Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Simple question about dimension

  1. Dec 4, 2013 #1
    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

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

    So do we now say that [itex] s(x) [/itex] has an infinite dimensional parameter space?

    When I think of x being one dimensional, I think of [itex] y = mx + b [/itex] 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.
     
  2. jcsd
  3. Dec 9, 2013 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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 [itex]R^2[/itex], gives a function y= ax+ b.

    And, yes,
    [tex]s(x)= \frac{a_0}{2}+ \sum_0^\infty \left[a_ncos(nx)+ b_nsin(nx)\right][/tex]
    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 [itex]a_n[/itex] and [itex]b_n[/itex] sequences have an infinite number of parameters. But "infinite" should be enough for anybody!)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook