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David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Thu Jul 13, 2006 10:35 am    Post subject: Re: Absolutely continuous, L^2 question

On Wed, 12 Jul 2006 15:39:57 EDT, James <james545@gmail.com> wrote:

 Quote: In article 5036829.1152728209310.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to 0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that f is AC on [0,a] where a is greater than x+h. So you have that f is uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h ---> 0 for all x in [0,a] doesn't give you what you want. If you want to say that f is uniformly continuous beyond a, then when you write |f(x+h) - f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s for all x in [0,a]. But what I am saying is that if you want to check uniform continuity at y a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo). If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | <= [int_[x,y] |f'|^2)^(1/2)]*|y-x|^(1/2) <= [int_[0,oo) |f'|^2)^(1/2)]*|y-x|^(1/2). So there is a constant C such that |f(y) - f(x)| <= C*|y-x|^(1/2) for all x, y in [0,oo). It's really the first part of your response that I am having trouble with. You say "If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | It seems to me that what you are basically saying is that f is absolutely continuous on all [0,oo). In order to say that |f(y) - f(x)| = |int_[x,y] f' |, you need to first say that f(y) = int_[0,y] f' and f(x) = int[0,x] f'. In order to say those two things you need to say that f is absolutely continuous on [0,z] where z is greater than y and x. So ok, given x and y, there is a z that makes this work.

Right. So what the heck is the problem?

 Quote: If you pick x' and y', then there is a z' that makes this work. So it changes.

So what? It's still true that for any x and y

f(y) - f(x) = int_x^y f'(t) dt,

because f is AC on [x,y]. How does the fact that if you change x and y
then you change x and y affect this?

 Quote: I am missing a logical step here. Side question : If f is absolutely continuous on [0,x] for all x > 0, then does this imply that f is absolutely continuous on [0,oo)?

No. And that doesn't matter one bit - all we use above is the fact
that it's AC on [x,y].

 Quote: James

************************

David C. Ullrich
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

Posted: Wed Jul 12, 2006 8:10 pm    Post subject: Re: Absolutely continuous, L^2 question

In article
<30772584.1152733227614.JavaMail.jakarta@nitrogen.mathforum.org>, James
<james545@gmail.com> wrote:

 Quote: In article 5036829.1152728209310.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to 0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that f is AC on [0,a] where a is greater than x+h. So you have that f is uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h ---> 0 for all x in [0,a] doesn't give you what you want. If you want to say that f is uniformly continuous beyond a, then when you write |f(x+h) - f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s for all x in [0,a]. But what I am saying is that if you want to check uniform continuity at y a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo). If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | <= [int_[x,y] |f'|^2)^(1/2)]*|y-x|^(1/2) <= [int_[0,oo) |f'|^2)^(1/2)]*|y-x|^(1/2). So there is a constant C such that |f(y) - f(x)| <= C*|y-x|^(1/2) for all x, y in [0,oo). It's really the first part of your response that I am having trouble with. You say "If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | It seems to me that what you are basically saying is that f is absolutely continuous on all [0,oo). In order to say that |f(y) - f(x)| = |int_[x,y] f' |, you need to first say that f(y) = int_[0,y] f' and f(x) = int[0,x] f'. In order to say those two things you need to say that f is absolutely continuous on [0,z] where z is greater than y and x. So ok, given x and y, there is a z that makes this work. If you pick x' and y', then there is a z' that makes this work. So it changes. I am missing a logical step here. Side question : If f is absolutely continuous on [0,x] for all x > 0, then does this imply that f is absolutely continuous on [0,oo)?

You're straining at a gnat. You're told f is a.c. on [0,x] for all x.
That's all you need. You don't need a.c. on all of R.

It doesn't matter that x and y can change. IN THIS SUBARGUMENT, THEY
DON'T. x and y are fixed, and we claim that for THESE x and y,
f(y)-f(x) = \int_x^y f'. (And you don't need to do this by subtracting
integrals; the fact that f is a.c. on [0,y] IMPLIES, a fortiori, that f
is a.c. on [x,y] (look at the DEFINITION of a.c.); hence that f(y)-f(x)
= \int_x^y f'.)

--
Ron Bruck

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Joined: 24 Mar 2005
Posts: 790

Posted: Wed Jul 12, 2006 8:04 pm    Post subject: Re: Absolutely continuous, L^2 question

In article
<30772584.1152733227614.JavaMail.jakarta@nitrogen.mathforum.org>,
James <james545@gmail.com> wrote:

 Quote: In article 5036829.1152728209310.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to 0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that f is AC on [0,a] where a is greater than x+h. So you have that f is uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h ---> 0 for all x in [0,a] doesn't give you what you want. If you want to say that f is uniformly continuous beyond a, then when you write |f(x+h) - f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s for all x in [0,a]. But what I am saying is that if you want to check uniform continuity at y a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo). If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | <= [int_[x,y] |f'|^2)^(1/2)]*|y-x|^(1/2) <= [int_[0,oo) |f'|^2)^(1/2)]*|y-x|^(1/2). So there is a constant C such that |f(y) - f(x)| <= C*|y-x|^(1/2) for all x, y in [0,oo). It's really the first part of your response that I am having trouble with. You say "If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | It seems to me that what you are basically saying is that f is absolutely continuous on all [0,oo). In order to say that |f(y) - f(x)| = |int_[x,y] f' |, you need to first say that f(y) = int_[0,y] f' and f(x) = int[0,x] f'. In order to say those two things you need to say that f is absolutely continuous on [0,z] where z is greater than y and x. So ok, given x and y, there is a z that makes this work. If you pick x' and y', then there is a z' that makes this work. So it changes.

f is AC on every bounded subinterval [a,b] of [0,oo). Therefore
f(b) - f(a) = int_[a,b] f' for every bounded subinterval [a,b] of
[0,oo). I'm not sure how to make this clearer; I'm just using the
central basic theorem as it is stated.

 Quote: Side question : If f is absolutely continuous on [0,x] for all x > 0, then does this imply that f is absolutely continuous on [0,oo)?

Certainly not.
James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 12, 2006 7:54 pm    Post subject: Re: Absolutely continuous, L^2 question

 Quote: In article 5036829.1152728209310.JavaMail.jakarta@nitrogen.mathf orum.org>, James james545@gmail.com> wrote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to |0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that |f is AC on [0,a] where a is greater than x+h. So you have that f is |uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h |---> 0 for all x in [0,a] doesn't give you what you want. If you want to |say that f is uniformly continuous beyond a, then when you write |f(x+h) - |f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 |there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s |for all x in [0,a]. But what I am saying is that if you want to check |uniform continuity at y > a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo). Oh crap, I'm sick and tired of this thread. Look: for 0 < x < y, |f(y)^2 - f(x)^2| = |\int_x^y 2 f(t) f'(t) dt| (I'll leave the fact that f^2 is ac, hence (f^2)' = 2ff' a.e. to you), = (\int_x^y |f(t)|^2 dt)^(1/2) (\int_x^y int_x^y |f'(t)|^2 dt)^(1/2) = (\int_x^\infty |f|^2)^(1/2) (\int_x^\infty ^\infty |f'|^2)^(1/2) and here's the point: if g \in L^2, then \int_x^\infty |g|^2 --> 0 as x --> infinity. From this you should see that f(x) is a Cauchy net as x --> infinity, hence has a limit. Now what kind of limits can an L^2 function have?

Thank you, and you probably mean "(f(x))^2 is a Cauchy net as x ---> infinity".

 Quote: -- Ron Bruck Posted Via Usenet.com Premium Usenet Newsgroup p Services ------------------------------------------------------ ---- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY MITY ** ------------------------------------------------------ ---- http://www.usenet.com
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

Posted: Wed Jul 12, 2006 7:41 pm    Post subject: Re: Absolutely continuous, L^2 question

In article
<5036829.1152728209310.JavaMail.jakarta@nitrogen.mathforum.org>, James
<james545@gmail.com> wrote:

 Quote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to |0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that |f is AC on [0,a] where a is greater than x+h. So you have that f is |uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h |---> 0 for all x in [0,a] doesn't give you what you want. If you want to |say that f is uniformly continuous beyond a, then when you write |f(x+h) - |f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 |there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s |for all x in [0,a]. But what I am saying is that if you want to check |uniform continuity at y > a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo).

Oh crap, I'm sick and tired of this thread. Look: for 0 < x < y,

|f(y)^2 - f(x)^2| = |\int_x^y 2 f(t) f'(t) dt|

(I'll leave the fact that f^2 is ac, hence (f^2)' = 2ff' a.e. to you),

<= (\int_x^y |f(t)|^2 dt)^(1/2) (\int_x^y |f'(t)|^2 dt)^(1/2)

<= (\int_x^\infty |f|^2)^(1/2) (\int_x^\infty |f'|^2)^(1/2)

and here's the point: if g \in L^2, then \int_x^\infty |g|^2 --> 0 as
x --> infinity. From this you should see that f(x) is a Cauchy net as
x --> infinity, hence has a limit. Now what kind of limits can an L^2
function have?

--
Ron Bruck

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James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 12, 2006 7:39 pm    Post subject: Re: Absolutely continuous, L^2 question

 Quote: In article 5036829.1152728209310.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to 0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that f is AC on [0,a] where a is greater than x+h. So you have that f is uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h ---> 0 for all x in [0,a] doesn't give you what you want. If you want to say that f is uniformly continuous beyond a, then when you write |f(x+h) - f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s for all x in [0,a]. But what I am saying is that if you want to check uniform continuity at y a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo). If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | <= [int_[x,y] |f'|^2)^(1/2)]*|y-x|^(1/2) <= [int_[0,oo) |f'|^2)^(1/2)]*|y-x|^(1/2). So there is a constant C such that |f(y) - f(x)| <= C*|y-x|^(1/2) for all x, y in [0,oo).

It's really the first part of your response that I am having trouble with. You say
"If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' |

It seems to me that what you are basically saying is that f is absolutely continuous on all [0,oo). In order to say that |f(y) - f(x)| = |int_[x,y] f' |, you need to first say that f(y) = int_[0,y] f' and f(x) = int[0,x] f'. In order to say those two things you need to say that f is absolutely continuous on [0,z] where z is greater than y and x. So ok, given x and y, there is a z that makes this work. If you pick x' and y', then there is a z' that makes this work. So it changes.

I am missing a logical step here. Side question : If f is absolutely continuous on [0,x] for all x > 0, then does this imply that f is absolutely continuous on [0,oo)?

James
science forum Guru

Joined: 24 Mar 2005
Posts: 790

Posted: Wed Jul 12, 2006 7:13 pm    Post subject: Re: Absolutely continuous, L^2 question

In article
<5036829.1152728209310.JavaMail.jakarta@nitrogen.mathforum.org>,
James <james545@gmail.com> wrote:

 Quote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).) Dear Wade, I don't see it. You say there is a universal delta on all of [0,oo)? |f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to 0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that f is AC on [0,a] where a is greater than x+h. So you have that f is uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h ---> 0 for all x in [0,a] doesn't give you what you want. If you want to say that f is uniformly continuous beyond a, then when you write |f(x+h) - f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s for all x in [0,a]. But what I am saying is that if you want to check uniform continuity at y a, I think that this delta (or s) changes. In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo).

If 0 <= x < y, you have |f(y) - f(x)| = |int_[x,y] f' | <=
[int_[x,y] |f'|^2)^(1/2)]*|y-x|^(1/2) <= [int_[0,oo)
|f'|^2)^(1/2)]*|y-x|^(1/2). So there is a constant C such that
|f(y) - f(x)| <= C*|y-x|^(1/2) for all x, y in [0,oo).
James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 12, 2006 6:16 pm    Post subject: Re: Absolutely continuous, L^2 question

 Quote: In article 1820258.1152720081570.JavaMail.jakarta@nitrogen.mathf orum.org>, James wrote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x -- oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x]. I don't think you know what his argument is. (Yes, f is UC on [0,oo).)

I don't see it. You say there is a universal delta on all of [0,oo)?

|f(x+h) - f(x)| = |int_[x,x+h] f' | <= ||f'||_2 * h^(1/2) ---> 0 as h goes to 0. But to justify |f(x+h) - f(x)| = |int_[x,x+h] f' | you need to say that f is AC on [0,a] where a is greater than x+h. So you have that f is uniformly continuous on [0,a]. The fact that |f(x+h) - f(x)| --> 0 as h ---> 0 for all x in [0,a] doesn't give you what you want. If you want to say that f is uniformly continuous beyond a, then when you write |f(x+h) - f(x)| ---> 0 as h --> 0 (uniformly in x), this means that for all eps > 0 there is an s > 0 (i.e. delta > 0) with |f(x+h) - f(x)| < eps for all h < s for all x in [0,a]. But what I am saying is that if you want to check uniform continuity at y > a, I think that this delta (or s) changes.

In any case, whether or not my babbling above makes any sense, please share with me how you are getting your universal delta for the uniform continuity of f on all of[0,oo).

Thank you,

James
science forum Guru

Joined: 24 Mar 2005
Posts: 790

Posted: Wed Jul 12, 2006 5:49 pm    Post subject: Re: Absolutely continuous, L^2 question

In article
<1820258.1152720081570.JavaMail.jakarta@nitrogen.mathforum.org>,
James <james545@gmail.com> wrote:

 Quote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x. Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x].

I don't think you know what his argument is. (Yes, f is UC on
[0,oo).)
James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 12, 2006 4:00 pm    Post subject: Re: Absolutely continuous, L^2 question

 Quote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x.

Are you asserting that f is uniformly continuous on all of [0,oo)? Your argument shows that f is uniformly continuous on [0,x] since f is AC on [0,x].

 Quote: Unfortunately our f is absolutely continuous only on [0,x] for all x > 0, but I think I'd need it to be on [0,oo). Even then, it is f^2 that would need to be uniformly continuous to imply that f^2 is in L^1[0,oo), so f^2 goes to 0. 2) f is absolutely continuous, so lim f(x) = lim int_[0,x] f ' = int_[0,oo] f '. I couldn't show that the latter is 0. I would appreciate any help you can give, James ************************ David C. Ullrich
eugene
science forum Guru

Joined: 24 Nov 2005
Posts: 331

Posted: Wed Jul 12, 2006 3:05 pm    Post subject: Re: Absolutely continuous, L^2 question

James wrote:
 Quote: James wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo. Unfortunately our f is absolutely continuous only on n [0,x] for all x > 0, but I think I'd need it to be on [0,oo). Even then, it is f^2 that would need to be uniformly continuous to imply that f^2 is in L^1[0,oo), so f^2 goes to 0. 2) f is absolutely continuous, so lim f(x) = lim int_[0,x] f ' = int_[0,oo] f '. I couldn't show that the latter is 0. I would appreciate any help you can give, James Just a remark: Doesn't it folows from that f,f' in L^2 [0,oo] that f and f' are bounded and from |f(x) - f(y)| <= sup_R |f'| (x-y), sup|f| oo you get that f^2 is uniformly continuous ? Why does it follow that f,f' are bounded? I also wouldn't see why f^2 is uniformly continuous in your case either.

Yes, you'd better follow David :
|f(x+h) - f(x) | <= int_[x,x+h] |f'(t)| dt <= || f' ||_2 * sqrt(h)
which implies the uniform continuity, form which for any given e > 0
you can chose a > 0 such that
| f(x) -f(y) | < e whenever |x-y| < a, so there exist a > 0 such that
|f(x)| = | f(x) - f(0) | < e whenever |x| < a .
James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 12, 2006 2:24 pm    Post subject: Re: Absolutely continuous, L^2 question

 Quote: James wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo. Unfortunately our f is absolutely continuous only on n [0,x] for all x > 0, but I think I'd need it to be on [0,oo). Even then, it is f^2 that would need to be uniformly continuous to imply that f^2 is in L^1[0,oo), so f^2 goes to 0. 2) f is absolutely continuous, so lim f(x) = lim int_[0,x] f ' = int_[0,oo] f '. I couldn't show that the latter is 0. I would appreciate any help you can give, James Just a remark: Doesn't it folows from that f,f' in L^2 [0,oo] that f and f' are bounded and from |f(x) - f(y)| <= sup_R |f'| (x-y), sup|f| oo you get that f^2 is uniformly continuous ?

Why does it follow that f,f' are bounded? I also wouldn't see why f^2 is uniformly continuous in your case either.
James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 12, 2006 2:17 pm    Post subject: Re: Absolutely continuous, L^2 question

 Quote: On Wed, 12 Jul 2006 09:00:56 EDT, James james545@gmail.com> wrote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo. Well, that fact that f' is in L^2 shows that f is uniformly continuous: |f(x+h) - f(x)| = ______ <= _______, which tends to 0 as h -> 0, uniformly in x.

I already knew that f is uniformly continuous. It is assumed that f is absolutely continuous. That isn't my problem...

 Quote: Unfortunately our f is absolutely continuous only on [0,x] for all x > 0, but I think I'd need it to be on [0,oo). Even then, it is f^2 that would need to be uniformly continuous to imply that f^2 is in L^1[0,oo), so f^2 goes to 0. 2) f is absolutely continuous, so lim f(x) = lim int_[0,x] f ' = int_[0,oo] f '. I couldn't show that the latter is 0. I would appreciate any help you can give, James ************************ David C. Ullrich
David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Wed Jul 12, 2006 2:02 pm    Post subject: Re: Absolutely continuous, L^2 question

On Wed, 12 Jul 2006 09:00:56 EDT, James <james545@gmail.com> wrote:

 Quote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x > 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo.

Well, that fact that f' is in L^2 shows that f is uniformly
continuous:

|f(x+h) - f(x)| = ______ <= _______,

which tends to 0 as h -> 0, uniformly in x.

 Quote: Unfortunately our f is absolutely continuous only on [0,x] for all x > 0, but I think I'd need it to be on [0,oo). Even then, it is f^2 that would need to be uniformly continuous to imply that f^2 is in L^1[0,oo), so f^2 goes to 0. 2) f is absolutely continuous, so lim f(x) = lim int_[0,x] f ' = int_[0,oo] f '. I couldn't show that the latter is 0. I would appreciate any help you can give, James

************************

David C. Ullrich
eugene
science forum Guru

Joined: 24 Nov 2005
Posts: 331

Posted: Wed Jul 12, 2006 1:29 pm    Post subject: Re: Absolutely continuous, L^2 question

James wrote:
 Quote: This question has been killing me. Please help with any insight: Let f be absolutely continuous on [0,x] for all x > 0. Let f, f ' be in L^2[0,oo). Let f(0) = 0. Prove that lim f(x) = 0 x --> oo There was a part (a) to this problem that I got : Prove that int_[0,x] |f f '| <= .5* (int_[0,x] |f '| )^2. But I don't see how this helps for part (b). I have tried several ways to prove that lim f(x) = 0 as x --> oo. 1) I have proven that if f is in L^1[0,oo) and f is uniformly continuous, then lim f(x) = 0 as x --> oo. Unfortunately our f is absolutely continuous only on [0,x] for all x > 0, but I think I'd need it to be on [0,oo). Even then, it is f^2 that would need to be uniformly continuous to imply that f^2 is in L^1[0,oo), so f^2 goes to 0. 2) f is absolutely continuous, so lim f(x) = lim int_[0,x] f ' = int_[0,oo] f '. I couldn't show that the latter is 0. I would appreciate any help you can give, James

Just a remark: Doesn't it folows from that f,f' in L^2 [0,oo] that f
and f' are bounded and from |f(x) - f(y)| <= sup_R |f'| (x-y), sup|f| <
oo you get that f^2 is uniformly continuous ?

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