Passive approximation and optimization using b-splines

Research output: Contribution to journalArticle


A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L p -norm where 1 ≤ p ≤ ∞. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Hölder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.


External organisations
  • Linnaeus University
  • KTH Royal Institute of Technology
  • Stockholm University
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematical Analysis
  • Electrical Engineering, Electronic Engineering, Information Engineering


  • Approximation, B-splines, Convex optimization, Herglotz functions, Passive systems, Sum rules
Original languageEnglish
Pages (from-to)436-458
Number of pages23
JournalSIAM Journal on Applied Mathematics
Issue number1
Publication statusPublished - 2019
Publication categoryResearch