Hi all,
In my numerical analysis class we have been discussing lagrange and chebyshev interpolation
my professor derived a bound for the chebyshev interpolation [math] Q_nf(x) [/math] that is
[math] |Q_nf(x)| \leq ... \leq ||f||_{\infty} \int_0^{\pi} |\frac{\sin((n+1/2)\theta)}{2\sin(\theta /2)} | d\theta [/math] where [ math] \theta = \arccos x [/math]
they then conclude that [math] ||Q_n|| = \int_0^{\pi} |\frac{\sin((n+1/2)\theta)}{2\sin(\theta /2)} | d\theta [/math]
So my question is how do you make the jump
obviously we can take the sup over [math] Q_nf(x) [/math] to get [math] \frac{||Q_nf(x)||_{\infty}}{||f||_{\infty}} \leq \int_0^{\pi} |\frac{\sin((n+1/2)\theta)}{2\sin(\theta /2)} | d\theta [/math] but how do we get the final result about operator norm? seems more like we're bounding [math] ||Q_n|| [/math] instead of equating it
What is Q_n(f)(x)?That supremum norm is taken over what set?
>>7984381
[math] Q_n f(x) = \sum_{k=0}^n <f,\phi_k> \phi_k(x) [/math] where [math] <g,h> = \int_a^b g(x)h(x)w(x) dx [/math] where [math] w(x) [/math] is some weight function on given fixed points [math] x_0 \leq x_1 \leq ... \leq x_n, x_i \in [a,b] [/math]
>>7984381
so the sup is taken over [math] x \in [a,b] [/math]
>>7984405
ah fuck i forgot [math] {\phi_0,...,\phi_n}[/math] form an orthonormal basis
>>7984405
So w(x) is a step funcion, what do you mean by "weight function on given fixed points"?
>>7984447
The same guy, I thinkt the thing is that you have to find a norm 1 function that will realise the supremum norm. I would try to plug a chybyshev polynomial for f.
>>7984447
i don't think it's a step function,... it's a term to account for how much you want to count each point in the integral
https://en.wikipedia.org/wiki/Orthogonality#Orthogonal_functions
>>7984456
i don't quite understand what you mean
my prof went from the bound i did to the conclusion about the operator norm without skipping a beat so i think i either dont quite understand the mateiral or am overlooking something
>>7984468
Then show his proof, maybe we can workout something from it.
>>7984479
ok uses basic def to get going
>>7984502
then develops its a bit into an intergral over cos
then turns the cos term into the sin term we want
finishing turning it into a sign
then thats the last we saw of it in class
>>7984508
>sign
meant sin
then on an assignment he says that we showed the result mentioned in OP, in class, when all we did was bound [math] |Q_nf(x)| [/math] from above
>>7984479
i really don't think the derivation adds anything...
my confusion is just about how bounding the norm of a function can be turned into a sup norm
>>7984515
Holy fuck, I am retarted, >>7984504
The inequality on the top of the page always hodls, so we need only to find a function f witch sup norm equal to 1, such that in that ineaquality we just have equality. But just look on >>7984502
If you plug in f(x)=1 on whole [a,b], then the inequality from >>7984504
becomes an equality.
>>7984358
Your numerical analysis class sounds so much better than mine which is basically designed for engineers.
>>7984586
Well, the idea was that if you put f(x)=1, then you get
[math]Q_nf(x)=\int\sum\phi_k(y)\phi_k(x)w(y)f(y)dy =\int\sum\phi_k(y)\phi_k(x)w(y)dy[/math]
but you actually want to get
[math]Q_nf(x)=\int\sum\phi_k(y)\phi_k(x)w(y)f(y)dy =\int|\sum\phi_k(y)\phi_k(x)w(y)|dy[/math]
So you should use [math]f(x)=sgn(\sum\phi_k(x)(w(x)).[/math] Buuuut that function is not contnouos. I think we can use Urynsohns lemma to approximate [math]f(x)=sgn(\sum\phi_k(x)(w(x)).[/math] with cont. functions g_j(x) in such way that
[math]\lim_{j\to\infty}\int\sum\phi_k(y)\phi_k(x)w(y)g_j (y)dy =\int|\sum\phi_k(y)\phi_k(x)w(y)|dy [/math]
>>7984771
The same guy. This will actually work if we assume that the function [math]w\in C^1[a,b][\math]
>>7984753
That is called numerical METHODS not numerical analysis. It is all an engineer needs.
>>7984358
what the fuck is this? I am kindly asking someone to explain this. Is this meme or is it significant?
>>7985567
Numerical analysis. It's applied math, so it's pretty important. It tells you how good an approximation is.
>>7985567
basically you're approximating a function based on n points
you can read more here:
https://en.wikipedia.org/wiki/Interpolation
>>7984771
>>7984802
update, i talked to my prof today and this is the general idea
you have to explicitly find an f (and i think yours works here although i haven't looked too closerly) to conclude (he skipped the step when concluding on our assignment
