Question

Generalization of even/odd garbage for dan, check it @dan815

Answer 1

I found a garbage can

Answer 2

\[\LARGE \sum_{n=1}^N \sum_{k=1}^N \frac{ e^{i 2\pi n /N} f( e^{i 2\pi k /N}x)}{N}\]

Answer 3

f(x)=sumsum...etc...

Answer 4

@sahrya

Answer 5

Do it for N=2 or N=1 and then see that N=3 also works as expressions for f(x).

Answer 6

@Heart_Broken_Kid

Answer 7

If you do the k summation first it will be kind of awkward. Maybe it should be flipped, so the n and k are different so that you do the other summation first.

Answer 8

@Obama

Answer 9

He will help us with this problem guys, dun worry :)

Answer 10

N = 1, not equal, f(x)?

Answer 11

So for N=1 we have:\[\LARGE f(x)=\frac{ f(x)}{1}\]
N=2 we have: \[\LARGE f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}\] See how f(x) is now expressed as the sum of an even and odd function? Let's go deeper. N=3 and let's say: \[\Huge e^{i 2 \pi /3}= \omega\] \[\large f(x)=\frac{f(x)+f( \omega x)+ f(\omega^2 x)}{3}+\frac{f(x)+\omega f( \omega x)+ \omega^2 f(\omega^2 x)}{3}+ \\ \large +\frac{f(x)+\omega ^2 f( \omega x)+ \omega f(\omega^2 x)}{3}\]

Make sure you can show to yourself that that's true. I hope the equation I wrote is actually describing these, they might be off slightly, but maybe you can see where I'm going with this.

Answer 12

Nice nice!

Answer 13

ive seen this before :o

Answer 14

I sort of stumbled across this when I noticed e^x is its own derivative and sinx is its own 4th derivative and wanted to find a function that was its own nth derivative. But it sort of seems like we think and talk about even and odd numbers as ways of reasoning but what about every third number reasoning? It doesn't even have a name as far as I know

even/odd is 2n and 2n+1 but what about 3n, 3n+1, and 3n+2? How about even higher divisibility sort of things? Anyways just throwing around ideas haha, feel free to throw your stuff here too and talk about ideas.

Answer 15

cool :D
but ..
according to D.A
2n , 2n+1 ( on Z)
gives all sets of numbers >=2

3n,3n+1, 3n+2
gives all sets of numbers >=3
but you want know which odd which even , how ever if u used D.A for 4,6,...ect
its the same to 2 but it would start from 4,6,..
so , being 2n,2n+1 do not miss any case

Answer 16

Well I'm not saying I'm interested in even/odd anymore. I'm saying this is like a new version of even/odd, you see?

Answer 17

im interested in what u doing :o
i was working on if we could see even /odd around any line hehe

Answer 18

what do you mean around any line? I'm curious, describe it or draw pictures.

Answer 19

I just noticed, so if a function is even, then it satisfies: \[\LARGE f(x)=f(-x)\] and if it's odd it satisfies: \[\LARGE f(x)=-f(-x)\]

Now if we go into this new thing we have three new types of functions we can have: \[\LARGE f(x)=f( \omega x) = f( \omega^2 x)\] another is: \[\LARGE f(x)=\omega f( \omega x) = \omega ^2 f( \omega^2 x)\] and the final is \[\LARGE f(x)=\omega^2 f( \omega x) = \omega f( \omega^2 x)\]

So pretty interesting, and just like all functions can be represented as the sum of an even and an odd function, so too can a function be represented as the sum of 3 functions that follow these identities.

Answer 20

odd / even around any line like this