16. Concavity (Convexity)

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Concavity (Convexity)

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III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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Concavity (Convexity)

Definition. A set $S$ is said to be concave if for every vector $x$ and $y$ in $S$, $\ is in $S$ for all $0\. In other words, all points on the line between two points of $S$ lie in $S$.

Definition. A functional is a mapping from a vector space to the real numbers $\.

Thus, for example, every norm is a functional.

Definition. A linear functional is a functional $f$ such that for each $x$ and $y$ in the linear space $X$, and for all scalars$\ and $\, we have $f(\.

Definition. The norm of a linear functional $f$ is defined on the normed linear space $X$ by


Definition. A functional $f$ defined on a concave subset $S$ of a vector space$X$ is said to be concave on $S$ if for every vector $x$ and $y$ in $S$,


A concave functional has the property that its values along a line segment lie below or on the line between its values at the end points. The functional is strictly concave on $S$ if strict inequality holds above for $\. Finally, $f$ isuniformly concave on $S$ if there exists $c>0$ such that for all $x,y\,

We have

Definition. A local minimizer of a real-valued function $f(x)$ is any $x^\ such that $f(x^\ in some neighborhood of $x$.

Definition. A global minimizer of a real-valued function $f(x)$ on a set $S$is any $x^\ such that $f(x^\ for all $x\.

Definition. A cluster point $x$ of a sequence $x_n$ is any point such that every neighborhood of $x$ contains at least one $x_n$.

Definition. The concave hull of a set $S$ in a metric space is the smallest concave set containing $S$.

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