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Passive approximation and optimization using b-splines

and Toft, Joachim LU (2019) In SIAM Journal on Applied Mathematics 79(1). p.436-458
Abstract

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... (More)

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.

(Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Approximation, B-splines, 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
0036-1399
DOI
10.1137/17M1161026
language
English
LU publication?
yes
id
1bba6978-afcb-42ef-a52c-9b93a16e28e1
2019-04-11 09:59:03
date last changed
2019-06-03 13:58:02
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abstract     = {<p><br>
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                             <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 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.                         <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         = {0036-1399},
keyword      = {Approximation,B-splines,Convex optimization,Herglotz functions,Passive systems,Sum rules},
language     = {eng},
number       = {1},
pages        = {436--458},
publisher    = {SIAM Publications},
series       = {SIAM Journal on Applied Mathematics},
title        = {Passive approximation and optimization using b-splines},
url          = {http://dx.doi.org/10.1137/17M1161026},
volume       = {79},
year         = {2019},
}

```