The Backus-Gilbert method for signals in reproducing kernel Hilbert spaces and wavelet subspaces

The Backus-Gilbert (BG) method is an inversion procedure for a moment problem when moments of a function and related kernel functions are known. In this paper, we consider the BG method when, in addition, the signal to be recovered is known a priori to be in certain reproducing kernel Hilbert spaces (RKHS), such as wavelet subspaces. We show that better performance may be achieved over the original BG method. In particular, under the D-criterion the BG method with RKHS information for a sampled signal in wavelet subspaces can completely recover the original signal, while the one without any additional information can only provide a constant-valued signal.