Elliptic trigonometric functions as basis for function expansion ?

[Avaro667]

Hey everyone .
So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me ! It really changed the way I'm thinking !

So i was wondering whether it could be possible to go even further and decompose a signal or function f(x) ,if you prefer, using elliptic trigonometric functions like Sn,Cn . Has anyone heard of such thing before ? I tried to search it myself but i didn't find anything relevant . I would be really interested knowing possible applications or even discovering more cool connections and meanings . Any thoughts on the subject will be much appreciated,thank you!

 

[Infrared]

What are these "elliptic trigonometric functions like Sn, Cn"?

 

[Stephen Tashi]

Has anyone heard of such thing before ? I tried to search it myself but i didn't find anything relevant .

Many ways of expressing functions as a sum of other functions have been studied. The situation is analagous to expressing a vector as a sum of basis vectors. Different basis vectors can be used. It is convenient if the basis vectors are orthogonal. With an orthogonal basis the coefficent that is used for each basis vector can be found by projecting the vector onto that basis vector. In dealing with functions, projecting function $f(x)$ on basis function $b_i(x)$ is done by $\int f(x)b_i(x) dx$. For othogonal basis functions $b_i, b_j$, $\int b_i(x) b_j(x) = 0$. https://en.wikipedia.org/wiki/Orthogonal_functions

I don't know if there is an often used basis consisting of elliptic functions. A quick web search turns up http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1838-07.pdf section 5.5, but I don't understand that paper.

Simpler types of functions have been studied - for example, orthogonal polynomials https://en.wikipedia.org/wiki/Orthogonal_polynomials

 

[Avaro667]

What are these "elliptic trigonometric functions like Sn, Cn"?

I'm sure you will find a more strict definition than i can give right now. But the main idea is I'm talking about like ordinary trigonometric functions but instead of being defined on a circle they're defined on an ellipse .

 

[Avaro667]

Many ways of expressing functions as a sum of other functions have been studied. The situation is analagous to expressing a vector as a sum of basis vectors. Different basis vectors can be used. It is convenient if the basis vectors are orthogonal. With an orthogonal basis the coefficent that is used for each basis vector can be found by projecting the vector onto that basis vector. In dealing with functions, projecting function $f(x)$ on basis function $b_i(x)$ is done by $\int f(x)b_i(x) dx$. For othogonal basis functions $b_i, b_j$, $\int b_i(x) b_j(x) = 0$. https://en.wikipedia.org/wiki/Orthogonal_functions

I don't know if there is an often used basis consisting of elliptic functions. A quick web search turns up http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1838-07.pdf section 5.5, but I don't understand that paper.

Simpler types of functions have been studied - for example, orthogonal polynomials https://en.wikipedia.org/wiki/Orthogonal_polynomials

Thank you Stephen i will try to take a look on this paper ! I'm surprised it doesn't seem like people have even tried this .Of course there's a good chance this transform just make things worse,but my intuition right now says to confirm it .