If you want the whole story you need to learn some "real analysis" first (measure theory, topological vector spaces, etc). There are many places you can find the theory of distributions worked out in detail - the two that are standard texts where I come from would be Folland "Real Analysis" (or maybe it's "Real Analysis and Applications" or something) and Rudin "Functional Analysis".
Website? Hmm, google...
It seems that wikipedia
discusses the topic, although I doubt that there's a complete exposition of the theory there, in an "encyclopedia" they must have just statements of the main results.
The description of
on google sounds like it might be what you want, but that actual pdf is just a table of contents. I didn't see how to find the actual notes on the site, but maybe you can if you hunt around.
Otoh I wouldn't be surprised if there is no web site that actually contains the whole story.
I have "Intermediate Real Analysis" by Emanuel Fischer, but if it discusses those topics it does so by entirely different names -- I rather suspect that it leaves off where your other texts start.
So far any time I've felt a need for rigor around the delta function (distribution, whatever) I've just constructed some real function with area one that's either parameterized by height (or width), found my result, then taken the limit as the parameter goes to infinity (or zero). It's probably not entirely kosher, but it's served my purposes.
Gareth, this really is ridiculous. So many people have posted answers to you that it isn't practical to even know where to start. You, on the other hand, are the one claiming they're incorrect, so the burden really is on you to explain why you think so.
This just oozes irony. Gushes. Floods.
Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions.
However, the sampling waveforms in practice are not represented by an area of 1.
If anyone wishes to claim that their sampled signals are represented in some way by the Diracian, then they must mentally model an invisible scaling factor to bring the magnitudes of their sampled waveforms into the order of magnitude of the attributes of the Dracaena.
I enquired as to how others came to terms with this blatant discrepancy.
It has been explained to you over and over why there is no "blatant discrepancy", that nobody is using the model you claim they are. You are arguing with a strawman and you know it.
If the measurement process is modeled with delta-functions, the model also includes an integration. The delta-functions appear under integrals.
If the model is just using multiplication by a sampling function, then it uses spikes of finite height, usually 1.
Nobody is modeling sampling as multiplication by a delta function without integration. Your repeated questions about how can this model work are pointless, since you are asking about a model nobody is using.
And for the record, nobody is using the term "Diracian" as far as I have ever seen.
I don't know that book but based on the title that seems likely - "real analysis" covers a lot of ground.
Can't say for sure without seeing exactly what you've done, but that could very well be just fine (although it's not _really_ ok unless you can explain why...). For example:
Say f_n(t) = n for 0 < t < 1/n, 0 for other t. Then f_n -> delta "in the sense of distributions" as n -> infinity. What convergence "in the sense of distributions" means is that if g is an infinitely differentiable function then
(*) int f_n g -> int delta g = g(0) as n -> infinity.
(Here int is the integral from -infinity to infinity.)
If all you're doing is things that look like (*) then the things you're doing are ok.
Coming from you that's very funny. Hint: there's no such word as "Diracian". Even though you use the word all the time. (Ok, it appears that there is such a word. But it's not a noun referring to the delta function, execpt in your posts.)
Well, if you want to work with the Fourier transform of a sampled signal, the integral over the sampled function must be non-zero, and you just have isolated points with content. Something has to give, and it turned out that it was easiest to sacrifice strict point-wise defined functions since they don't survive integration and differention unmolested, anyway. And if one wants to have an entity that will do the right things under integral transforms, it is probably easiest to define it by its behavior for integration in the first place: distributions. Stieltjes integrals might be related, though.
Texts and thier authors are often wrong in one aspect or another as you approach absolute limits... as with most instruments they read most accurately in the mid range. The usual electricians meter of course wouldnt read a tenth of a volt with any accuracy and he doesnt need it to... it wouldnt read a trillionth of a volt at all..
A meter that would read a trillionth of a volt would be toast at 480 volts.. its not practical to design a meter for both uses... the same applies with mathematics or any science. Techiques used to measure spin of subatomic particles are not useful in measuring items the size of molecules,
To trash one or the other as worthless because it doesnt address both is not good logic nor good argument.
so its a matter of whats most applicable for a given range. In your case you are at an extreme limit for any kind of workability for that solution to be the best choice....you need to pick, or invent another approach.
is a short
If a person chose to apply that notion he wouldnt use zero in the calculation obviously, but instead a wild guess on the duration, something real close to zero, but not zero... or an actual number based on what is discoverable regarding the signal... nanoseconds or whatever.
If you are any sort of mathematician you know that the use of imaginary, or trial numbers is quite common and accepted...you can plot a curve using these imaginaries and from that can see the actual trend line...and extrapolate from there in many but not of course all cases...especially as you approach quantum levels... then it all changes, often unpredicatably.
We know only a faint trace of what there is to know in the world... maybe 0.00001% at best... and much of that, in a million years will be shown to poorly founded.
If it were me, I would then examine the results to see if they made any sense...and I would also use various other means...in the end arriving at the best number the project warranted...narrowing it down as the function shaped up on a graph. Absolute pure perfect accuracy though is almost never attainable for reasons you are no doubt familiar with.
If you see some person writing a texts that seems to indicate that his his approach is somehow workable in all cases and situations to near perfect accuracy... then all you have to do is what you have done...point out the loopholes in his logic... and do not use his approach in those situation...find another.
If every author used 15 pages of text to point out the unworkable ranges of his solution texts would be too large to be useful...so in a text referring to one aspect of level of application you will see time tested formula's for that particular range of applications.... not some exotic other, inapplicable range of course.
You are spendiing time focusing on the inapplicable range... thats either legitimate or insane, depending on your intent.
If you are looking to trash people for believing texts..well you wont have to look far or long for targets... its human nature to believe whats in a text...and as any student of history knows, that is virtually always in error at some level of later advance, very often 100% in error.
Spending your life searching for those who believe without thinking... is a waste of your time...beyond satisfying yourself that one must always look deeper...depending on his need for accuracy or truth in the matter.
One as always does the best he personally can...for some thats not very good..... others have many options...and take a broad view, and integrate the results of several calculations and observations, and trials with some common sense to achieve at the least, a workable solution.
I find graphical solutions work well in many areas. If by use of mathematics, or direct observation, you can develop a verifiable curve on either side of your particular situation...then the points inbetween in almost all cases will fall on the curve.
For some reason you appear frightened of researching information, such as following a url, and integrating the knowledge gained with what you already know in order to take things into new territory. But this is how science advances, and the study for a degree of PhD requires that the current position be adequately researched as a prerequisite to moving on to one's particular research topic. This is called the Literature Survey, and it is a fundamental part of the PhD. Fail to perform this adequately, and your PhD is doomed.
As you will not follow urls, I append a short article to help you, and as you are prone to ISP failures, I may repost this from time-to-time.
Note particularly the paragraph headed "RESEARCH METHODS".
For the information of other readers, the url is
GENERAL COMMENTS ON RESEARCH
PREAMBLE: Research, by its very nature, is a step into the unknown and therefore open-ended; there are no guarantees. As such your supervisor(s) will not know the answer to your research questions (research is not the same as coursework). This step is usually guided by the results of previous researchers in the field. Such previous work "sets the scene"/points you in the right direction/tells you where to look. Steady, methodical and persistent effort on your part is then necessary to reach your research goal, often employing the scientific/experimental method(s) (e.g. hypothesis testing). Of itself, this might not be sufficient; genuine insight, serendipity and unexpected "connections" from seemingly unrelated areas are often necessary. These can neither be anticipated nor manifested at will. Many scientific breakthroughs come from the most unexpected sources.
RESEARCH METHODS: In order to (a) become familiar with your chosen area of research, and (b) to ensure you don't "reinvent the wheel" and commence working on a topic which has been previously researched, it is essential to become familiar with the published literature in the field. A good way of doing this is to write your own literature survey/review article, perhaps even presenting a seminar/conference paper on your findings. This helps you not only to familiarise yourself with previous work, but also to highlight what has yet to be done/what problems remain to be solved in your chosen field. It also helps to identify areas in which you are perhaps weak and need to learn and/or improve your skills. The first six months of a 3-year PhD programme should be devoted to a literature survey; the second six months to replicating previous work. By the end of the first year, it should become clear as to how the earlier work can be extended/improved, thus enabling a detailed research proposal to be formulated. Naturally, the remaining two years are spent in following these ideas (and periodically backtracking and revising your research plan in the light of your findings).
NOTE: For Research Masters (and undergraduate Honours), it is quite valid to work on a topic which has been researched previously, but from a different perspective/extending it in some manner. For a PhD, an original contribution to knowledge is required - establishing what has been done previously and identifying a substantial problem to tackle is even more critical here. Successfully applying new/different (and better) techniques to problems previously solved by other means is still a valid approach for a PhD however. In order to conduct a literature survey, you will need to hone your library skills, specifically: (i) how to track down survey papers/introductory books, (ii) developing the art of quickly reading and evaluating abstracts (at least - entire papers if appropriate), (iii) identification of the classic references in the field, and subsequently tracking them down (in hard copy form, either within the UoW Library, or via Inter-Library Loans), (iv) use of the UoW on-line Library resources, as well as more general searching of the World Wide Web, & (v) the ability to critically evaluate what's been done previously. In short, who are the key researchers in the field? What are the seminal works/books/survey papers? What are the most important journals in your chosen area?
NOTE: It is very important to keep abreast of the latest developments in the field, especially if someone publishes what you are currently working on. If this happens, you may need to take a significant change of direction with your work. Thus periodic updates of your literature survey will be necessary during the course of your study.