Talk:Generalized function

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It took me some time to notice the edit of the introduction by C.M., but I do not agree at all:

  • neither in Schwartz distributions, nor any other theory mentioned (nonstandard, Colombeau), the generalized functions are R- or C-valued. So the first added phrase is at least misleading.
  • one can well generalize other structures than numerical functions: this is the case in QFT, the main application field of these theories (where the objects to be generalized are operators on Fock spaces)
  • the second phrase "what is generalized is some operator aspect of functions" is not very illuminating (imho). I don't understand what it could mean.

Summarizing, it appears to me that neither of the two phrases is very useful, and I think that the original formulation (see given "diff" link above) is at the same time quite easily understandable, sufficiently precise and still general enough to be correct.

Of course I greatly appreciate the added sections on "early history". MFH: Talk 14:34, 23 May 2005 (UTC)

First point - I was just trying to say that in generalised function, one does mean generalised numerical function (not for f:X → Y for any sets). Perhaps this could be expressed better, though. Second point, on operators. This is not so much for pure mathematicians, as for computer scientists, physicists and so on. It is meant to be read with the discussion at operator; is an operator not just some kind of function? Well, not exactly. So I defend this as a general thought. More work is probably needed on the introduction. Charles Matthews 15:20, 23 May 2005 (UTC)

still in favour of orig.formulation[edit]

Once again, I advocate in favour of my earlier formulation, "making discont. functions differentiable and, which is related, describing point charges", rather than :

Generalized functions are especially useful in making
discontinuous functions more like smooth functions,
and (going to extremes) describing physical phenomena such as
point charges.
  • Indeed distributions are really in no point like smooth functions, they don't even have a value in each point. The only truth is that they are infinitely differentiable.
  • Second, I insist that "making discont. fcts derivable" is related to "describing point charges", if not the same, and the latter is not "going to extremes". Indeed, what is a point change (say, unit charge at the origin)? It is a "density" \rho such that \int_{-\infty}^x \rho(t) dt is equal to zero for all x<0, and equal to one for all x>0. So the point charge \rho is exactly the derivative of the most simple discont. fct which is Heaviside's fct.

Next paragraph:

A common feature of some of the approaches

(somehow contradition in itself, imho)

is that they build on operator aspects of everyday, numerical functions.

I can't see what this explains ("they build on .. aspects": I think this it quite obscure - unless being more precise, I'm against putting this here), and once again, it is not true, since distibutions are in the main field of their application (QFT) operator-valued, and thus not numerical functions.

I'm sorry if I appear too critical (or even unpolite, really sorry for that, I can't help!), and I have not the slightest doubt in your mathematical competence (on the contrary), but I don't appreciate too much changes which are not clearly an improvement of the previous formulation. I would really appreciate if the latter would be left untouched in its place, and if s.o. wants to add explanations, they would be added in a separate section.

Once again, I really prefer leaving the intro in my original formulation, quite short and "easy to read" at a first glance. I would have nothing against putting your precisions in a following "Motivations" or "Basic ideas" section or so, and even develop them further.

Thanks in advance for your comprehension! MFH: Talk 13:28, 27 May 2005 (UTC)

My only comment is that the introduction is not for you - an expert - but for someone who has really no idea about the subject. What might be imprecise to you is possibly more helpful to a novice. I have noticed, all over the mathematics articles, that the technical content improves but the accessibility gets worse. There is no unique way to write a good introduction - it is journalism, not academic writing. I have met this problem for example at Euler angles, Gröbner basis and so on. It is nothing to do with you: you are the expert in this case. I just advocate not having the expert as judge of the introduction. Charles Matthews 13:43, 27 May 2005 (UTC)

OK, I'm looking in various sources, to see what some authors say. Dieudonné, Treatise on Analysis Vol. III Ch.17 introduction, says something about the 'operator' aspect of distributions being more fundamental. The Soviet encyclopedia emphasises the charge distribution analogy: you might be happy with a paraphrase of what appears there. I'd like to look in a few more books: I don't actually have an academic library to consult. Charles Matthews 15:31, 27 May 2005 (UTC)

The Japanese encyclopedia has some interesting history: Hadamard on fundamental solutions of wave equations (1932), M. Riesz on Riemann-Liouville integrals (fractional calculus) from 1938, Bochner (1932) and Carleman (1944) on Fourier transforms of functions of polynomial growth. Sobolev used integration by parts, and was studying the Cauchy problems for hyperbolic equations. A few other people, too. Charles Matthews 15:40, 27 May 2005 (UTC)

Another WP's article about the same topics[edit]

The article distribution (mathematics) seems to be focused on the same topics of this article. —The preceding unsigned comment was added by 80.180.174.104 (talk) 02:11, 10 March 2007 (UTC).

Article still limited[edit]

There are many approaches to generalised functions out there. The panorama presented by this wikiarticle is very limited thus far. I added Egorov's approach very hurriedly. I also agree on the fact that the main property desired in a generalised function in differentiability to any order 85.226.147.21 (talk) 11:00, 18 April 2009 (UTC)

---

This may be of use to someone who would want to expand the article: While researching Dirac delta I encountered Bracewell's treatment in

Ron Bracewell 1965 The Fourier Transform and Its Applications, McGraw-Hill Book Company, NY, no ISBN, LCCCN 64-23272.

This text is geared to EE's at the graduate level. His treatment begins on p. 87-94 with a major subchapter: "The Concept of generalized function" in the chapter "The impulse symbol, δ(x)". He develops it through sub-sections "Particularly well-behaved functions", "Regular sequences", "Generalized functions", "Algebra of generalized functions", "Differentiation of ordinary functions". He provides the following commentary about the history and a set of interesting references (including the Schwartz used prominently in the article). Here's his take on the history of the development of "generalized functions":

"A satisfactory mathematical formulation of the theory of impulses has been evolved along these lines [i.e. limit as tau --> 0 of the integral that I wrote above] and is expounded in the books of Lighthill1 and Friedman2. Lighthill credits Temple with simplifying the mathematical presentation; Temple3 in turn credits the Polish mathematician Mikusinski4 with introducing the presentation in terms of sequences in 1948. Schwartz's two volumes5 on the theory of distributions unify "in one systematic theory a number of partial and special techniques proposed for the analytical inerpretation of 'imporper' or 'ideal' fucntions and symbolic methods6. // the idea of sequences was current in physical circles before 1948, however7.
  • 1: M. J. Lighthill, "An introduction to Fourier Analsys and Generalised Functions", Cambridge University Press, Cambridge UK 1958.
  • 2: B. Friedman, "Principles and Techniques of Applied Mathematics," John wiley & Sons, New York, 1956.
  • 3: G. Temple, Theories and Appliations of Generalized Functions, J. Lond. Math. Soc., vol. 28, p. 181, 1953.
  • 4 J. G.-Mikusinski, Sur la methode de generalisation de Laurent Schwartz et sur la convergence faible, Fundamenta Mathematicae, vol. 35, p. 235, 1948.
  • 5 L. Schwartz, "Theorie des distributions," vols. 1 and 2, Herman & Cie, Paris, 1950 and 1951.
  • 6 Temple, op. cit. p. 175
  • 7 B. van der Pol, Discontinuous Pheonmena in Radio Communication, J. Inst. Elec. Engrs, vol. 81, p. 381, 1937.
  • 8 Schwartz, op. cit. p. 22.
From my underlining ca 1970-1971 I see that I at read it, but I didn't retain a word of it. Meaning: I can be of little assitance re improving this article. But maybe this sourcing can help someone who can improve it. Bill Wvbailey (talk) 21:44, 4 February 2012 (UTC)