Minimum Property of Fourier Coefficients

• May 2nd 2011, 04:11 AM
raed
Minimum Property of Fourier Coefficients
Dear Colleagues,

Because of LATEX technical problems I attached a file containing the problem.

Best Regards.
• May 2nd 2011, 08:13 AM
Opalg
If $\displaystyle y = \textstyle\sum\beta_je_j$ then $\displaystyle \textstyle\|x-y\|^2 = \bigl\langle x-\sum\beta_je_j, x-\textstyle\sum\beta_je_j\bigr\rangle$, so that

$\displaystyle \textstyle\|x-y\|^2 = \|x\|^2 - 2\text{Re}\sum\bigl(\langle x,e_j\rangle\beta_je_j\bigr)\overline{\beta_j} + \sum|\beta_je_j|^2.$

Also, $\displaystyle \textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = \sum|\langle x,e_j\rangle|^2 - 2\text{Re}\sum\bigl(\langle x,e_j\rangle\beta_je_j\bigr)\overline{\beta_j} + \sum|\beta_je_j|^2.$

Subtract that equation from the previous one to see that $\displaystyle \|x-y\|^2$ is smallest when $\displaystyle \textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = 0.$
• May 2nd 2011, 10:29 AM
raed
Thank you for your reply. But I proved that -as I mentioned in the attachments- and I need your help in proving the converse.

Best Regard.
• May 2nd 2011, 11:35 AM
Opalg
Quote:

Originally Posted by raed
Thank you for your reply. But I proved that -as I mentioned in the attachments- and I need your help in proving the converse.

Surely the equation $\displaystyle \textstyle\|x-y\|^2 = \|x\|^2 - \sum|\langle x,e_j\rangle|^2 + \sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2$ proves both the result and the converse?