Thread: Minimum Property of Fourier Coefficients

1. Minimum Property of Fourier Coefficients

Dear Colleagues,

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

Best Regards.

2. If $y = \textstyle\sum\beta_je_j$ then $\textstyle\|x-y\|^2 = \bigl\langle x-\sum\beta_je_j, x-\textstyle\sum\beta_je_j\bigr\rangle$, so that

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

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

Subtract that equation from the previous one to see that $\|x-y\|^2$ is smallest when $\textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = 0.$

3. 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.

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