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Passive Approximation and Optimization Using B-splinesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true}); PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt910",{id:"formSmash:j_idt910",widgetVar:"widget_formSmash_j_idt910",onLabel:"Hide others and affiliations",offLabel:"Show others and affiliations"});
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Number of Authors: 72019 (English)In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 79, no 1, p. 436-458Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2019. Vol. 79, no 1, p. 436-458
##### Keywords [en]

approximation, Herglotz functions, B-splines, passive systems, convex optimization, sum rules
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-167692DOI: 10.1137/17M1161026ISI: 000460127100021OAI: oai:DiVA.org:su-167692DiVA, id: diva2:1301654
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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 <= infinity. 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 Holder 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.

doi
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