Don Lancaster's "magic sinewaves" are functions from discrete time to {-1, 0, +1} intended for use, among other things, in modulating an H-bridge to approximate a sine wave for power electronics. I think there's a way to use them to get a reasonably good and extremely efficient frequency-selective sparse filter (see Sparse filters).
The simplest approximation of a sine wave of period, say, 4 samples, is a square wave: [-1, -1, +1, +1, -1, -1, +1, +1, ...]. But that isn't a very good approximation; it has a pretty loud third harmonic. (This is above Nyquist, so suppose we're using a zero-order hold here.) A better approximation, though lower amplitude, includes a 0 period instead: [0, -1, 0, +1, 0, -1, 0, +1, ...]. This has less harmonic distortion. But if we expand this out with a zero-order hold it's still obviously imperfect: [0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, +1, +1, +1, +1, +1, 0, 0, 0, 0, ...]. We can twiddle the samples around a little to kind of dither the transition; for example, [0, -1, 0, -1, -1, -1, 0, -1, 0, 0, 0, +1, 0, +1, +1, +1, 0, +1, 0, 0, 0, ...]. This way, although we don't get lower total harmonic distortion, we can push more of it up to higher frequencies, which makes it easier to filter out. Again, this gives us lower amplitude for the fundamental.
If you want a sine wave, why not just use an actual sine wave, like [0, -0.3, -0.6, -0.8, -1, -1, -1, -0.8, -0.6, -0.3, 0, 0.3, 0.6, 0.8, 1, 1, 1, 0.8, 0.6, 0.3...]? In the case of power electronics with an H-bridge, it's because your H-bridge wastes no power when it's off, a tiny bit of power when it's fully on (either forward or reverse), and a massive amount of power when it's partway on. So for an efficient system you want to switch between fully on and fully off as quickly (and infrequently!) as possible.
For sample rates that are higher compared to the signal frequency, it's easy to push all the harmonic distortion many octaves away from the signal, making it easy to filter with efficient passive filters; this is pretty much the same principle behind delta-sigma DACs, though those usually use {0, 1} rather than {-1, 0, 1}.
The example "reference signal" waveforms above have period 20. If we were to multiply them elementwise by some signal, we would downconvert one phase of the frequency component with period 20 to DC, and then we could extract it by merely summing. The actual sine wave would give us the pure frequency component, while the {-1, 0, +1} signals mix in some pretty significant harmonic distortion. But multiplying by them is trivial: you add or subtract samples from your running total, no multiplication required. You can do this a second time with a second reference signal in quadrature with the first to get a complete measurement of the amplitude and phase of a given frequency.
If you maintain, say, 60 totals buckets, such that bucket i contains a total of the samples x[j] such that j % 60 == i, you can run this analysis over the period-20 component of your whole signal by doing 60 additions and subtractions; so, too, for components whose periods are other factors of 60, such as 30, 15, 12, 10, 6, 5, and 4. If instead of just maintaining 60 such buckets, you calculate a feedback comb filter y[n] = x[n] + y[n - 60], you can calculate such totals for any given segment of the signal by subtracting 60 subsequent y values at the beginning of that segment from 60 corresponding y values at its end; and the usual tricks to get a triangular or otherwise approximately gaussian temporal window apply, as described in some other notes here.
You could, instead, do this kind of correlation by generating a dithered approximation "reference signal" dynamically, for example by running a delta-sigma conversion of the output of a free-running Goertzel or Minsky oscillator, and use that to decide whether to process the current sample by adding, subtracting, or neither, to each of your I and Q accumulators. This has the great advantage that you can dynamically vary the frequency of your local oscillator, thus tracking chirp signals, such as whistles (see Whistle detection.) A lightweight low-pass filter applied to the input signal should be sufficient to eliminate the high-frequency signals that could otherwise produce spurious correlations, but with delta-sigma conversion of more than first order, I think the dither noise might be sufficiently random to not need this.
Delta-sigma conversion in this case might be nothing more than a matter of the Bresenham line-drawing algorithm.
If the number of transitions per period is relatively small, as with magic-sinewave waveforms designed to reduce switching losses by switching less frequently, it may be more efficient to work from a prefix sum of the signal rather than the signal itself. For example, the initial reference waveform suggested above has only four transitions per period; rather than doing five additions and five subtractions per period, you could do two additions and two subtractions from the prefix sum. Of course the prefix sum itself requires 20 additions per period to compute, so this is only an improvement if you can share the prefix sum with some other computation, such as the detection of a second frequency.