Timeline for Why distributions as functionals? [closed]
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Feb 19, 2014 at 20:02 | comment | added | Alexander Shamov | @Dirk: Well, the way that quantity affects the measurement device is quite complicated and nonlinear, right? Of course we can argue that "everything is approximately linear unless it's too large", but isn't the whole point of distributions to capture things that are exactly "too large", i.e. "singular"? | |
| Feb 19, 2014 at 19:45 | comment | added | Dirk | @Alexander Shamov: That's some point... But well, "average" sounds pretty linear to me. More practically: A light sensor is basically collecting all the light that hits it and hence, measures the average (in the sense of arithmetic mean) light in that region. I do not know enough about other sensors but linearity sounds very reasonable to me. | |
| Feb 19, 2014 at 19:08 | comment | added | Alexander Shamov | @Dirk: And this averaging procedure is linear just because... | |
| Feb 19, 2014 at 14:21 | comment | added | paul garrett | In fact, one need not emphasize generalized functions (a.k.a. distributions) as "functionals", but as elements of the completion of Schwartz functions with respect to the weak *-topology, showing that they extend Schwartz functions, emphasizing this extensions property rather than the duality. | |
| Feb 19, 2014 at 14:18 | history | edited | paul garrett | CC BY-SA 3.0 |
Corrected spelling of L. Schwartz' name, clarified language, ... perhaps.
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| Jan 19, 2014 at 10:42 | comment | added | Dirk | Ah sheesh, it should read neatly (not "nearly") in the first sentence. | |
| Jan 14, 2014 at 8:05 | comment | added | Dirk | Although closed: There is a fundamental physical motivation for distributions which is nearly explained in Strichartz book "A Guide to Distribution Theory and Fourier Transforms", chapter 1. In a nutshell: If $f$ shall represent a physical quantity (like temparature, pressure,…) then it does not seem plausible from a physical point of view to talk about values of $f$ and some point $x$ because true point measurements are not possible. However, averaged measurements are possible and this gives rise to testing the quantity against functions and there you are. | |
| Jan 14, 2014 at 1:02 | history | closed |
Steven Gubkin Will Jagy Andrey Rekalo Qiaochu Yuan Theo Johnson-Freyd |
Needs more focus | |
| Jan 14, 2014 at 0:52 | comment | added | S. Carnahan♦ | These objects appeared naturally long before Schwartz made them rigorous. See the history section of en.wikipedia.org/wiki/Dirac_Delta_function for motivation. There are other generalizations, e.g., hyperfunctions, so Schwartz's solution is not unique by any means. | |
| Jan 14, 2014 at 0:48 | comment | added | C-star-W-star | ...hmm ...and what does it meanin this case? | |
| Jan 14, 2014 at 0:33 | review | Close votes | |||
| Jan 14, 2014 at 1:02 | |||||
| Jan 14, 2014 at 0:28 | review | First posts | |||
| Jan 14, 2014 at 0:30 | |||||
| Jan 14, 2014 at 0:25 | comment | added | Steven Gubkin | I think fundamentally, it comes down to the philosophy embodied by the yoneda lemma. To understand "blah" look at its relationship to all other things, or maybe just one particular thing. Often such a representable functor captures ALL data about a thing. Then generalized blahs come from functors which were not representable to start with. In other words, studying the category of presheaves on a given category is almost always worthwhile. | |
| Jan 14, 2014 at 0:22 | comment | added | Steven Gubkin | I voted to close because it seems to discussiony, but maybe I was mistaken to do that. There could be some really good answers to this question. | |
| Jan 14, 2014 at 0:11 | history | asked | C-star-W-star | CC BY-SA 3.0 |