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The main accomplishment of sampled-data control theory in the last decade is that it successfully derives a digital (discrete-time) control law that makes the overall analog (continuous-time) performance optimal. The same hybrid nature of designing a digital filter for analog signals is also prevalent in digital signal processing. A crucial observation is that the perfect band-limiting hypothesis can be inadequate for many practical situations. In practice, the original analog signals (sounds, images, etc.) are neither fully band-limited nor even close to be bandlimited in the current processing standards. This is the problem of interpolating high-frequency components, which in turn is that of recovering the intersample behavior. Sampled-data control theory provides an optimal platform for such problems. This paper provides a new problem formulation, design procedure, and various applications in sound processing/compression and image processing.