Discretization and Perturbation of Wavelet-like Families

We estimate the effects of discretization and dilation/translation errors on a familiar class of Bessel families {psi((Q))}(Q is an element of Dd) indexed over D-d, the dyadic cubes in R-d. We show that, if the psi((Q))s satisfy an appropriately scaled uniform Holder alpha condition for 0 < alpha <= 1, replacing each psi((Q)) by its averages over arbitrary (but sufficiently small) dyadic subcubes of Q yields a family {<(psi)over tilde>((Q))}(Q is an element of Dd) arbitrarily close to {psi((Q))}(Q is an element of Dd) in the sense of Bessel bounds. For alpha = 1 we get the same result by replacing each {psi((Q))}(Q is an element of Dd) with averages over arbitrary identical subcubes of Q. We show that, when alpha = 1, the signal-processing properties of {(psi) over tilde ((Q))}(Q is an element of Dd) are not strongly affected when subjected to small errors in dilation and translation.