Movatterモバイル変換

Jump to content

Homogeneous function

From Wikipedia, the free encyclopedia

Function with a multiplicative scaling behaviour

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(July 2018) (Learn how and when to remove this message)

For homogeneous linear maps, seeGraded vector space § Homomorphisms.

Inmathematics, ahomogeneous function is afunction of several variables such that the following holds: If each of the function's arguments is multiplied by the samescalar, then the function's value is multiplied by some power of this scalar; the power is called thedegree of homogeneity, or simply thedegree. That is, ifk is an integer, a functionf ofn variables is homogeneous of degreek if

f(sx_{1},\ldots ,sx_{n})=s^{k}f(x_{1},\ldots ,x_{n})

for every $x_{1},\ldots ,x_{n},$ and $s\neq 0.$ This is also referred to akth-degree orkth-order homogeneous function.

For example, ahomogeneous polynomial of degreek defines a homogeneous function of degreek.

The above definition extends to functions whosedomain andcodomain arevector spaces over afieldF: a function $f:V\to W$ between twoF-vector spaces ishomogeneous of degree $k {\displaystyle k}$ if

f(s\mathbf {v} )=s^{k}f(\mathbf {v} )

1

for all nonzero $s\in F$ and $v\in V.$ This definition is often further generalized to functions whose domain is notV, but acone inV, that is, a subsetC ofV such that $\mathbf {v} \in C$ implies $s\mathbf {v} \in C$ for every nonzero scalars.

In the case offunctions of several real variables andreal vector spaces, a slightly more general form of homogeneity calledpositive homogeneity is often considered, by requiring only that the above identities hold for $s>0,$ and allowing any real numberk as a degree of homogeneity. Every homogeneous real function ispositively homogeneous. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.

Anorm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is theabsolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition ofprojective schemes.

Definitions

The concept of a homogeneous function was originally introduced forfunctions of several real variables. With the definition ofvector spaces at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since atuple of variable values can be considered as acoordinate vector. It is this more general point of view that is described in this article.

There are two commonly used definitions. The general one works for vector spaces over arbitraryfields, and is restricted to degrees of homogeneity that areintegers.

The second one supposes to work over the field ofreal numbers, or, more generally, over anordered field. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore calledpositive homogeneity, the qualificativepositive being often omitted when there is no risk of confusion. Positive homogeneity leads to considering more functions as homogeneous. For example, theabsolute value and allnorms are positively homogeneous functions that are not homogeneous.

The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.

General homogeneity

LetV andW be twovector spaces over afieldF. Alinear cone inV is a subsetC ofV such that $sx\in C$ for all $x\in C$ and all nonzero $s\in F.$

Ahomogeneous functionf fromV toW is apartial function fromV toW that has a linear coneC as itsdomain, and satisfies

f(sx)=s^{k}f(x)

for someintegerk, every $x\in C,$ and every nonzero $s\in F.$ The integerk is called thedegree of homogeneity, or simply thedegree off.

A typical example of a homogeneous function of degreek is the function defined by ahomogeneous polynomial of degreek. Therational function defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; itscone of definition is the linear cone of the points where the value of denominator is not zero.

Homogeneous functions play a fundamental role inprojective geometry since any homogeneous functionf fromV toW defines a well-defined function between theprojectivizations ofV andW. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degree) play an essential role in theProj construction ofprojective schemes.

Positive homogeneity

When working over thereal numbers, or more generally over anordered field, it is commonly convenient to considerpositive homogeneity, the definition being exactly the same as that in the preceding section, with "nonzeros" replaced by "s > 0" in the definitions of a linear cone and a homogeneous function.

This change allows considering (positively) homogeneous functions with any real number as their degrees, sinceexponentiation with a positive real base is well defined.

Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of theabsolute value function andnorms, which are all positively homogeneous of degree1. They are not homogeneous since $|-x|=|x|\neq -|x|$ if $x\neq 0.$ This remains true in thecomplex case, since the field of the complex numbers $\mathbb {C}$ and every complex vector space can be considered as real vector spaces.

Euler's homogeneous function theorem is a characterization of positively homogeneousdifferentiable functions, which may be considered as thefundamental theorem on homogeneous functions.

Examples

A homogeneous function is not necessarilycontinuous, as shown by this example. This is the function $f {\displaystyle f}$ defined by $f(x,y)=x$ if $xy>0$ and $f(x,y)=0$ if $xy\leq 0.$ This function is homogeneous of degree 1, that is, $f(sx,sy)=sf(x,y)$ for any real numbers $s, x, y . {\displaystyle s,x,y.}$ It is discontinuous at $y=0,x\neq 0.$

{\displaystyle f} — A homogeneous function is not necessarilycontinuous, as shown by this example. This is the function $f {\displaystyle f}$ defined by $f(x,y)=x$ if $xy>0$ and $f(x,y)=0$ if $xy\leq 0.$ This function is homogeneous of degree 1, that is, $f(sx,sy)=sf(x,y)$ for any real numbers $s, x, y . {\displaystyle s,x,y.}$ It is discontinuous at $y=0,x\neq 0.$

Simple example

The function $f(x,y)=x^{2}+y^{2}$ is homogeneous of degree 2: $f(tx,ty)=(tx)^{2}+(ty)^{2}=t^{2}\left(x^{2}+y^{2}\right)=t^{2}f(x,y).$

Absolute value and norms

Theabsolute value of areal number is a positively homogeneous function of degree1, which is not homogeneous, since $|sx|=s|x|$ if $s>0,$ and $|sx|=-s|x|$ if $s<0.$

The absolute value of acomplex number is a positively homogeneous function of degree $1 {\displaystyle 1}$ over the real numbers (that is, when considering the complex numbers as avector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.

More generally, everynorm andseminorm is a positively homogeneous function of degree1 which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.

Linear Maps

Anylinear map $f:V\to W$ betweenvector spaces over afieldF is homogeneous of degree 1, by the definition of linearity: $f(\alpha \mathbf {v} )=\alpha f(\mathbf {v} )$ for all $\alpha \in {F}$ and $v\in V.$

Similarly, anymultilinear function $f:V_{1}\times V_{2}\times \cdots V_{n}\to W$ is homogeneous of degree $n, {\displaystyle n,}$ by the definition of multilinearity: $f\left(\alpha \mathbf {v} _{1},\ldots ,\alpha \mathbf {v} _{n}\right)=\alpha ^{n}f(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})$ for all $\alpha \in {F}$ and $v_{1}\in V_{1},v_{2}\in V_{2},\ldots ,v_{n}\in V_{n}.$

Homogeneous polynomials

Main article:Homogeneous polynomial

Monomials in $n {\displaystyle n}$ variables define homogeneous functions $f:\mathbb {F} ^{n}\to \mathbb {F} .$ For example, $f(x,y,z)=x^{5}y^{2}z^{3}\,$ is homogeneous of degree 10 since $f(\alpha x,\alpha y,\alpha z)=(\alpha x)^{5}(\alpha y)^{2}(\alpha z)^{3}=\alpha ^{10}x^{5}y^{2}z^{3}=\alpha ^{10}f(x,y,z).\,$ The degree is the sum of the exponents on the variables; in this example, $10=5+2+3.$

Ahomogeneous polynomial is apolynomial made up of a sum of monomials of the same degree. For example, $x^{5}+2x^{3}y^{2}+9xy^{4}$ is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Given a homogeneous polynomial of degree $k {\displaystyle k}$ with real coefficients that takes only positive values, one gets a positively homogeneous function of degree $k/d$ by raising it to the power $1/d.$ So for example, the following function is positively homogeneous of degree 1 but not homogeneous: $\left(x^{2}+y^{2}+z^{2}\right)^{\frac {1}{2}}.$

Min/max

For every set of weights $w_{1},\dots ,w_{n},$ the following functions are positively homogeneous of degree 1, but not homogeneous:

$\min \left({\frac {x_{1}}{w_{1}}},\dots ,{\frac {x_{n}}{w_{n}}}\right)$ (Leontief utilities)
$\max \left({\frac {x_{1}}{w_{1}}},\dots ,{\frac {x_{n}}{w_{n}}}\right)$

Rational functions

Rational functions formed as the ratio of twohomogeneous polynomials are homogeneous functions in theirdomain, that is, off of thelinear cone formed by thezeros of the denominator. Thus, if $f {\displaystyle f}$ is homogeneous of degree $m {\displaystyle m}$ and $g {\displaystyle g}$ is homogeneous of degree $n, {\displaystyle n,}$ then $f/g$ is homogeneous of degree $m-n$ away from the zeros of $g . {\displaystyle g.}$

Non-examples

The homogeneousreal functions of a single variable have the form $x\mapsto cx^{k}$ for some constantc. So, theaffine function $x\mapsto x+5,$ thenatural logarithm $x\mapsto \ln(x),$ and theexponential function $x\mapsto e^{x}$ are not homogeneous.

Euler's theorem

Roughly speaking,Euler's homogeneous function theorem asserts that the positively homogeneous functions of a given degree are exactly the solution of a specificpartial differential equation. More precisely:

Euler's homogeneous function theorem—Iff is a(partial) function ofn real variables that is positively homogeneous of degreek, andcontinuously differentiable in some open subset of $\mathbb {R} ^{n},$ then it satisfies in this open set thepartial differential equation $k\,f(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}{\frac {\partial f}{\partial x_{i}}}(x_{1},\ldots ,x_{n}).$

Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degreek, defined on a positive cone (here,maximal means that the solution cannot be prolongated to a function with a larger domain).

Proof

For having simpler formulas, we set $\mathbf {x} =(x_{1},\ldots ,x_{n}).$ The first part results by using thechain rule for differentiating both sides of the equation $f(s\mathbf {x} )=s^{k}f(\mathbf {x} )$ with respect to $s, {\displaystyle s,}$ and taking the limit of the result whens tends to1.

The converse is proved by integrating a simpledifferential equation.Let $\mathbf {x}$ be in the interior of the domain off. Fors sufficiently close to1, the function ${\textstyle g(s)=f(s\mathbf {x} )}$ is well defined. The partial differential equation implies that $sg'(s)=kf(s\mathbf {x} )=kg(s).$ The solutions of thislinear differential equation have the form $g(s)=g(1)s^{k}.$ Therefore, $f(s\mathbf {x} )=g(s)=s^{k}g(1)=s^{k}f(\mathbf {x} ),$ ifs is sufficiently close to1. If this solution of the partial differential equation would not be defined for all positives, then thefunctional equation would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degreek. $\square$

As a consequence, if $f:\mathbb {R} ^{n}\to \mathbb {R}$ is continuously differentiable and homogeneous of degree $k, {\displaystyle k,}$ its first-orderpartial derivatives $\partial f/\partial x_{i}$ are homogeneous of degree $k-1.$ This results from Euler's theorem by differentiating the partial differential equation with respect to one variable.

In the case of a function of a single real variable ( $n=1$ ), the theorem implies that a continuously differentiable and positively homogeneous function of degreek has the form $f(x)=c_{+}x^{k}$ for $x>0$ and $f(x)=c_{-}x^{k}$ for $x<0.$ The constants $c_{+}$ and $c_{-}$ are not necessarily the same, as it is the case for theabsolute value.

Application to differential equations

Main article:Homogeneous differential equation

The substitution $v=y/x$ converts theordinary differential equation $I(x,y){\frac {\mathrm {d} y}{\mathrm {d} x}}+J(x,y)=0,$ where $I {\displaystyle I}$ and $J {\displaystyle J}$ are homogeneous functions of the same degree, into theseparable differential equation $x{\frac {\mathrm {d} v}{\mathrm {d} x}}=-{\frac {J(1,v)}{I(1,v)}}-v.$

Generalizations

Homogeneity under a monoid action

The definitions given above are all specialized cases of the following more general notion of homogeneity in which $X {\displaystyle X}$ can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of amonoid.

Let $M {\displaystyle M}$ be amonoid with identity element $1\in M,$ let $X {\displaystyle X}$ and $Y {\displaystyle Y}$ be sets, and suppose that on both $X {\displaystyle X}$ and $Y {\displaystyle Y}$ there are defined monoid actions of $M . {\displaystyle M.}$ Let $k {\displaystyle k}$ be a non-negative integer and let $f:X\to Y$ be a map. Then $f {\displaystyle f}$ is said to behomogeneous of degree $k {\displaystyle k}$ over $M {\displaystyle M}$ if for every $x\in X$ and $m\in M,$ $f(mx)=m^{k}f(x).$ If in addition there is a function $M\to M,$ denoted by $m\mapsto |m|,$ called anabsolute value then $f {\displaystyle f}$ is said to beabsolutely homogeneous of degree $k {\displaystyle k}$ over $M {\displaystyle M}$ if for every $x\in X$ and $m\in M,$ $f(mx)=|m|^{k}f(x).$

A function ishomogeneous over $M {\displaystyle M}$ (resp.absolutely homogeneous over $M {\displaystyle M}$ ) if it is homogeneous of degree $1 {\displaystyle 1}$ over $M {\displaystyle M}$ (resp. absolutely homogeneous of degree $1 {\displaystyle 1}$ over $M {\displaystyle M}$ ).

More generally, it is possible for the symbols $m^{k}$ to be defined for $m\in M$ with $k {\displaystyle k}$ being something other than an integer (for example, if $M {\displaystyle M}$ is the real numbers and $k {\displaystyle k}$ is a non-zero real number then $m^{k}$ is defined even though $k {\displaystyle k}$ is not an integer). If this is the case then $f {\displaystyle f}$ will be calledhomogeneous of degree $k {\displaystyle k}$ over $M {\displaystyle M}$ if the same equality holds: $f(mx)=m^{k}f(x)\quad {\text{ for every }}x\in X{\text{ and }}m\in M.$

The notion of beingabsolutely homogeneous of degree $k {\displaystyle k}$ over $M {\displaystyle M}$ is generalized similarly.

Distributions (generalized functions)

Main article:Homogeneous distribution

A continuous function $f {\displaystyle f}$ on $\mathbb {R} ^{n}$ is homogeneous of degree $k {\displaystyle k}$ if and only if $\int _{\mathbb {R} ^{n}}f(tx)\varphi (x)\,dx=t^{k}\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx$ for allcompactly supported test functions $\varphi$ ; and nonzero real $t . {\displaystyle t.}$ Equivalently, making achange of variable $y=tx,$ $f {\displaystyle f}$ is homogeneous of degree $k {\displaystyle k}$ if and only if $t^{-n}\int _{\mathbb {R} ^{n}}f(y)\varphi \left({\frac {y}{t}}\right)\,dy=t^{k}\int _{\mathbb {R} ^{n}}f(y)\varphi (y)\,dy$ for all $t {\displaystyle t}$ and all test functions $\varphi .$ The last display makes it possible to define homogeneity ofdistributions. A distribution $S {\displaystyle S}$ is homogeneous of degree $k {\displaystyle k}$ if $t^{-n}\langle S,\varphi \circ \mu _{t}\rangle =t^{k}\langle S,\varphi \rangle$ for all nonzero real $t {\displaystyle t}$ and all test functions $\varphi .$ Here the angle brackets denote the pairing between distributions and test functions, and $\mu _{t}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ is the mapping of scalar division by the real number $t . {\displaystyle t.}$

Glossary of name variants

This sectionpossibly containsoriginal research. Pleaseimprove it byverifying the claims made and addinginline citations. Statements consisting only of original research should be removed.(December 2021) (Learn how and when to remove this message)

Let $f:X\to Y$ be a map between twovector spaces over a field $\mathbb {F}$ (usually thereal numbers $\mathbb {R}$ orcomplex numbers $\mathbb {C}$ ). If $S {\displaystyle S}$ is a set of scalars, such as $\mathbb {Z} ,$ $[0,\infty ),$ or $\mathbb {R}$ for example, then $f {\displaystyle f}$ is said to behomogeneous over $S {\displaystyle S}$ if ${\textstyle f(sx)=sf(x)}$ for every $x\in X$ and scalar $s\in S.$ For instance, everyadditive map between vector spaces ishomogeneous over the rational numbers $S:=\mathbb {Q}$ although itmight not behomogeneous over the real numbers $S:=\mathbb {R} .$

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

(Strict)Positive homogeneity:^[1] $f(rx)=rf(x)$ for all $x\in X$ and allpositive real $r>0.$
- When the function $f {\displaystyle f}$ is valued in a vector space or field, then this property islogically equivalent^{[proof 1]} tononnegative homogeneity, which by definition means:^[2] $f(rx)=rf(x)$ for all $x\in X$ and allnon-negative real $r\geq 0.$ It is for this reason that positive homogeneity is often also called nonnegative homogeneity. However, for functions valued in theextended real numbers $[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \},$ which appear in fields likeconvex analysis, the multiplication $0\cdot f(x)$ will be undefined whenever $f(x)=\pm \infty$ and so these statements are not necessarily always interchangeable.^{[note 1]}
- This property is used in the definition of asublinear function.^[1]^[2]
- Minkowski functionals are exactly those non-negative extended real-valued functions with this property.
Real homogeneity: $f(rx)=rf(x)$ for all $x\in X$ and all real $r . {\displaystyle r.}$
- This property is used in the definition of areallinear functional.
Homogeneity:^[3] $f(sx)=sf(x)$ for all $x\in X$ and all scalars $s\in \mathbb {F} .$
- It is emphasized that this definition depends on the scalar field $\mathbb {F}$ underlying the domain $X . {\displaystyle X.}$
- This property is used in the definition oflinear functionals andlinear maps.^[2]
Conjugate homogeneity:^[4] $f(sx)={\overline {s}}f(x)$ for all $x\in X$ and all scalars $s\in \mathbb {F} .$
- If $\mathbb {F} =\mathbb {C}$ then ${\overline {s}}$ typically denotes thecomplex conjugate of $s {\displaystyle s}$ . But more generally, as withsemilinear maps for example, ${\overline {s}}$ could be the image of $s {\displaystyle s}$ under some distinguished automorphism of $\mathbb {F} .$
- Along withadditivity, this property is assumed in the definition of anantilinear map. It is also assumed that one of the two coordinates of asesquilinear form has this property (such as theinner product of aHilbert space).

All of the above definitions can be generalized by replacing the condition $f(rx)=rf(x)$ with $f(rx)=|r|f(x),$ in which case that definition is prefixed with the word"absolute" or"absolutely." For example,

Absolute homogeneity:^[2] $f(sx)=|s|f(x)$ for all $x\in X$ and all scalars $s\in \mathbb {F} .$
- This property is used in the definition of aseminorm and anorm.

If $k {\displaystyle k}$ is a fixed real number then the above definitions can be further generalized by replacing the condition $f(rx)=rf(x)$ with $f(rx)=r^{k}f(x)$ (and similarly, by replacing $f(rx)=|r|f(x)$ with $f(rx)=|r|^{k}f(x)$ for conditions using the absolute value, etc.), in which case the homogeneity is said to be"of degree $k {\displaystyle k}$ " (where in particular, all of the above definitions are"of degree $1 {\displaystyle 1}$ ").For instance,

Real homogeneity of degree $k {\displaystyle k}$ : $f(rx)=r^{k}f(x)$ for all $x\in X$ and all real $r . {\displaystyle r.}$
Homogeneity of degree $k {\displaystyle k}$ : $f(sx)=s^{k}f(x)$ for all $x\in X$ and all scalars $s\in \mathbb {F} .$
Absolute real homogeneity of degree $k {\displaystyle k}$ : $f(rx)=|r|^{k}f(x)$ for all $x\in X$ and all real $r . {\displaystyle r.}$
Absolute homogeneity of degree $k {\displaystyle k}$ : $f(sx)=|s|^{k}f(x)$ for all $x\in X$ and all scalars $s\in \mathbb {F} .$

A nonzerocontinuous function that is homogeneous of degree $k {\displaystyle k}$ on $\mathbb {R} ^{n}\backslash \lbrace 0\rbrace$ extends continuously to $\mathbb {R} ^{n}$ if and only if $k>0.$

See also

Homogeneous space
Triangle center function – Point in a triangle that can be seen as its middle under some criteriaPages displaying short descriptions of redirect targets

Notes

^However, if such an $f {\displaystyle f}$ satisfies $f(rx)=rf(x)$ for all $r>0$ and $x\in X,$ then necessarily $f(0)\in \{\pm \infty ,0\}$ and whenever $f(0),f(x)\in \mathbb {R}$ are both real then $f(rx)=rf(x)$ will hold for all $r\geq 0.$

Proofs

^Assume that $f {\displaystyle f}$ is strictly positively homogeneous and valued in a vector space or a field. Then $f(0)=f(2\cdot 0)=2f(0)$ so subtracting $f(0)$ from both sides shows that $f(0)=0.$ Writing $r:=0,$ then for any $x\in X,$ $f(rx)=f(0)=0=0f(x)=rf(x),$ which shows that $f {\displaystyle f}$ is nonnegative homogeneous.

References

^^a ^bSchechter 1996, pp. 313–314.
^^a ^b ^c ^dKubrusly 2011, p. 200.
^Kubrusly 2011, p. 55.
^Kubrusly 2011, p. 310.

Sources

Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.".Analysis II (in German) (2nd ed.). Springer Verlag. p. 188.ISBN 3-540-09484-9.
Kubrusly, Carlos S. (2011).The Elements of Operator Theory (Second ed.). Boston:Birkhäuser.ISBN 978-0-8176-4998-2.OCLC 710154895.
Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
Schechter, Eric (1996).Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press.ISBN 978-0-12-622760-4.OCLC 175294365.

External links

"Homogeneous function",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Eric Weisstein."Euler's Homogeneous Function Theorem".MathWorld.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Homogeneous_function&oldid=1317825628"

Hidden categories:

[8]ページ先頭

©2009-2025 Movatter.jp