Homogeneous function: Difference between revisions

Revision as of 19:08, 10 April 2023

Revision as of 19:10, 10 April 2023

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The following commonly encountered special cases and variations of this definition have their own terminology:

#({{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}}}) {{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}}}:{{sfn|Schechter|1996|pp=313-314}} <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>

#* When the function <math>f</math> is valued in a vector space or field, then this property is [[Logical equivalence|logically equivalent]]<ref group=proof name=posHomEquivToNonnegHom /> to {{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}}}, which by definition means:{{sfn|Kubrusly|2011|p=200}} means: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|non-negative}} real <math>r \geq 0.</math> It is for this reason that positive homogeneity is often also called nonnegative homogeneity. However, for functions valued in the [[extended real numbers]] <math>[-\infty, \infty] = \Reals \cup \{\pm \infty\},</math> which appear in fields like [[convex analysis]], the multiplication <math>0 \cdot f(x)</math> will be undefined whenever <math>f(x) = \pm \infty</math> and so these statements are not necessarily always interchangeable.<ref group=note>However, if such an <math>f</math> satisfies <math>f(rx) = r f(x)</math> for all <math>r > 0</math> and <math>x \in X,</math> then necessarily <math>f(0) \in \{\pm \infty, 0\}</math> and whenever <math>f(0), f(x) \in \R</math> are both real then <math>f(r x) = r f(x)</math> will hold for all <math>r \geq 0.</math></ref>

#* This property is used in the definition of a [[sublinear function]].{{sfn|Schechter|1996|pp=313-314}}{{sfn|Kubrusly|2011|p=200}}

#* [[Minkowski functional]]s are exactly those non-negative extended real-valued functions with this property.

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#{{em|{{visible anchor|Homogeneity|Homogeneous}}}}:{{sfn|Kubrusly|2011|p=55}} <math>f(sx) = s f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>

#* It is emphasized that this definition depends on the scalar field <math>\mathbb{F}</math> underlying the ___domain <math>X.</math>

#* This property is used in the definition of [[linear functional]]s and [[linear map]]s.{{sfn|Kubrusly|2011|p=200}}

#{{em|[[Semilinear map|{{visible anchor|Conjugate homogeneity|Conjugate homogeneous}}]]}}: <math>f(sx) = \overline{s} f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>

#* If <math>\mathbb{F} = \Complex</math> then <math>\overline{s}</math> typically denotes the [[complex conjugate]] of <math>s</math>. But more generally, as with [[semilinear map]]s for example, <math>\overline{s}</math> could be the image of <math>s</math> under some distinguished automorphism of <math>\mathbb{F}.</math>