Induction and Fundamental Theorem of Calculus for Bounded Linear Operators

[dirk_mec1]

Homework Statement

http://img389.imageshack.us/img389/9272/33055553mf5.png

The Attempt at a Solution

Via induction: for n=1 equality holds now assume that V_n=Jⁿ.
I introduce a dummy variable b and the fundamental theorem of calculus and change order of integration:

V_{n+1}f(t) = \int_a^t \frac{(t-s)^n}{n!} f(s) \mbox{d}s = \int_a^t \frac{t-s}{n} \frac{(t-s)^{n-1} } {(n-1)!} \mbox{d}s = \int_a^t \int_s^t \frac{1}{n} \frac{(t-s)^{n-1} } {(n-1)!} f(s) \mbox{d}b\ \mbox{d}s = \int_a^t \frac{1}{n} \int_a^b \frac{(t-s)^{n-1} } {(n-1)!} f(s) \mbox{d}s\ \mbox{d}b = \int_a^t \frac{1}{n} J^n f(b) \mbox{d}b<br />

<br /> = \frac{1}{n} J(Jf) = \frac{1}{n} J^{n+1}f


But how do I get rid of the 1/n?

 

[Palindrom]

First of all I would use a different "dummy" variable as b is the right end of the interval. Let's us x.

For the matter at hand, your fifth equality is wrong: you can't use the induction hypothesis here, since in order for it to work you have to have \left(x-s\right)^{n-1} in the numerator and not \left(t-s\right)^{n-1}.

I would take a slightly different approach (although induction is obviously the way to go).

 

[fikus]

you should use integration by parts to integrate
V_{n+1}f(t) = \int_a^t \frac{(t-s)^n}{n!} f(s) \mbox{d}s
try with
\frac{(t-s)^n}{n!} = u , f(s)= dv
It works for me.

 

[dirk_mec1]

First of all I would use a different "dummy" variable as b is the right end of the interval. Let's us x.

For the matter at hand, your fifth equality is wrong: you can't use the induction hypothesis here, since in order for it to work you have to have \left(x-s\right)^{n-1} in the numerator and not \left(b-s\right)^{n-1}.

Yes you're right, I should have used for example x. I still don't see the mistake it shouldn't read b-a it should read t-s.

I would take a slightly different approach (although induction is obviously the way to go).

Which is?

you should use integration by parts to integrate
V_{n+1}f(t) = \int_a^t \frac{(t-s)^n}{n!} f(s) \mbox{d}s
try with
\frac{(t-s)^n}{n!} = u , f(s)= dv
It works for me.

Yes that worked, thanks!

 

[Palindrom]

I don't see that I need to convince you anymore seeing as you solved it, but the induction hypothesis reads

J^{n}f\left(t\right)=\int_{a}^{t}\frac{\left(t-s\right)^{n-1}}{\left(n-1\right)!}f\left(s\right)ds

Or, since the argument is now x and not t,

J^{n}f\left(x\right)=\int_{a}^{x}\frac{\left(x-s\right)^{n-1}}{\left(n-1\right)!}f\left(s\right)ds

Right?

 

[dirk_mec1]

I don't see that I need to convince you anymore seeing as you solved it, but the induction hypothesis reads

J^{n}f\left(t\right)=\int_{a}^{t}\frac{\left(t-s\right)^{n-1}}{\left(n-1\right)!}f\left(s\right)ds

Or, since the argument is now x and not t,

J^{n}f\left(x\right)=\int_{a}^{x}\frac{\left(x-s\right)^{n-1}}{\left(n-1\right)!}f\left(s\right)ds

Right?

Yes you're right I see my mistake now thanks.

So I found this:

<br /> V_{n+1}f = \int_a^t \frac{(t-s)^n}{n!}\ f(s)\ \mbox{d}s = \left[ Jf(s) \frac{(t-s)^n}{n!} \right] _a^t + \int_a^t \frac{(t-s)^{n-1}}{(n-1)!)}\ Jf(s) \mbox{d}s = V_n(Jf) = J^n (Jf) =J^{n+1}f <br />

 

[dirk_mec1]

I'm also stuck at the second assignment:

http://img122.imageshack.us/img122/9168/97996319or7.png With definitions:

http://img383.imageshack.us/img383/715/34385925su8.png

Should I use Holders inequality?

 

[Pere Callahan]

Show that for any function f mit sup norm 1, you have ||J^nf||\leq||J^n\underline 1|| where \underline 1 is the constant one function. Calculate ||J^n\underline 1|| and conclude!

 

[dirk_mec1]

Show that for any function f mit sup norm 1, you have ||J^nf||\leq||J^n\underline 1|| where \underline 1 is the constant one function. Calculate ||J^n\underline 1|| and conclude!

Suppose ||J^nf||\leq||J^n\underline 1|| then how can I calculate ||J^n\underline 1||?

Is it this:

\int_a^b \int_a^t \left| \frac{(t-s)^{n-1}}{(n-1)!}\ \right| \mbox{d}s\ \mbox{d}t

 

[Pere Callahan]

What is the norm you are using? Normally, C([a,b]) is endowed with the sup norm. Are you using the L_1 norm?

 

[dirk_mec1]

What is the norm you are using? Normally, C([a,b]) is endowed with the sup norm. Are you using the L_1 norm?

Yes I'm using the L_1 norm but what do you mean by 'C([a,b]) is endowed by the sup norm' that's only valid if we use the infinity norm, right?

So you're saying that is should be:

<br /> \left| \left| \int_a^t \frac{(t-s)^{n-1}}{(n-1)!}\ \mbox{d}s\ \right| \right| _{\infty}<br />

 

[Palindrom]

In your assignment it says "calculate the norm in the Banach space...". The space C[a,b] with the L^1 norm is not a Banach space, and so if you take this norm the space BL(C[a,b]) is not a Banach space either. This should imply that they meant the supremum norm (and in fact, like Pere correctly noted, unless explicitly said otherwise the space C[a,b] is taken with the supremum norm).

 

[dirk_mec1]

In your assignment it says "calculate the norm in the Banach space...". The space C[a,b] with the L^1 norm is not a Banach space, and so if you take this norm the space BL(C[a,b]) is not a Banach space either. This should imply that they meant the supremum norm (and in fact, like Pere correctly noted, unless explicitly said otherwise the space C[a,b] is taken with the supremum norm).

So the expression below is wrong?

<br /> <br /> \left| \left| \int_a^t \frac{(t-s)^{n-1}}{(n-1)!}\ \mbox{d}s\ \right| \right| _{\infty}<br /> <br />

Then what should it be?

 

[Pere Callahan]

So the expression below is wrong?

<br /> <br /> \left| \left| \int_a^t \frac{(t-s)^{n-1}}{(n-1)!}\ \mbox{d}s\ \right| \right| _{\infty}<br /> <br />

Then what should it be?

This is correct; you just need to evaluate the integral and take the \sup_{t\in[a,b]}

 

[dirk_mec1]

This is correct; you just need to evaluate the integral and take the \sup_{t\in[a,b]}

Indeed the answer is deduced from this expression but how do you prove that:

<br /> ||J^nf||\leq||J^n\underline 1||<br /> ?

 

[Pere Callahan]

<br /> ||J^nf||=\sup_{t\in[a,b]}\left|\int_a^t{ds\frac{(t-s)^{n-1}}{(n-1)!}f(s}\right|<br /> \leq\int_a^b{ds\frac{(t-s)^{n-1}}{(n-1)!}|f(s)|}<br />
For any f with ||f||<=1, you can bound ||f(s)|| by 1, and you get what I wrote.