# Thread: orthonormal set of functions

1. ## orthonormal set of functions

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!!

2. Originally Posted by coley0412
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!!

Remember your linear algebra and the Gram-Schmidt process to turn any lin. ind. set of vectors into an orthonormal set of vectors.

Tonio

3. yes i understand that but i'm looking for the formula or an example to do it for a function not a vector.

4. Originally Posted by coley0412
yes i understand that but i'm looking for the formula or an example to do it for a function not a vector.
What tonio said is true in ANY innerproduct space. The Gram-Schmidt process will work in any inner product space and is the solution to your question!

try it!!!

Gram

5. ok i understand how to do part a but i'm having trouble finding any material on part b. is there a formula for this? i do not see it mentioned in my textbook.

i have come across this:

$\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]^{2}dx$

would this be an appropriate formula?