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This paper examines the amplitude fluctuations of band-limited functions with bounded zeroth absolute moments, and the problems associated with estimating the level crossing profiles of these functions. Level crossings have received increased attention as features for pattern recognition because of their capability to provide information related to both amplitude and frequency behavior. A detailed analysis of the average rate of change of band-limited functions is presented, including a derivation of the least upper bound on functions for which the zeroth absolute moments of the functions are bounded. The average rate of change of a function over an interval in which one endpoint of the interval is an extremum of the function is similarly bounded and used to establish a sampling rate which guarantees that between successive samples of a band-limited function with bounded zeroth absolute moment the function itself does not deviate from the amplitude interval defined by the samples by more than some predefined amplitude change. Based on these results, a theorem is developed which defines the sampling rate required to ensure that, in the estimation of level crossing profiles, no more than 2m (m >> 1) level crossings of m levels are missed per extremum of the sampled function. It is shown that the sampling rate defined by this theorem reduces to the well-known Nyquist rate for the special case of zero crossing analysis.