orthonormal set of functions

T

tremor

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Hi, I'm hoping someone can help me get started with this problem as I'm not entirely sure what they're looking for:

We wish to approximate the function \(\displaystyle x(t) = e^{t}\)
over the interval (0, 1), using a second order polynomial.
a). From the set of linearly independent function,\(\displaystyle [1, t, t^{2}]\)form an orthonormal set of functions. The inner product is defined as \(\displaystyle <f(t), g(t)>=\int_0^1\! f(t)g(t) \, \mathrm{d}x\)
b). Based on this set of orthonormal functions, fit the best approximation in the least square error sense, that is minimize the norm of the error between the function x(t) and its approximation.

I think I understand how to approximate the function but I'm not sure how to get an orthonormal set of functions or part b. Any help would be appreciated - thanks!!
 
That makes absolutely no sense. Without the basis, how could you find the projection?

Use your Gramm-Schmidt Orthogonalization process and find the Orthogonal Basis. Great place to start.
 
ok so i found the orthogonal basis and i'm trying for part b. i've tried finding a good formula for this but i haven't been able to locate one. the best i found was this:

http://en.wikipedia.org/wiki/Approximation_theory

but that does not seem to take into account the orthogonal basis...only the original function and the second order polynomial. is this the correct method or is there a better way? thanks!
 
Try googling Fourier series. The topic of orthonormal sets often arises in the study of them.

Here is a link.

http://online.redwoods.cc.ca.us/instruc ... al1028.pdf


But, you can use Gram-Schmidt to find the orthonormal basis. Then, the m.s.e is \(\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]^{2}dx\)

Here's a start, using \(\displaystyle [1,t,t^{2}]\) with Gram-Schmidt.

\(\displaystyle v_{1}=1\)

\(\displaystyle (u_{2},v_{1})=\int_{0}^{1}tdt=\frac{1}{2}\)

\(\displaystyle v_{2}=\frac{t-\frac{1}{2}}{||t-\frac{1}{2}||}=\sqrt{12}(t-\frac{1}{2})\)

\(\displaystyle v_{2}=\sqrt{3}(2t-1)\)

\(\displaystyle (u_{3},v_{1})=\int_{0}^{1}t^{2}dt=\frac{1}{3}\)

\(\displaystyle (u_{3},v_{2})=\sqrt{3}\int_{0}^{1}(2t^{3}-t^{2})dt=\frac{\sqrt{3}}{6}\)

\(\displaystyle v_{3}=\frac{t^{2}-\frac{1}{3}-\frac{1}{2}(2t-1)}{||t^{2}-\frac{1}{3}-\frac{1}{2}(2t-1)||}=6\sqrt{5}\left(t^{2}-t+\frac{1}{6}\right)\)

\(\displaystyle v_{3}=\sqrt{5}(6t^{2}-6t+1)\)

\(\displaystyle v_{1}, \;\ v_{2}, \;\ v_{3}\) is the orthonormal basis.

As I said before, one sees this in Fourier series/functional analysis.
 
Well, I see your Orthonormal Basis was handed to you on a silver platter.

Now you just need a projection.

A couple of questions arise.

1) Do you have a text book?
2) If not, how and why were you presented with this problem?
3) If so, why on Earth are you looking up the basis on the internet?
 
i do have a textbook...this is not covered in any detail so i had to look elsewhere. i did not ask anyone to hand me an answer on a silver platter...i'm pretty sure i stated that i had calculated that already. i was simply asking if someone knew a formula that would help in part b. this is a help forum, is it not?? if you don't want to help then that is no big deal. thanks anyways.
 
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