Passive approximation and optimization using bsplines
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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 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.
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY
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Original language  English 

Pages (fromto)  436458 
Number of pages  23 
Journal  SIAM Journal on Applied Mathematics 
Volume  79 
Issue number  1 
Publication status  Published  2019 
Publication category  Research 
Peerreviewed  Yes 