## Error level saturation for Popoff’s generalized derivative operator.(English)Zbl 1277.26005

The generalized derivative of a function $$f$$ introduced by K. Popoff is defined by
$f^p(x)=\lim_{h\rightarrow 0} P_h f(x),$
provided that this limit exists, where
$P_h f(x)= \frac{2}{h^2}\int_{0}^{h}[f(x+t)-f(x)]dt.$
Suppose that $$f$$ has a continuous bounded second derivative on some open interval $$I$$. For $$\varepsilon>0$$, by $$f^\varepsilon$$ is denoted a bounded integrable perturbation of $$f$$ satisfying $$|f(t)-f^\varepsilon(t)|\leq\varepsilon$$ for $$t\in I$$. It is shown that
$|P_{\sqrt{\varepsilon}}f^{\varepsilon}(x)-f^{\prime}(x)|=O(\sqrt{\varepsilon});$
and on the other hand, the following saturation result is established: If for some choice $$h(\varepsilon)\rightarrow 0$$, as $$\varepsilon\rightarrow 0$$,
$|P_{h(\varepsilon)}f^{\varepsilon}(x)-f^{\prime}(x)|=o(\sqrt{\varepsilon})$
for each $$x\in I$$, then $$f$$ is a linear function on $$I$$.

### MSC:

 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 41A25 Rate of convergence, degree of approximation

### Keywords:

Popoff derivative; saturation; error level
Full Text:

### References:

 [1] C. W. Groetsch, Lanczos’ generalized derivative , Amer. Math. Monthly 105 (1998) 320-326. · Zbl 0927.26003 · doi:10.2307/2589707 [2] K. Popoff, Sur une extension de la notion de dérivée , C. R. Acad. Sci. Paris, 207 (1938) 110-112. · Zbl 0019.05602
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