Thread: how to find the inner product formulla

1. how to find the inner product formulla

once i solve that one is the derivative of the other

but here its much harder to guess the formulla
http://i47.tinypic.com/ixt74i.jpg

what is the general method?

2. Fourier series

It is suffice to prove the following statement:
$\displaystyle \lim_{n\to \infty} \int_{0}^{\pi} f(x)\sin{nx}=0 \text { and }\lim_{n\to \infty} \int_{0}^{\pi} f(x)\cos{nx}=0$
These conclusion follows immediately from the Fourier series in $\displaystyle L^{2}\text{ space}$.

3. i want to find the formula for the inner product which defines such norm.

what it has to do with proving that its foorier coefficients
?

4. The Trigonometric function system is a complete orthonormal base for the Hilbert Space $\displaystyle L^{2}$ .

5. i am looking for answer like
<f,g>=\int f(x)g(x) + etc..

an actual formula

i cant translate your words into such formula

6. To prove the first statement : Extend the domain of f to $\displaystyle [-\pi ,\pi]$ such that f is odd function.
To prove the second statement : Extend the domain of f to $\displaystyle [-\pi ,\pi]$ such that f is even function.
The inner product is defined as:
$\displaystyle < f , g >=\int_{-\pi}^{\pi} f *g dx$
in both case.

7. what prove?

i dont ask to prove anything

i ask how to find the formula which defines this norm

8. norm ? sorry , there may be some mistakes , I didn't see any norm in your question and picture.

9. ohh sorry its the wrong foto
i will change it in a moment

10. http://i47.tinypic.com/ixt74i.jpg

a little change

"which defines the minimal"

11. i need to find alpha beta and gama
so this expression will be minimal

i know how to solve such stuff
usually
i have a vector and a subspace to make a projection of the vector

so i make an orthogonal basis and then i make a projection of that vector
into my space

and then the difference between that vector and the original vector is the minimal

but in order to do all that i need the
inner product formula which defines this norm.

usually i figured out the formula by guessing

but here i cant guess

so i am asking if there is a general method
?

12. Product Hilbert space

Since $\displaystyle \mathbb{C}$ and $\displaystyle L^{2}$ are both complex Hilbert Space,
Then $\displaystyle \mathbb{C}\times L^{2}$ is also Hilbert Space under the inner product defined by:
$\displaystyle < (r,f(x)) , (s,g(x)) >= r*\overline{s}+\int_{0}^{1}f(x)\overline{g(x)}dx$
The norm is induced from the inner product.
The minimal problem is equivalent to find the distance between point $\displaystyle (1,3x^2)$ and The closed subspace spanned by $\displaystyle (1,1)$ and $\displaystyle (1,2x)$ and $\displaystyle (1,0)$.
Indeed, Since $\displaystyle \alpha$ is a free variable, you can eliminate the $\displaystyle |1-(\alpha+\beta+\gamma)|^{2}$ part, it doesn't affect the final minimal value (but it has something to do with minimal point).

13. how??

what is (r,f(x)) ?

what is (s,g(x))?

what is s? what is r?

14. Here $\displaystyle \mathbb{C}\times L^2$ is the cartesian product of $\displaystyle \mathbb{C}$ (the complex number field) and $\displaystyle L^2$, and r , s are any complex numers, f(x) and g(x) are any elements of $\displaystyle L^2$.
Do you know what is Cartesian Product?

15. its the sum of the multiplication on coordinates with the same index.

how to get the formula from your definition

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