Does a Fourier Series Have an Infinite Dimensional Parameter Space?

[autobot.d]

Want to understand a concept here about dimensions of a function.

Using example 1: a simple Fourier series from http://en.wikipedia.org/wiki/Fourier_series

$s(x) = \frac{a_0}{2} + \sum ^{\infty}_{0}[a_n cos(nx) + b_n sin(nx)]$

So do we now say that $s(x)$ has an infinite dimensional parameter space?

When I think of x being one dimensional, I think of $y = mx + b$ which to me is a 2 dimensional parameter problem for y...

Trying to figure out what exactly is meant by a "high dimensional pde" which lives in 2-d (as an example)...

I assume this would be the same answer for a problem that considers the Karhunen-Loeve transform.

 

[HallsofIvy]

The "dimension of the parameter space" is the number of independent, real valued parameters. Yes, "y= mx+ b" depends upon the two "parameters", m and b, and has parameter space of dimension two. In particular, any point (a, b) in $R^2$, gives a function y= ax+ b.

And, yes,
$$s(x)= \frac{a_0}{2}+ \sum_0^\infty \left[a_ncos(nx)+ b_nsin(nx)\right]$$
has infinite dimensional parameter space because there are an infinite number of parameters.

(In fact some people would say it has a "doubly infinite parameter space" since both the $a_n$ and $b_n$ sequences have an infinite number of parameters. But "infinite" should be enough for anybody!)