Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of H\"older type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.