Trigonometric system and orthogonality

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

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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 $\frac{0}{0}$?

Answers 2

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
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.

July 12, 2019 10:18 AM

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