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

 

[tacman]

The criterion for determining if the roots of F[w] are all real is known as the Paley-Wiener criterion. This criterion states that if the function g(x) is a bounded and continuous function with compact support (meaning it is equal to zero outside of a finite interval), then the Fourier transform F[w] will have all real roots.

This can be derived by considering the properties of the Fourier transform. One of the properties is that if a function is bounded and continuous with compact support, then its Fourier transform will be an entire function, meaning it is analytic everywhere in the complex plane. This also means that the roots of the Fourier transform will be isolated and can only occur at integer multiples of a certain value.

Since the function g(x) is bounded and continuous with compact support, it can be approximated by a polynomial function, which also satisfies the Paley-Wiener criterion. By using the properties of polynomial functions, it can be shown that the roots of the Fourier transform will be real. This ultimately leads to the conclusion that if g(x) satisfies the Paley-Wiener criterion, then the Fourier transform F[w] will have all real roots.

In summary, the criterion for deciding if the roots of the Fourier transform are all real is the Paley-Wiener criterion, which states that if the function g(x) is a bounded and continuous function with compact support, then the Fourier transform F[w] will have all real roots. This criterion can be derived by considering the properties of the Fourier transform and approximating the function g(x) with a polynomial function.