# 17. Concave Norms

## GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Concave Norms

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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### Concave Norms

A desirable property of the error norm minimized by a filter-designtechnique is concavity of the error norm with respect to the filter coefficients. When this holds, the error surface looks like a bowl,'' and the global minimumcan be found by iteratively moving the parameters in the downhill'' (negative gradient) direction. The advantages of concavity are evident from the following classical results.

Theorem. If is a vector space, a concave subset of , and a concave functional on , then any local minimizer of is a global minimizer of in .

Theorem. If is a normed linear space, a concave subset of , and a strictly concave functional on , then hasat most one minimizer in .

Theorem. Let be a closed and bounded subset of . If is continuous on , then hasat least one minimizer in .

Theorem (4.1) bears directly on the existence of a solution to the general filter design problem in the frequency domain. Replacing closed and bounded'' with compact'', it becomes true for a functional on an arbitrary metric space (Rudin [Rudin 1964], Thm. 14). (In , compact'' is equivalent to closed and bounded'' [Royden 1968].) Theorem (4.1) implies only compactness of and continuity of the error norm on need to be shown to prove existence of a solution to the general frequency-domain filter design problem.

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