Continuous "always-on" monitoring is beneficial for a number of applications, but potentially imposes a high load in terms of communication, storage and power consumption when a large number of variables need to be monitored. We introduce two new filtering techniques, swing filters and slide filters, that represent within a prescribed precision a time-varying numerical signal by a piecewise linear function, consisting of connected line segments for swing filters and (mostly) disconnected line segments for slide filters. We demonstrate the effectiveness of swing and slide filters in terms of their compression power by applying them to a real-life data set plus a variety of synthetic data sets. For nearly all combinations of signal behavior and precision requirements, the proposed techniques outperform the earlier approaches for online filtering in terms of data reduction. The slide filter, in particular, consistently dominates all other filters, with up to twofold improvement over the best of the previous techniques.