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

Real roots of Fourier transform

  1. May 29, 2006 #1
    If we define the function:

    [tex] F[w]=\int_{-\infty}^{\infty}dxe^{-iwx}g(x) [/tex]

    my question is..what would be the criterion to decide if F[w] has all the roots real (w=w*) and how is derived?..thanks.
     
  2. jcsd
  3. Jun 7, 2006 #2
    If anybody knows the answer to this question, I would be very interested in learning the answer also!

    At least in the case of a similar transformation, a Laplace like transform,

    Int[e^(-s*sqrt(t))*sin(pi*sqrt(t)),0, infinity] = T{sin(pi*sqrt(t))} = 4*pi*s/(s^2 + pi^2)^2

    Int[e^(-s*sqrt(t+c))*sin(pi*sqrt(t+c)),0, infinity] = T{sin(pi*sqrt(t+c))}

    = q(s)*e^[-sqrt(c)*s]/(s^2 + pi^2)^2, where q(s) is a cubic polynomial in s, c is an arbitrary positive constant, and t is greater than or equal to 0.

    It seems that the information about the zeros of the sine functions above is contained in the residue at the poles of there s-like-transforms, T{f(t)} = F(s), while the kind of function f(t) is, is indicated by the position and order of the poles of F(s). This is only a guess, based on the fact that the transformations above yield functions of s with a common denominator of (s^2 + pi^2)^2, but a different expression in the variable s in the numerator. The two original sine functions in the real variable t have zeros in different locations, but the position and order of the singularities of their s-like-transforms above are identical. As a result, I would presume that all the information about the zeros of the original sine functions in t would have to be tied up in the residues of the singularities of the s-like-transforms

    What are your thoughts?

    Inquisitively,

    Edwin G. Schasteen
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Real roots of Fourier transform
  1. Fourier transformation (Replies: 0)

  2. Fourier transform (Replies: 2)

  3. Fourier transforms (Replies: 1)

  4. Fourier transform (Replies: 1)

Loading...