While the recovery of a time series f(x) from any finite sample set (f(xi); i=1. . .,N), is always non-unique, the smoothness implied by an assumption of band-limitedness allows certain definite quantitative inferences to be made about the solution set. Band-limitedness allows sample values to be expressed as a set of linear functionals of the unknown f(x), specifically integrals of f(x) weighted by known data kernel functions. The form of the data kernels depends upon the assumed spectral cutoff. The theory of Backus and Gilbert (1968) is applied to find weighted averages common to all solutions f(x) which satisfy these data constraints. The weighting function is a linear combination of data kernels which is designed to be as concentrated as possible at a target x=x0, and thereby gives an average as strongly dependent on solution values as near x0 as possible, while being relatively insensitive to values elsewhere. An apparent paradox arises when standard Backus-Gilbert theory is applied: for some values of x0, a smaller assumed spectral cutoff results in averaging functions which are much broader than the corresponding ones for a larger cutoff. This contradicts the intuitive notion that the increased smoothness implied by a smaller cutoff should result in increased knowledge of f(x), here reflected in narrower averaging functions. Band-limited solutions compatible with a smaller cutoff are a proper subset of those with a larger cutoff, and resolving power should not be lost when attention is restricted to these smoother solutions. The resolution of the paradox comes from the recognition that this argument fails because the data constraints admit a larger solution set than desired, including solutions violating the assumed spectral cutoff; Backus-Gilbert theory yields averages shared by all solutions, not just those which are properly band-limited.