On ill-posedness measures and space change in Sobolev scales
Let \(A:X \rightarrow Y\) be a non-degenerate, compact, linear operator between infinite-dimensional, separable Hilbert spaces \(X\), \(Y\). Hence the range is non-closed and the equation (1) \(Ax = y\) is ill-posed. Let \(\{ s_i(A)\}\) be the (ordered) sequence of positive singular values of \(A\). Then the (finite or infinite) interval \[ [\underline{\mu}(A), \bar{\mu}(A)] = \Big[ \liminf_{i \to \infty} {{\log s_i(A)} \over {\log i}}, \limsup_{i \to \infty} {{\log s_i(A)} \over {\log i}} \Big] \] is defined as the interval of ill-posedness of (1) and \(\underline{\mu} (A)\) as the degree of ill-posedness. This interval is compared with alternative interval definitions obtained from compactness measures as \(\varepsilon\)-entropy or \(\varepsilon\)-capacity. For the specific case \(X = L^2(0,1)\) the space-change problem of shifting the space \(X\) along a Sobolev scale is treated. In particular, the change of the ill-posedness interval is analyzed in detail when the solutions are restricted to \(W^1_2[0,1]\). In addition, some inter-dependencies between ill-posedness measures, embedding operators, Hilbert and Sobolev scales are discussed.
scientific article; zbMATH DE number 1129180
scientific article