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On the Singular Value Decomposition of n-Fold Integration Operators

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 310)

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.

Keywords

  • Integration operators
  • Singular systems
  • N-fold integration
  • Boundary value problems
  • Symbolic determinant

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Acknowledgements

Ronny Ramlau was supported by the Austrian Science Fund (FWF): SFB F68-N36 and DK W1214. Christoph Koutschan was supported by the Austrian Science Fund (FWF): P29467-N32 and F5011-N15. Bernd Hofmann was supported by German Research Foundation (DFG): HO 1454/12-1.

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Correspondence to Ronny Ramlau .

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Ramlau, R., Koutschan, C., Hofmann, B. (2020). On the Singular Value Decomposition of n-Fold Integration Operators. In: Cheng, J., Lu, S., Yamamoto, M. (eds) Inverse Problems and Related Topics. ICIP2 2018. Springer Proceedings in Mathematics & Statistics, vol 310. Springer, Singapore. https://doi.org/10.1007/978-981-15-1592-7_11

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