Functions as Vectors: Understanding Fourier Series and Orthogonal Functions

[futurebird]

Functions as "vectors"

I'm reading a book called "Fourier series and orthogonal functions" by Davis since it seemed pretty readable (at first) and since I don't really know what is going on with these Fourier series, yet.

The book suggests that one can think of functions as vectors. After all a sequence is a function whose domain is the set of real numbers. And a vectors is a list of coordinates (x, y, z) or as many dimensions as you like.

This "functions can be vectors" idea seems pretty central to the ideas in the book, and I'm getting confused about how ... for example ... a continuous function could be a vector. A continuous function is defined at every singe point and there's nothing discrete about the domain.

Is the idea that through use of series we create a list-like correspondence that allows us to think of the function as a vector? I'm really confused. What kind of dimensional space would contain continuous functions as vectors? Wouldn't it need to have infinite dimensions?

 

[D H]

The concept of a function as a vector is integral to physics beyond the freshman/sophomore level. You are correct that the space is infinite. The key thing that is needed to think of a function as a vector is to define an inner product, typically by means of an integral over some interval.

 

[futurebird]

The concept of a function as a vector is integral to physics beyond the freshman/sophomore level. You are correct that the space is infinite. The key thing that is needed to think of a function as a vector is to define an inner product, typically by means of an integral over some interval.

Okay the inner product is like a dot porduct and then we use a different integral to take the magnitude. I was having a hard time believing that the space was really infinite dimensional, but now that I'm just going with that idea this chapter is starting to make more sense.

 

[D H]

Once you get past that mental block of an infinite-dimensional space things become both easier and a lot harder. Things can get really, really ugly in infinite dimensions.

Fortunately, physicists don't look at the really ugly stuff. Some examples from a physics perspective include Fourier series, Sturm-Liouville systems of orthogonal polynomials, and spherical harmonics. These are all well-defined, well-behaved, and avoid most of the ugly stuff (e.g., anything that involves the axiom of choice.)