Trigonometric system and orthogonality

by MinYoung Kim   Last Updated July 12, 2019 10:20 AM - source

I'm having trouble understanding the 2nd line. So, just looking at the 3rd line (4.9), I can see that this is true, ie, each term in the series will $$= 1$$ if $$j=k$$ by orthogonality of the trig system, and summing that N times gives N if $$j = k$$, but all terms 0 if $$j \neq k$$

But then, using the partial sum equation in line 2, if $$j = k$$, the partial sum is $$\frac{0}{0}$$?

Tags :

You have to handle the case $$j=k$$ separately. It is obvious that the sum is $$N$$ when $$j=k$$ because each term is just $$1$$.

The line just above (4.9) is not valid when $$j=k$$.

Kavi Rama Murthy
July 12, 2019 09:49 AM

Well large part of the issue seems that the first line is completly false. The identity does not hold when r=1 and also does not seem to hold when r is negative. Since this is a proof such information about r should be contained in hypothesis, and in that case the case of j=k has to be treated separately.

TheCoolDrop
July 12, 2019 10:18 AM