Time and frequency domain scattering for the one-dimensional wave equation

We give a constructive method for extending time domain data for the inverse scattering problem for the one-dimensional wave equation. We show that a reflection operator on L-2(-T, T) with T finite is essentially a Hankel operator and then modify the Nehari extension of the kernel of the reflection operator to obtain a reflection operator on L-2(R) that is consistent with conservation of energy. This extension result allows frequency domain techniques to be used when the time domain data are only available for finite time, and we demonstrate this by using the frequency domain characterization of reflection coefficients to obtain a new proof of the characterization of reflection operators on L-2(-T, T). We also give an a priori estimate for the operator norm of the reflection operator on L-2(-T, T) and use the theory of Toeplitz operators to show how the singular values of these reflection operators are related to the reflection coefficient.