Estimate of aliasing error for non-smooth signals prefiltered by quasi-projections into shift-invariant spaces

An ideal analog-to-digital (A/D) converter prefilters a signal by an ideal lowpass filter. Recent research on A/D conversion based on shift-invariant spaces reveals that prefiltering signals by quasiprojections into shift-invariant spaces provides more flexible choices in designing an A/D conversion system of high accuracy. This paper focuses on the accuracy of such prefiltering, in which the aliasing error e_f^λ is found to behave like e_f^λ ₂ = O(λ^-α) with respect to the dilation λ of the underlying shift-invariant space, provided that the input signal f is Lipschitz-α continuous. A formula to calculate the coefficient of the decay rate is also figured out in this paper.