Calculation of the derivative of an inaccurately defined function by means of feedback laws (Q267928)

The ill-posed problem of calculating the derivative \(\dot x\) of an inaccurately defined function \(x:[0,\,\vartheta] \to {\mathbb R}^n\) is considered. For equispaced nodes \(\tau_j = \frac{j}{m}\,\vartheta\) \((j=0,\ldots,m)\), the function values \(x(\tau_j)\) are approximately measured by vectors \(\xi_j^h \in {\mathbb R}^n\) with \(\|x(\tau_j) - \xi_j^h\|_2 \leq h\), where \(h\in (0,\,1)\) is the noise level. For approximate calculation of \(\dot x\), the author applies the feedback control method with a model. Thus the problem of numerical differentiation is replaced by the control problem of a virtual dynamical system which is defined by a vector differential equation \({\dot w} = v^h\) with special control \(v^h\). Under convenient assumptions, it is shown that \(v^h\) converges to \(\dot x\) in \(L_2([0,\vartheta],{\mathbb R}^n)\) as \(h \to 0\). This method is robust with respect to noise.