Technological advances have led to an increase in the collection of high-dimensional, nearly continuously sampled signals. Evolutionary correlations between such signals are salient to many studies, as they provide important information about associations between different dynamic processes and can be used to understand how these processes relate to larger complex mechanisms. Despite the large number of methods for analyzing functional data that have been explored in the past twenty-five years, there is a dearth of methods for analyzing functional correlations. This dissertation introduces new methods for addressing three questions pertaining to functional correlations. First, we address the problem of estimating a single functional correlation by developing a smoothing spline estimator and accompanying bootstrap procedure for forming confidence intervals. Next, we consider the problem of testing the equivalence of two functional correlations from independent samples by developing a novel adaptive Neyman testing procedure. Lastly, we address the problem of testing the equivalence of two functional correlations from dependent samples by extending the adaptive Neyman test to this more complicated setting, and by embedding the problem in a state-space framework to formulate a practical Kalman filter-based algorithm for its implementation. These methods are motivated by questions in sleep medicine and chronobiology and are used to analyze the dynamic coupling between delta EEG power and high frequency heart rate variability during sleep
