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We present methods for velocity estimation from discrete and quantized position samples using adaptive windowing. Previous methods necessitate trade-offs between noise reduction, control delay, estimate accuracy, reliability, computational load, transient preservation, and difficulties with tuning. In contrast, a first-order adaptive windowing method is shown to be optimal in the sense that it minimizes the velocity error variance while maximizes the accuracy of the estimates, requiring no tradeoff. Variants of this method are also discussed. The effectiveness of the proposed technique is verified in simulation and by experiments on the control of a haptic device.