Increasing stability for near field from the scattering amplitude

Click on the DOI link to access the article (may not be free).We obtain stability estimates for the near field of a radiating solution of the Helmholtz equation from the far field (scattering amplitude). This estimates contain the best possible Lipschitz term, a Holder term, and terms which decay as powers of the frequency k for large k under some a priori bounds. These estimates contain only explicit constants and show increasing stability of recovery of the near field from scattering amplitude with growing k. Proofs are elementary and are based on new explicit bounds for Hankel functions. We give first applications to increasing stability in (linearized) inverse scattering by obstacles