Homogeneous function: Difference between revisions

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==Glossary of name variants==

{{or section|date=December 2021}}

Let <math>f : X \to Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{F}</math> (usually the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex</math>). If <math>S</math> is a set of scalars, such as <math>\Z,</math>, <math>[0, \infty),</math>, or <math>\R</math> for example, then <math>f</math> is said to be {{em|{{visible anchor|homogeneous over}} <math>S</math>}} if

<math display=inline>f(s x) = s f(x)</math> for every <math>x \in X</math> and scalar <math>s \in S.</math>.

For instance, every [[additive map]] between vector spaces is {{em|{{visible anchor|homogeneous over the rational numbers}}}} <math>S := \Q</math> although it [[Cauchy's functional equation|might not be {{em|{{visible anchor|homogeneous over the real numbers}}}}]] <math>S := \R.</math>.

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>.

#* This property is often also called {{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}}} because for a function valued in a vector space or field, it is [[Logical equivalence|logically equivalent]] to: <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>.<ref group=proof>Assume that <math>f</math> is strictly positively homogeneous and valued in a vector space or a field. Then <math>f(0) = f(2 \cdot 0) = 2 f(0)</math> so subtracting <math>f(0)</math> from both sides shows that <math>f(0) = 0.</math>. Writing <math>r := 0,</math>, then for any <math>x \in X,</math>, <math>f(r x) = f(0) = 0 = 0 f(x) = r f(x),</math> which shows that <math>f</math> is nonnegative homogeneous.</ref> However, for a function valued in the [[extended real numbers]] <math>[-\infty, \infty] = \R \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 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}}

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

#{{em|{{visible anchor|Real homogeneity|Real homogeneous}}}}: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>.

#* This property is used in the definition of a {{em|real}} [[linear functional]].

#{{em|{{visible anchor|Homogeneity|Homogeneous}}}}: <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.

#{{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>.

#* Along with [[Additive map|additivity]], this property is assumed in the definition of an [[antilinear map]]. It is also assumed that one of the two coordinates of a [[sesquilinear form]] has this property (such as the [[inner product]] of a [[Hilbert space]]).

All of the above definitions can be generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = |r| f(x),</math>, in which case that definition is prefixed with the word {{nowrap|"{{em|absolute}}"}} or {{nowrap|"{{em|absolutely}}."}}

For example,

<ol start=5>

<li>{{em|{{visible anchor|Absolute homogeneity|Absolute homogeneous|Absolutely homogeneous}}}}: <math>f(sx) = |s| f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>.

* This property is used in the definition of a [[seminorm]] and a [[Norm (mathematics)|norm]].

</li>

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For instance,

<ol start=6>

<li>{{em|{{visible anchor|Real homogeneity of degree}} <math>k</math>}}: <math>f(rx) = r^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>.

</li>

<li>{{em|{{visible anchor|Homogeneity of degree}} <math>k</math>}}: <math>f(sx) = s^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>.

</li>

<li>{{em|{{visible anchor|Absolute real homogeneity of degree}} <math>k</math>}}: <math>f(rx) = |r|^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>.

</li>

<li>{{em|{{visible anchor|Absolute homogeneity of degree}} <math>k</math>}}: <math>f(sx) = |s|^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>.

</li>

</ol>

A nonzero [[continuous function]] that is homogeneous of degree <math>k</math> on <math>\R^n \backslash \lbrace 0 \rbrace</math> extends continuously to <math>\R^n</math> if and only if <math>k > 0.</math>.

== See also ==