/sci/ - Science & Math

[Anonymous No.4544759]

I posted this question yesterday, with no answer provided. Maybe I'll be luckier today.

Let <span class="math">X=\{x_1,x_2,\ldots\}[/spoiler] be a set in <span class="math">l_2[/spoiler]. A set <span class="math">U=\{u_1,u_2,\ldots\}[/spoiler] is called the backward orthogonalization of X if U is orthogonal and for every <span class="math">n \geq 1[/spoiler]: <span class="math">\overline{Sp}\{x_n,x_{n+1},\ldots\} = \overline{Sp}\{u_n,u_{n+1},\ldots\}[/spoiler].

Let <span class="math">x_k=e_k+e_{k+1}[/spoiler] when <span class="math">e_k[/spoiler]'s coordinates are all zero but kth, which is 1. Find the backward orthogonalisation of such an X.


I am pretty sure I have to find scalars such as <span class="math">u_k=x_k+\sum_{j=k+1}^\infty \alpha(k,j)x_j[/spoiler], but I still haven't quite figured out what they were.

Thanks for any help.

[Anonymous No.4544769]

this has been asked on sci every day for like a year.

[Anonymous No.4544785]

>>4544769
I wrote it down yesterday, and that's it.
If it had been posted before, interesting.

[Anonymous No.4544810]

>>4544785
>I wrote it down yesterday, and that's it.

Explain this
http://archive.installgentoo.net/sci/thread/4526066