Their "memprocessor" is a set of standard analog computing elements - four analog multipliers, two op amps, and a "controlled inverting differentiator". General purpose analog computers are rare today, but you could set this up on the patchboard of an analog computer from the 1950s or 1960s.

The idea that analog computers are in some sense more powerful than Turing-complete digital computers comes back now and then. Here's a paper on that from 1998:

http://www.eetimes.com/document.asp?doc_id=1138111

"The neural network that represents the analog computer proves to be inherently richer than the standard Turing model."

The idea that having variables which can take on an infinite range of real-valued values fascinates some people. But you can't, not really. Resolution is limited by noise. Noise is inescapable, since electrons are discrete. Infinite resolution is impossible in a granular universe. Real numbers are a convenient fiction which cannot be realized in hardware. ("God created the integers; all else is the work of man." - Kronecker)

The author of the paper almost gets this. They mention the Nyquist-Shannon sampling theorem, so they know about Shannon. But they don't seem to be considering the problems of the noise floor. They also write "the multimeter of the hardware implementation analogically measures the integral over a continuous time interval, directly providing the result, thus avoiding the need of sampling the waveform and computing the integral". Amusingly, they're using a digital multimeter.