4.1. Defining p-adic numbers 26 4.2. Misc. -not sure what to do with them (add to an appendix about p-adic numbers?) 29 4.3. p-adic expansions 31 4.4. Inverse limits 32 4.5. Haar measure and local fields 32 4.6. Some basic properties of `-spaces. 33 4.7. Distributions on `-spaces 35 4.8. Distributions supported on a subspace 35 5. Vector valued distributions 37 Date: October 7, 2018. 1 GENERALIZED FUNCTIONS LECTURES 2 5.1. Smooth measures 37 5.2. Generalized functions versus distributions 38 5.3. Some linear algebra 39 5.4. Generalized functions supported on a subspace 41 6.
    [Show full text]
  • Schwartz Functions on Nash Manifolds and Applications to Representation Theory
    SCHWARTZ FUNCTIONS ON NASH MANIFOLDS AND APPLICATIONS TO REPRESENTATION THEORY AVRAHAM AIZENBUD Joint with Dmitry Gourevitch and Eitan Sayag arXiv:0704.2891 [math.AG], arxiv:0709.1273 [math.RT] Let us start with the following motivating example. Consider the circle S1, let N ⊂ S1 be the north pole and denote U := S1 n N. Note that U is diffeomorphic to R via the stereographic projection. Consider the space D(S1) of distributions on S1, that is the space of continuous linear functionals on the 1 1 1 Fr´echet space C (S ). Consider the subspace DS1 (N) ⊂ D(S ) consisting of all distributions supported 1 at N. Then the quotient D(S )=DS1 (N) will not be the space of distributions on U. However, it will be the space S∗(U) of Schwartz distributions on U, that is continuous functionals on the Fr´echet space S(U) of Schwartz functions on U. In this case, S(U) can be identified with S(R) via the stereographic projection. The space of Schwartz functions on R is defined to be the space of all infinitely differentiable functions that rapidly decay at infinity together with all their derivatives, i.e. xnf (k) is bounded for any n; k. In this talk we extend the notions of Schwartz functions and Schwartz distributions to a larger geometric realm. As we can see, the definition is of algebraic nature. Hence it would not be reasonable to try to extend it to arbitrary smooth manifolds. However, it is reasonable to extend this notion to smooth algebraic varieties.
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]
  • Mathematical Problems on Generalized Functions and the Can–
    Mathematical problems on generalized functions and the canonical Hamiltonian formalism J.F.Colombeau [email protected] Abstract. This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a “formal way”- as done by Heisenberg and Pauli - i.e. without mathematical definitions and then, of course, without mathematical rigour) the Heisenberg-Pauli calculations on the simplest model close to physics. The problem for mathematicians is to give a mathematical sense to these calculations, which is possible without any knowledge in physics, since they mimick exactly usual calculations on C ∞ functions and on bounded operators, and can be considered at a purely mathematical level, ignoring physics in a first step. The mathematical tools to be used are nonlinear generalized functions, unbounded operators on a Hilbert space and computer calculations. This text is the improved written version of a talk at the congress on linear and nonlinear generalized functions “Gf 07” held in Bedlewo-Poznan, Poland, 2-8 September 2007. 1-Introduction . The Heisenberg-Pauli calculations (1929) [We,p20] are a set of 3 or 4 pages calculations (in a simple yet fully representative model) that are formally quite easy and mimick calculations on C ∞ functions. They are explained in detail at the beginning of this text and in the appendices. The H-P calculations [We, p20,21, p293-336] are a basic formulation in Quantum Field Theory: “canonical Hamiltonian formalism”, see [We, p292] for their relevance. The canonical Hamiltonian formalism is considered as mainly equivalent to the more recent “(Feynman) path integral formalism”: see [We, p376,377] for the connections between the 2 formalisms that complement each other.
    [Show full text]
  • Introduction to the Fourier Transform
    Chapter 4 Introduction to the Fourier transform In this chapter we introduce the Fourier transform and review some of its basic properties. The Fourier transform is the \swiss army knife" of mathematical analysis; it is a powerful general purpose tool with many useful special features. In particular the theory of the Fourier transform is largely independent of the dimension: the theory of the Fourier trans- form for functions of one variable is formally the same as the theory for functions of 2, 3 or n variables. This is in marked contrast to the Radon, or X-ray transforms. For simplicity we begin with a discussion of the basic concepts for functions of a single variable, though in some de¯nitions, where there is no additional di±culty, we treat the general case from the outset. 4.1 The complex exponential function. See: 2.2, A.4.3 . The building block for the Fourier transform is the complex exponential function, eix: The basic facts about the exponential function can be found in section A.4.3. Recall that the polar coordinates (r; θ) correspond to the point with rectangular coordinates (r cos θ; r sin θ): As a complex number this is r(cos θ + i sin θ) = reiθ: Multiplication of complex numbers is very easy using the polar representation. If z = reiθ and w = ½eiÁ then zw = reiθ½eiÁ = r½ei(θ+Á): A positive number r has a real logarithm, s = log r; so that a complex number can also be expressed in the form z = es+iθ: The logarithm of z is therefore de¯ned to be the complex number Im z log z = s + iθ = log z + i tan¡1 : j j Re z µ ¶ 83 84 CHAPTER 4.
    [Show full text]
  • Tempered Generalized Functions Algebra, Hermite Expansions and Itoˆ Formula
    TEMPERED GENERALIZED FUNCTIONS ALGEBRA, HERMITE EXPANSIONS AND ITOˆ FORMULA PEDRO CATUOGNO AND CHRISTIAN OLIVERA Abstract. The space of tempered distributions S0 can be realized as a se- quence spaces by means of the Hermite representation theorems (see [2]). In this work we introduce and study a new tempered generalized functions algebra H, in this algebra the tempered distributions are embedding via its Hermite ex- pansion. We study the Fourier transform, point value of generalized tempered functions and the relation of the product of generalized tempered functions with the Hermite product of tempered distributions (see [6]). Furthermore, we give a generalized Itˆoformula for elements of H and finally we show some applications to stochastic analysis. 1. Introduction The differential algebras of generalized functions of Colombeau type were developed in connection with non linear problems. These algebras are a good frame to solve differential equations with rough initial date or discontinuous coefficients (see [3], [8] and [13]). Recently there are a great interest in develop a stochastic calculus in algebras of generalized functions (see for instance [1], [4], [11], [12] and [14]), in order to solve stochastic differential equations with rough data. A Colombeau algebra G on an open subset Ω of Rm is a differential algebra containing D0(Ω) as a linear subspace and C∞(Ω) as a faithful subalgebra. The embedding of D0(Ω) into G is done via convolution with a mollifier, in the simplified version the embedding depends on the particular mollifier. The algebra of tempered generalized functions was introduced by J. F. Colombeau in [5] in order to develop a theory of Fourier transform in algebras of generalized functions (see [8] and [7] for applications and references).
    [Show full text]
  • Applications of Fourier Transforms to Generalized Functions
    Applications of Fourier Transforms to Generalized Functions WITPRESS WIT Press publishes leading books in Science and Technology. Visit our website for the current list of titles. www.witpress.com WITeLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visit the WIT eLibrary athttp://library.witpress.com Applications of Fourier Transforms to Generalized Functions M. Rahman Halifax, Nova Scotia, Canada M. Rahman Halifax, Nova Scotia, Canada Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-564-9 Library of Congress Catalog Card Number: 2010942839 The texts of the papers in this volume were set individually by the authors or under their supervision. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. The Publisher does not necessarily endorse the ideas held, or views expressed by the Editors or Authors of the material contained in its publications.
    [Show full text]
  • Generalized Functions: Homework 1
    Generalized Functions: Homework 1 Exercise 1. Prove that there exists a function f Cc∞(R) which isn’t the zero function. ∈ Solution: Consider the function: 1 e x x > 0 g = − 0 x 0 ( ≤ 1 We claim that g C∞(R). Indeed, all of the derivatives of e− x are vanishing when x 0: ∈ n → d 1 1 lim e− x = lim q(x) e− x = 0 x 0 n x 0 → dx → ∙ where q(x) is a rational function of x. The function f = g(x) g(1 x) is smooth, as a multiplication of two smooth functions, has compact support∙ − (the interval 1 1 [0, 1]), and is non-zero (f 2 = e ). Exercise 2. Find a sequence of functions fn n Z (when fn Cc∞ (R) n) that weakly converges to Dirac Delta function.{ } ∈ ∈ ∀ Solution: Consider the sequence of function: ˜ 1 1 fn = g x g + x C∞(R) n − ∙ n ∈ c where g(x) Cc∞(R) is the function defined in the first question. Notice that ∈ 1 1 the support of fn is the interval n , n . Normalizing these functions so that the integral will be equal to 1 we get:− 1 ∞ − f = d(n)f˜ , d(n) = f˜ (x)dx n n n Z −∞ For every test function F (x) C∞(R), we can write F (x) = F (0) + x G(x), ∈ c ∙ where G(x) C∞(R). Therefore we have: ∈ c 1/n ∞ F (x) f (x)dx = F (x) f (x)dx ∙ n ∙ n Z 1Z/n −∞ − 1/n 1/n = F (0) f (x)dx + xG(x) f (x)dx ∙ n ∙ n 1Z/n 1Z/n − − 1/n = F (0) + xG(x) f (x)dx ∙ n 1Z/n − 1 1/n ∞ ∞ So F (x) fn(x)dx F (x) δ(x)dx = 1/n xG(x) fn(x)dx.
    [Show full text]
  • Nonlinear Generalized Functions: Their Origin, Some Developments and Recent Advances
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Cadernos Espinosanos (E-Journal) S~aoPaulo Journal of Mathematical Sciences 7, 2 (2013), 201{239 Nonlinear Generalized Functions: their origin, some developments and recent advances Jean F. Colombeau Institut Fourier, Universit´ede Grenoble (retired) Abstract. We expose some simple facts at the interplay between math- ematics and the real world, putting in evidence mathematical objects " nonlinear generalized functions" that are needed to model the real world, which appear to have been generally neglected up to now by mathematicians. Then we describe how a "nonlinear theory of gen- eralized functions" was obtained inside the Leopoldo Nachbin group of infinite dimensional holomorphy between 1980 and 1985 **. This new theory permits to multiply arbitrary distributions and contains the above mathematical objects, which shows that the features of this theory are natural and unavoidable for a mathematical description of the real world. Finally we present direct applications of the theory such as existence-uniqueness for systems of PDEs without classical so- lutions and calculations of shock waves for systems in non-divergence form done between 1985 and 1995 ***, for which we give three examples of different nature (elasticity, cosmology, multifluid flows). * work done under support of FAPESP, processo 2011/12532-1, and thanks to the hospitality of the IME-USP. ** various supports from Fapesp, Finep and Cnpq between 1978 and 1984. *** support of "Direction des Recherches, Etudes´ et Techniques", Minist`ere de la D´efense,France, between 1985 and 1995. [email protected] 1. Mathematics and the real world.
    [Show full text]
  • Global Algebras of Nonlinear Generalized Functions with Applications in General Relativity
    São Paulo Journal of Mathematical Sciences 7, 2 (2013), 143171 Global algebras of nonlinear generalized functions with applications in general relativity Eduard A. Nigsch Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria; phone: +43 1 4277 50760 E-mail address: [email protected] Clemens Sämann Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria; phone: +43 1 4277 50632 E-mail address: [email protected] Abstract. We give an overview of the development of algebras of gen- eralized functions in the sense of Colombeau and recent advances con- cerning dieomorphism invariant global algebras of generalized func- tions and tensor elds. We furthermore provide a survey on possible applications in general relativity in light of the limitations of distribu- tion theory. Keywords: Colombeau algebra; nonlinear generalized function; distri- butional geometry; Geroch Traschen class; smoothing kernel; full and spe- cial Colombeau algebra; general relativity; functional analytic approach; kernel theorem 2010 Mathematics Subject Classication: Primary 46F30, Sec- ondary 46T30, 83C35, 53C50 1. Introduction In physics there is the natural need to reduce the complexity of real-world physical systems for the formulation of mathematical models and therefore to work with idealizations. A well-known example of this is the notion of a point particle, which represents an object whose properties like mass or electric charge are concentrated in a single point. 143 144 E. A. Nigsch and C. Sämann Such a point charge is classically represented by Dirac's delta function δ, originally introduced as a useful notation for dealing with a certain kind of innities [20, p.
    [Show full text]
  • Problem Set 4. Generalized Functions
    Problem Set 4. GeneralizedProblem functions.set 3 MathMath 212a 212a OctoberSept. 30, 2, 2014,2008 due Due Oct. Oct. 15 1 The purpose of this exercise set, which is mainly computational, is to give you some experience with the ideas associated to generalized functions. Recall that in our study of the Fourier transform we introduced the space consisting of all functions on R which have derivatives of all order which vanishS rapidly at infinity, and that on this space we have a countable family of norms m n f m,n := x D f , ∞ where m and n are non-negative integers. Convergence in means convergence with respect to all of these norms. So a linear functionS on this space is continuous if (fk) 0 whenever fk m,n 0 for every m and n.Welet denote this dual| space,| → the space of continuous → linear functions on . S For example, consider the “Heaviside function” S 1 for x>0 H(x):= . 0 for x 0 ≤ This function does not belong to , but it defines a continuous linear function on by S S ∞ ∞ H, f = f(x)H(x)dx = f(x)dx. 0 −∞ Of course, any element of certainly defines a linear function on by the same procedure, the linear functionS associated to g is just the HilbertS space scalar product ∈ S ∞ f (f,g)= f(x)g(x)dx. → −∞ This also works for any g L2(R), but H L2(R). From the Riesz representa- tion theorem we thus know∈ that the linear function∈ given by H is not continuous with respect to the L2 norm, but it is continuous relative to the topology given above by the countable family of norms .
    [Show full text]
  • Regularity Properties of Distributions and Ultradistributions
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 12, Pages 3531{3537 S 0002-9939(01)06013-0 Article electronically published on June 28, 2001 REGULARITY PROPERTIES OF DISTRIBUTIONS AND ULTRADISTRIBUTIONS S. PILIPOVIC´ AND D. SCARPALEZOS (Communicated by Jonathan M. Borwein) Abstract. We give necessary and sufficient conditions for a regularized net of a distribution in an open set Ω which imply that it is a smooth function or Ck function in Ω. We also give necessary and sufficient conditions for an ultradistribution to be an ultradifferentiable function of corresponding class. Introduction Algebras of generalized functions are usually defined as factor algebras of cer- tain algebras of sequences (nets) of smooth functions (cf. [1], [2], [3], [8], [11]), and the classical spaces of functions, distributions and ultradistributions are em- bedded into the appropriate algebra through their regularizations; they are also equivalence classes of sequences (nets) of smooth functions. Solutions in algebras of generalized functions of PDE are also presented by such sequences (nets) and even differential operators (for example, with singular coefficients) are sometimes replaced by sequences (nets) of differential operators with smooth coefficients. So the natural question arises: Under what conditions is a given generalized function (a generalized solution of PDE) actually a classical function of appropriate class, distribution, or ultradistribution? The partial answers to such questions are given in Propositions 1, 2 and more gen- erally in Theorems 1, 2, respectively, in terms of asymptotic behaviour of sequences of functions provided that these sequences are evaluated on the ultraproduct ΩN: Denote by (θn) (respectively, (φn)) a δ-sequence, also called a sequence of molli- fiers, of smooth functions (respectively, of appropriate ultradifferentiable functions).
    [Show full text]