Theory of Distributions

Description

Ibragimov's A Practical Course in Differential Equations p. vi (PDF p. 7) and ch. 8 "Generalized functions or distributions" pp. 291ff. (PDF pp. 306ff.) mentions that he uses his text in a course on distribution theory. "Distribution" in this sense means "generalized function."

The author claims to present an "accessible introduction to some aspects of the theory of distributions'', "well suited for a one-semester lecture course''. For the reasons I will enumerate in the following, these objects have not been attained.
(1) Simple statements are proven, more difficult ones are not:

(1.1) The treatment of the convolution of distributions in D′(+Γ) with those in D′(S¯¯¯+) comprises 1 page (p. 115), and is cribbed from the book [V. S. Vladimirov, Generalized functions in mathematical physics, English translation, Mir, Moscow, 1979; [MR0564116](http://ams.rice.edu/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=564116&loc=fromrevtext)], where the exposition covers 8 pages [op. cit. (71–79)] with 5 figures. Moreover, the term "C-like'' (Definition 1.12., p. 9) is incomplete. It should read: "… if each straight line x=x0+te, t∈R, e∈prΓ, intersects S in one and only one point'', instead of "in one point only''. Otherwise, the distributions u=H(x)H(y)∈D′(Γ+), Γ=R¯¯¯¯2+, and v=1(x)H(y)∈D′(S+), S={(xy)∈R2;xy=1, x<0}, S¯¯¯+=S+Γ, yield a counterexample to convolvability (the y-axis does not intersect S, u and v are not convolvable). 
 (1.2) The proof of the Malgrange-Ehrenpreis theorem (pp. 179–182) is from [op. cit. (pp. 210–214)], with the omission, however, of Lemma 2 [op. cit. (p. 211)]. Instead, in the proof knowledge of the Paley-Wiener-Schwartz theorem is required (it does not occur in the book).

(2) Propositions which are not marked as such:

 (2.1) The characterization of the continuity of distributions by seminorms (p. 30). Moreover, this characterization is repeated on p. 43 as "Problem 2.12" and proven a second time. 
(2.2) The statement that the direct product of distributions is well defined (p. 99). 
(2.3) The statement that the convolution of distributions is well defined (p. 109). 
(2.4) The characterization of weakly bounded subsets of S′ (p. 151).

(3) Repetitions:

(3.1) see 2.1. 
(3.2) chΓ (pp. 1, 9). 
(3.3) Regularization (pp. 8, 116).

(4) Omissions:

(4.1) p. 113: TR has to be bounded. 
(4.2) S′(+Γ), S′(S¯¯¯+) are not defined (p. 154).

(5) Disorder:

(5.1) On p. 162, the Fourier transform of ei(Ax,x) should be calculated by an application of the definition of the Fourier transform for functions in S(Rn).
(5.2) In Example 2.16 (p. 36), singsuppP1x2 has to be determined. However, the distribution P1x2 is not defined until p. 41.

(6) Mistakes:

(6.1) Example 2.7, p. 29: the function Φ is not defined at 2x+3i.
(6.2) Example 2.19, p. 37: the singular distributions do not form a vector subspace. 
(6.3) Problem 2.33, p. 58: a∈Cn is not allowed. 
(6.4) Problem 2.37, p. 59: δ(x2−a2)=12|a|[δ(x−a)+δ(x+a)].
(6.5) Problem 3.33, p. 84: the expressions ∫bafn(x+t)dt, ∫baf(x+t)dt are not defined. 
(6.6) Problem 3.34, p. 84: convergence is not sufficient. 
(6.7) Problem 5.10, p. 107: the sign in the answer has to be cancelled. What is the difference between δt and δ(t)? 
 (6.8) Example 6.4, p. 113: the equation (H(x)∗P(x))′′′=P′′(x) for a polynomial P≠0 is wrong since H and P are not convolvable. 
 (6.9) p. 116: if X=Rn, u1∈D′(x), α∈C∞(X), the convolution u1∗α might not be defined. 
 (6.10) Definition 6.5, p. 117: read fx(x)=f(u)α+n(x) for α+n>0. 
 (6.11) Problem 6.6, p. 124: the integral ∫X2K(x1,x2)Φ(x2)dx2 is not defined. 
 (6.12) Problem 6.18, p. 128: the answer to 5. is not correct. 
 (6.13) Problem 6.22, p. 129: the answer is not correct. 
 (6.14) Problem 6.41, p. 142: a counterexample is f=1. 
 (6.15) Problem 8.14, p. 175: iP1ξ. 
 (6.16) Problem 8.19, p. 176: sinξtξ instead of sinξtξ. 
 (6.17) Problem 8.20, p. 176: replace S′(Rn) by S′(R).
 (6.18) Problem 9.10, p. 185: if n>1 the distribution H(t)tn/2e−i|x|24t is not defined correctly.

(7) Missing motivations, e.g.: why should one consider homogeneous distributions (Chapter 4, p. 87), the direct product (Chapter 5, p. 99) or the convolution (Chapter 6, p. 109)?
(8) Strange nomenclature:

 (8.1) "Exercises" are inserted in the text, whereas at the end of a section "Exercises'' are announced and then called "Problems'' (pp. 10, 40, 58). 
 (8.2) λ=∑ni=1xi∂i, on p. 96, is a differential operator, whereas in Problem 6.22 (p. 129), λ is a one-dimensional variable. 
 (8.3) A subspace of L. Schwartz' space OC is symbolized by ΘM (Definition 1.6, p. 5) alleging that this space is the space of multipliers of S and S′, which is not true.

Many parts of the book (convolution, fractional differentiation and integration, Fourier and Laplace transforms, … ) can be better studied in [op. cit.], and many of the exercises can also be found in [V. S. Vladimirov et al., Recueil de problèmes d'équations de physique mathématique, French translation, Mir, Moscow, 1976; MR0433009].
In the reviewer's opinion, the following books are of higher quality: [K. Vo-Khac, Distributions, analyse de Fourier, opérateurs aux dérivées partielles, tome 1,2, Vuibert, Paris, 1972; B. E. Petersen, Introduction to the Fourier transform & pseudodifferential operators, Monogr. Stud. Math., 19, Pitman, Boston, MA, 1983; MR0721328; J. Barros-Neto, An introduction to the theory of distributions, Marcel Dekker, Inc. New York, 1973; MR0461128; W. F. Donoghue, Distributions and Fourier transforms, Academic Press, New York, 1969; R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York, 1965; MR0221256; J. Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, MA, 1966; MR0205028; W. Rudin, Functional analysis, McGraw-Hill, New York, 1973; MR0365062; F. G. Friedlander (with additional material by M. Joshi), Introduction to the theory of distributions, second edition, Cambridge Univ. Press, Cambridge, 1998; MR1721032; L. V. Hörmander, The analysis of linear partial differential operators. I, Grundlehren Math. Wiss., 256, Springer, Berlin, 1983; MR0717035].
It remains unclear to the reviewer why the author cites the lecture notes of M. Kunzinger ["Distributionentheorie II'', Univ. Vienna, spring term, 1998; per bibl.] and of R. Steinbauer ["Locally convex vector spaces'', Univ. Vienna, fall term, 2008; per bibl.], which are available only as handwritten notes by personal contact with the authors, while the basic and monumental monograph of L. Schwartz [Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966; MR0209834] is not cited.

Reviewed by Norbert Ortner

"Uniquely, this work by Georgiev (Univ. of Sofia, Bulgaria) reads stylistically like a brief, basic, but exotic calculus text, replete with concrete sample calculations. Many researchers found distributions quite alien when they made the scene 65 years ago, but with this book, today’s mathematics and physics undergraduate students may well come to regard them as quite routine. Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners."

D. V. Feldman, Choice, Vol. 53 (10), June, 2016