Skip to main content

On the Singular Value Decomposition of n-Fold Integration Operators

  • 408 Accesses

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


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.


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

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-981-15-1592-7_11
  • Chapter length: 20 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-981-15-1592-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.99
Price excludes VAT (USA)
Hardcover Book
USD   219.99
Price excludes VAT (USA)
Fig. 11.1
Fig. 11.2
Fig. 11.3
Fig. 11.4


  1. H.W. Engl, Integralgleichungen (Springer, Wien/New York, 1997)

    CrossRef  Google Scholar 

  2. R. Gorenflo, S. Vessella, Abel Integral Equations (Springer, Berlin, 1991)

    CrossRef  Google Scholar 

  3. B. Hofmann, L. von Wolfersdorf, Some results and a conjecture on the degree of ill-posedness for integration operators with weights. Inverse Probl. 21(2), 427–433 (2005)

    MathSciNet  CrossRef  Google Scholar 

  4. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn. (Springer, New York, 2011)

    CrossRef  Google Scholar 

  5. R. Kress, Linear Integral Equations (Springer, Berlin, 1989)

    CrossRef  Google Scholar 

  6. P. Mathé, Saturation of regularization methods for linear ill-posed problems in Hilbert spaces. SIAM J. Numer. Anal. 42(3), 968–973 (2004)

    MathSciNet  CrossRef  Google Scholar 

  7. M.Z. Nashed, A new approach to classification and regularization of ill-posed operator equations. in Inverse and Ill-Posed Problems, Sankt Wolfgang, 1986, eds. by H.W. Engl, C.W. Groetsch (Academic Press, Boston, MA, 1987), pp. 53–75

    Google Scholar 

  8. N.J.A. Sloane, The on-line encyclopedia of integer sequences. OEIS Foundation Inc.,

  9. V.K. Tuan, R. Gorenflo, Asymptotics of singular values of fractional integral operators. Inverse Probl. 10(4), 949–955 (1994)

    Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ronny Ramlau .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

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.

Download citation