Adapting functions to experimental measurements
Let \(\phi_ 0: \mathbb{R}\to\mathbb{R}^ 2\) be a given \(C^ \infty\) embedding and \(\phi:\mathbb{R}\to\mathbb{R}^ 2\) be a \(C^ 1\) perturbation of \(\phi_ 0\). The author describes a class of estimation \(\tilde \phi\) for \(\phi_ 0\) defined on the basis of values \(v_ i=\phi(x_ i)\), \(i=1,\dots,n\) and unit normals \(W_ j\) to the curve \(\phi\) at points \(y_ j\), \(j=1,\dots,m\). The measurement \((v_ 1,\dots,v_ n;w_ 1,\dots,w_ m)\) is said to be singular if \(\tilde\phi'(y_ j)\) vanishes for some \(j\). The case of 3 and less points \(y_ j\) is considered in detail and conditions for \(\tilde \phi\) to be singular are analyzed.
scientific article; zbMATH DE number 24505