I am reading a book on Laplace transform, and in the section on the convergence of Laplace transform for various signals the following theorem is stated, without any proof :
If a signal's Fourier transform be zero in some frequencies, or have discontinuities, it will not have Laplace transform (like sinc function, or periodic signals)
I can not see the proof. For signals with a discontinuous Fourier transform I know that $\int_{-\infty}^{\infty} |x(t)|dt$ may not converge leading to discontinuities in the Fourier spectra (again, like sinc), but I don't think it will lead to not having a Laplace transform at all.
What is the proof for the above statement? (if correct)