On the Singular Value Decomposition of n-Fold Integration Operators

Abstract

In theory and practice  of inverse problems, linear operator equations $Tx=y$ with compact linear forward operators T having a non-closed range $\mathcal {R}(T)$ and mapping between infinite dimensional Hilbert spaces plays some prominent role. As a consequence of the ill-posedness of such problems, regularization approaches are required, and due to its unlimited qualification spectral cut-off is an appropriate method for the stable approximate solution of corresponding inverse problems. For this method, however, the singular system $\{\sigma _i(T),u_i(T),v_i(T)\}_{i=1}^\infty$ of the compact operator T is needed, at least for $i=1,2,...,N$, up to some stopping index N. In this note we consider n-fold integration operators $T=J^n\;(n=1,2,...)$ in $L^2([0,1])$ occurring in numerous applications, where the solution of the associated operator equation is characterized by the nth generalized derivative $x=y^{(n)}$ of the Sobolev space function $y \in H^n([0,1])$. Almost all textbooks on linear inverse problems present the whole singular system $\{\sigma _i(J^1),u_i(J^1),v_i(J^1)\}_{i=1}^\infty$ in an explicit manner. However, they do not discuss the singular systems for $J^n,\;n \ge 2$. We will emphasize that this seems to be a consequence of the fact that for higher n the eigenvalues $\sigma ^2_i(J^n)$ of the associated ODE boundary value problems obey transcendental equations, the complexity of which is growing with n. We present the transcendental equations for $n=2,3,...$ and discuss and illustrate the associated eigenfunctions and some of their properties.