(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Thanks very much for reading.

I actually have two problems, I hope it's ok to state both of the in the same thread.

1. Let V_{n}be the space of all functions having the n'th derivitve in the point x_{0}.

I've been given the semi-norm (holds all the norm axioms other than ||v|| = 0 => v = 0) defined by:

||f|| = [itex]\sum^{n}_{k=0}\frac{1}{k!}|f^{(k)}(x_{0})|[/itex]

I need to show that [itex]||fg|| \leq ||f|| ||g||[/itex]

2. I need to prove that if the set of polynomials {P_{i}}, i=0,...,n is orthonormal on [a,b] in respect to the inner product defined by: [itex]<f,g> = \int ^{b}_{a}f(x)g(x)dx[/itex]

then they hold the recurrance relation:

P_{k+1}(x) = (a_{k}x + b_{k})P_{k}(x) + c_{k}P_{k-1}(x)

3. The attempt at a solution

1. This one's been a nightmare for me. I've tried proving it straightforward, and then using induction, but I just get lost in a sea of indexes. I'm thinking there might be a more elegant way, but can't see it.

I'm of course trying to use the triangle inequality, which works great for the cases n=1 and n=2, but I just cannot generlize it.

I could write the whole development I've done here, it's like one whole page, but it'll take me ages and therefore think it's rather unnescesarry. I'll greatly appreciate a hint. If there's no other way but an ugly induction I think I'll skip it :-)

(I've used the fact that [itex](fg)^{(k)}(x_{0}) = \sum^{k}_{i=0}(\frac{k!}{i!(k-i)!})f^{(k-i)}g^{(i)}(x_{0})[/itex]

2. Well, I've been thinking polynomial division. If I assume the set of given P_{i}'s is of ascending degrees (0,1,2...), I can always divide P_{k+1}(x) in P_{k}(x) and write:

[itex]P_{k+1}(x) = (a_{k}x + b_{k})P_{k}(x) + L(x) [/itex]

Where deg(L(x)) < deg(P_{k}(x))

I don't know how to take it from here. How do I use the orthogonality? I would greatly appreciate hints :-)

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# Norms and orthogonal Polynomials

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