Real roots of Fourier transform

[eljose]

If we define the function:

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

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.

 

[Edwin]

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