Timeline for some strange orthogonal basis and an integral equation with it
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 25, 2010 at 21:04 | comment | added | Willie Wong | Nope. The span will be the same. | |
| May 25, 2010 at 20:48 | comment | added | vilvarin | It wouldn't even work if we extend the basis $p_j=p_(j+K)$, would it?(sorry) | |
| May 25, 2010 at 20:19 | vote | accept | vilvarin | ||
| May 25, 2010 at 19:50 | comment | added | Willie Wong | If you assume $f$ is in the span of $p_j$, then this is a non-issue, since the dimension is finite. If $f$ is not in the span of $p_j$, then it is also a non-issue, since trivially you cannot recover $f$ from a linear combination of $p_j$s. So I don't understand why this is a question. | |
| May 25, 2010 at 19:14 | comment | added | vilvarin | thank you very much. The last question I have is if Fourier inversion theorem is valid for this basis. It is not true in general case, isn't it? Probably i need to check this | |
| May 25, 2010 at 16:25 | comment | added | Willie Wong | .. I meant to write that $\langle f,p_j\rangle = \delta_{jk}a_{-1,K}$ and $a_{-1,K}p_K \neq f$ unless all other $a_{s,K} = 0$. Sorry for the typo. | |
| May 25, 2010 at 16:24 | comment | added | Willie Wong | ... so consider $f(x) = 1$, the constant function on $[0,1]$. Compute $\langle f, p_j\rangle = \delta_{jK} a_{-1,0}$ (in your notation where numbering of $j = 1... K$. But obviously $a_{-1,0} p_0 \neq f$ unless all other $a_{s,0} = 0$. But if you have some a priori knowledge that all the functions you care about lies in the span of the $p_j$s, then you don't even need Fourier analysis: you are just working with finite dimensional linear algebra on some inner product space. | |
| May 25, 2010 at 16:20 | comment | added | Willie Wong | It is a basis of a finite dimensional sub-space, so not all functions will be in the span of $p_j$. Recall that the Fourier coefficients can be defined as $\hat{f}(\xi) = \langle f(\cdot), \exp(2\pi i \xi \cdot)\rangle$. A general representation formula is that, for some Hilbert space $H$ with orthonormal bases $e_1,e_2, \ldots$ you can write $f = \sum \lambda_j e_j$ where $\lambda_j = \langle f,e_j\rangle$ (this essentially captures the Fourier transform and its inverse, completely analogous to the finite dimension inner product space case)... | |
| May 25, 2010 at 14:31 | comment | added | vilvarin | I am wondering if it is possible to use basis $p_j$ instead of$\exp(2\pi ij)$ to define Fourier transform, inverse fourier transform and so on. | |
| May 25, 2010 at 13:46 | history | answered | Willie Wong | CC BY-SA 2.5 |