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Topic: Question of generalized function definition
Replies: 7   Last Post: Mar 23, 2012 9:33 AM

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 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Question of generalized function definition
Posted: Mar 21, 2012 12:26 PM

On Wed, 21 Mar 2012 07:14:01 -0700 (PDT), vv <vanamali@netzero.net>
wrote:

>On Mar 21, 5:52 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>> Anyway, I don't understand the question. What's different from
>> what, and why do they appear to be the same?
>>
>> You say "phi(0) also seems to be a result of an argument that
>> is similar to what was used to obtain f(x0)."
>>
>> Yes, phi(0) is the _limit_ of a sequence f_n(phi), where
>> f_n is the same as the "phi_n" in the first paragraph
>> above. But that's the limit of a sequence - it's not
>> true that phi(0) is equal to int f phi for an actual
>> function f. That's not true here, and it's also not true
>> in the context of the first pargraph.
>>
>> So I really don't see exactly what difference you're
>> asking for an explanation of...

>
>Even while posting the query I knew I was not being clear; the
>argument didn't make sense because I haven't understood this stuff.
>Let me make one more attempt to state what is not clear. I am using
>the book's notation.
>
>The author says that Eq. (8), i.e., f(x0) being obtained from int
>f(x) phi_n(x) as the limit, can be justified from mean value theorem.
>phi_n(x) is located in a vanishingly small support region around x0,
>while maintaining unit area.
>
>The aurthor says that in Eq. (10), obtaining phi(0) from int phi(x)
>delta(x) requires the invention of generalized function and new
>machinery.

The difference is that (8) does not say that f(x_0) is equal to a
certain integral! It says that f(x_0) is the _limit_ of a _sequence
of integrals_. There's no such "limit as n -> infinity" in (10).

>
>What is confusing is why Eq. (10) not similar to Eq. (8) with x0 = 0.
>How confused I am is indicated by, "Is it because the roles of phi and
>f are interchanged?". If I am still not making myself clear, my
>apologies; maybe I'll come back later when I am able to ask the doubt
>more sensibly.
>
>--vv

Date Subject Author
3/21/12 Vanamali
3/21/12 David C. Ullrich
3/21/12 Vanamali
3/21/12 David C. Ullrich
3/21/12 Vanamali
3/22/12 David C. Ullrich
3/22/12 Vanamali
3/23/12 David C. Ullrich