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There is a question available on DSP SE that mentions types of interpolation used for signal reconstruction but there isn't any mention about the difference between piecewise-quadratic,piecewise-cubic and higher order polynomial interpolation

So how we can differentiate between them ?

Link of question is given below: Types of interpolation used for reconstruction in DSP?

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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 "good" and a "bad" interpolation. Then you can analyze different piecewise interpolation techniques with "typical" signals using the defined metrics and see how the performance is impacted and what is optimal.

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    $\begingroup$ but Hilmar, piecewise polynomial interpolation can be modeled as a form of LPF. the order of the polynomial is usually 1 less than the number of adjacent samples used in the interpolation. as the order (and number of adjacent samples) gets larger and larger, the apparent impulse response looks more and more like a sinc, and it can actually be divided by the sinc and an effective window determined. Duane Wise and i did a paper about this in the 90s. $\endgroup$
    – robert bristow-johnson
    Commented Apr 4, 2021 at 15:48

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