Passive approximation and optimization using bsplines
(2019) In SIAM Journal on Applied Mathematics 79(1). p.436458 Abstract
A passive approximation problem is formulated where the target function is an arbitrary complexvalued 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... (More)
(Less)
A passive approximation problem is formulated where the target function is an arbitrary complexvalued 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 Bspline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finitedimensional 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 Bspline 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.
 author
 Ivanenko, Yevhen ^{LU} ; Gustafsson, Mats ^{LU} ; Jonsson, B. L.G.; Luger, Annemarie ^{LU} ; Nilsson, Börje; Nordebo, Sven ^{LU} and Toft, Joachim ^{LU}
 organization
 publishing date
 2019
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Approximation, Bsplines, Convex optimization, Herglotz functions, Passive systems, Sum rules
 in
 SIAM Journal on Applied Mathematics
 volume
 79
 issue
 1
 pages
 23 pages
 publisher
 SIAM Publications
 external identifiers

 scopus:85063407473
 ISSN
 00361399
 DOI
 10.1137/17M1161026
 language
 English
 LU publication?
 yes
 id
 1bba6978afcb42efa52c9b93a16e28e1
 date added to LUP
 20190411 09:59:03
 date last changed
 20190430 04:11:15
@article{1bba6978afcb42efa52c9b93a16e28e1, abstract = {<p><br> A passive approximation problem is formulated where the target function is an arbitrary complexvalued 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 <br> <sup>p</sup><br> 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 Bspline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finitedimensional 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 Bspline 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. <br> </p>}, author = {Ivanenko, Yevhen and Gustafsson, Mats and Jonsson, B. L.G. and Luger, Annemarie and Nilsson, Börje and Nordebo, Sven and Toft, Joachim}, issn = {00361399}, keyword = {Approximation,Bsplines,Convex optimization,Herglotz functions,Passive systems,Sum rules}, language = {eng}, number = {1}, pages = {436458}, publisher = {SIAM Publications}, series = {SIAM Journal on Applied Mathematics}, title = {Passive approximation and optimization using bsplines}, url = {http://dx.doi.org/10.1137/17M1161026}, volume = {79}, year = {2019}, }