Question

Periodic Extension

Answer 1

I have no idea where to start.

Answer 2

@phi you helped me once, and you are the only name i can think of. can you help me with this? or do you know somebody else who can help me?

Answer 3

The first step is to sketch
\[ y= \frac{1}{\ell} x\]
from 0 to \( \ell\)
hint: at x=0, y=0
at x= ℓ, y=1
and it's a line.

then plot
y= (L-x)/(L- ℓ)

when you are done, it should look like a triangle.

A periodic extension means you repeat that pattern, in this case every L steps

an even extension means make a replica on the other side of the y-axis i,e. for x= -L to 0

an odd extension means make the shape negative ( see a sin wave for an example) to the left of the y-axis.

in either case, you then "extend" the pattern forever in both directions

Answer 4

is it something like this?

Answer 5

almost. but it should look more asymmetric.
it goes up for x= 0 to L/4 and then down from L/4 to L
that makes the 2nd part 3 times longer than the first part.