Is there a geometric interpretation of orthogonal functions?

[cmcraes]

Hi all. So to start I'll say I'm just dealing with functions of a real variable.

In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects"

So with that in mind, is there any geometric connection between two orthoganal functions on some interval? When I look at a plot of say, the legendre polynomials, nothing pops out at me as hinting there may be some orthoganality going on. Is there anything to see or do I have to look somewhere else ( like the integrals of the polynomials)? Thanks!

 

[FactChecker]

You have to look at the inner product of the two orthogonal functions, how ever the inner product is defined. The concept of "orthogonal" has been defined abstractly so that it can be used in many situations where it is not so easy to see. Given a general function, the inner products with each of the two orthogonal functions are completely independent of each other, just like in the XY plane, the (x,y) coordinates of a point are independent of each other. Orthogonal functions can be used in a basis the same way that (1,0) and (0,1) can be used as a basis in the XY plane.
So there is a geometric interpretation that makes sense for the inner product that is being used.

 

[cmcraes]

Yes I understand orthoganality of functions. I'm just wondering if for any specific sets of functions there happens to be any interesting geometric properties of those functions as well.

You have to look at the inner product of the two orthogonal functions, how ever the inner product is defined. The concept of "orthogonal" has been defined abstractly so that it can be used in many situations where it is not so easy to see. Given a general function, the inner products with each of the two orthogonal functions are completely independent of each other, just like in the XY plane, the (x,y) coordinates of a point are independent of each other. Orthogonal functions can be used in a basis the same way that (1,0) and (0,1) can be used as a basis in the XY plane.

 

[FactChecker]

Then I assume that you are looking at them in pairs, since that is how orthogonal is defined. I don't see anything in general beyond the inner product, however it is defined for a particular case. Consider any two functions that are not linearly dependent. Then the first one can be left alone and the second one can be modified to be orthogonal to the first one. Because the first one was an arbitrary function, there can be no way of identifying an orthogonal function.

 

[Stephen Tashi]

Is there anything to see or do I have to look somewhere else ( like the integrals of the polynomials)?

One sophisticated way that orthogonal functions appear is as matrix entries in the "representations" of continuous groups. The simple example is $sin(\theta)$ and $cos(\theta)$ as entries in matrices representing the rotation group in 2D. I don't know how much that concept helps with forming a visual picture. If you search on the keywords "Special functions" and "Group representations" you can find a lot of material. I don't know enough about it to recommend a good introductory text or website.

 

[FactChecker]

Orthogonality alone does not restrict the set of functions very much. It's easy to get orthogonal sets of functions. The real key to the well-known orthogonal sets is that they satisfy some other criteria such as being a solution to an important differential equation. The Legendre polynomials that you mention in the OP are azimuthally symmetric solutions of Legendre's differential equation. (Don't ask me. I don't know what that means or why that differential equation is important.)
See https://en.wikipedia.org/wiki/Legendre_polynomials.
There are a large variety of reasons that particular differential equations are important.

 

[chiro]

Hey cmcraes.

I'm going to assume that an orthogonal function is when you do Integral fg = 0 over some interval but if not let us know.

The idea is that the projection of one piece of information on-to another always gives nothing in common and an integral will give an infinite dimensional vector meaning that if you generalized this function to be an infinite dimensional vector and tried to "plot it" on an infinite dimensional plane [or some projective plane that made sense] then it would literally be independent in the same way that <0,0,1> is independent to <0,1,0> in a normal three dimensional space.

The functions themselves are represented as what is called a functional - but we use an infinite dimensional vector instead of a finite one for integrals and we also map a function to the vector meaning that instead of writing some symbolically as f(x) = blah, we instead use a combination of numbers in an infinite dimensional vector to represent it.