piecewise-quadratic vs piecewise-cubic vs higher order polynomial interpolation?
The mathematically correct way of signal reconstruction would be sinc interpolation https://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula . However that's not very practical so most real world implementation use an approximation. But it's still based on the same principal: it is essentially a low pass filter and NOT a piecewise polynomial interpolation.
That's how most of real world signal reconstruction is done.
Other interpolation methods may be useful in certain special cases for specific applications. To differentiate between these methods, you would have to formulate what application specific metrics distinguish between a &quot;good&quot; and a &quot;bad&quot; interpolation. Then you can analyze different piecewise interpolation techniques with &quot;typical&quot; signals using the defined metrics and see how the performance is impacted and what is optimal.