Non-uniform sampling in wavelet subspaces

Abstract

It is well known that the Shannon (1949) sampling theorem can be put into a wavelet context. But it has also been shown that for most wavelets, a sampling theorem for the associated subspaces exists. There is even a non-uniform sampling theorem as in the Shannon case. In general the bounds on the deviations from uniform are not as easy to specify in this case. No simple Kadec 1/4 theorem holds except in special cases (such as the Franklin case where the bound is 1/2). For a particular class, the Meyer (1990) wavelets, which are bandlimited but with a smooth spectrum, a similar bound is sometimes obtainable. Unfortunately, it is much smaller that 1/4.