Inverting noisy integral equations using wavelet expansions: a class of irregular convolutions
Peter Hall, Frits H. Ruymgaart, Onno van Gaans, Arnoud van Rooij · Lecture notes-monograph series · 2001
Suppose a random sample is observed from a density which is a known transformation of an unknown underlying density to be recovered.Expansion of this unknown density in a wavelet basis yields Fourier coefficients that can be reexpressed in terms of the sampled density and an extension of the adjoint of the inverse of the operator involved.This seems to yield a new approach to inverse estimation.Focusing on deconvolution optimal error rates are obtained in the case of certain irregular kernels like the boxcar that cannot easily be dealt with by classical techniques or by Donoho's (1995) wavelet-vaguelette method.