A stable finite difference ansatz for higher order differentiation of non-exact data

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Abstract
If standard central difference formulas are used to compute second or third order derivatives from measured data even quite precise data can lead to totally unusable results due to the basic instability of the differentiation process. Here an averaging procedure is presented and analysed which allows the stable computation of low order derivatives from measured data. The new method first averages the data, then samples the averages and finally applies standard difference formulas. The size of the averaging set acts like a regularisation parameter and has to be chosen as a function of the grid size h.