Haruki, Shigeru

Nakagiri, Shin-ichi

Kyoto University Research Information Repository

TitlePartial difference functional equations arising from theCauchy-Riemann equations (Mathematical models anddynamics of functional equations)Author(s)Haruki, Shigeru; Nakagiri, Shin-ichiCitation数理解析研究所講究録 (2004), 1372: 90-96Issue Date2004-04URL http://hdl.handle.net/2433/25496RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitysoPartial difference functional equations arising fromthe Cauchy-Riemann equationsShigeru Haruki ( )Department of $Appl_{\dot{i}}edM$athematics, Faculty of Science, Okayama University of Science, JAPAN.Shin-ichi Nakagiri ( )Department of Applied Mathematics, Faculty of Engineering, Kobe University, JAPAN.1. IntroductionLet $(G, +)$ be an additive Abelian group in which it is possible to devide by 2. Let $\mathbb{C}$ be thefield of complex numbers. The main aim of this note is to determine the general solution of thefollowing new functional equation(E.2) $f(x+t,y)-f$ ($x-$ t, $\mathrm{y}$) $=-i[f(x,y+t)-f(x, y-t)]$for all $x$ , $y$ , $t\in G,$ where $f$ : $G\cross Garrow \mathbb{C}$ and $i$ is the imaginary unit.Let $\mathbb{R}$ be the field of real numbers. For a function $f$ : $\mathbb{R}\mathrm{x}\mathbb{R}arrow \mathbb{C}$ we define the divided partialdifference operator $\triangle_{x,t}$ and $\triangle_{y,t}$ by $(\triangle_{x,t}f)(x, y)=[f(x+t,y)-f(x, y)]/t$ and $(\triangle_{y,t}f)(x, y)=$$[f(x, y+t)-f(x, y)]/t$, respectively. Then the partial difference functional equation$\triangle_{x,t}f=-ils_{y}$,$tf$may be considered as a descrete analogue of the Cauchy-Riemann equation$\frac{\partial f}{\partial x}=-i$.$\frac{\partial f}{\partial y}$ .The above partial difference functional equation may be rewritten in the form(E.2) $f(x+t, y)-f(x, y)=-i$ [$f(x,$ $y+t)-$ f(x $y)$ ], )-f$ (x, $y$ $=-i f ,y+t)-f(x, y)]for all $x$ , $y$ , $t\in \mathbb{R}$, which has a simple geometric interpretation on the plane. In previous paperof J. Aczel and S. Haruki 1981 [3], and S. Haruki 1986 [6] we consider equation (E.2) and show,among others, that (E.2) does not lead essentially beyond a linear function in the case when$\mathbb{R}$ is replaced by an arbitrary monoid $M$. Further, a paper of S. Haruki and C. T. Ng 1994 [7]obtains the general solution of (E.2) under more general algebraic structure than $M$ and C.It is natural to ask what happens if, instead of operators $\triangle_{x}$ , $t$ and $\triangle_{y,t}$ , we impose thedivided partial mean difference operators $\mathit{7}x,t$ and $)_{y,t}$ defined by $(\mathrm{v}\mathrm{x},\mathrm{t}\mathrm{f})(\mathrm{x},\mathrm{y})=[\mathrm{f}(\mathrm{x}+t, y)-$$f(x-t, y)]/(2t)$ and $(\nabla y,tf)(x,y)=[f(x, y+t)-f(x, y-t)]/(2t)$ . In this case we have thepartial difference functional equation$/_{x,t}f=-i)_{y,t}$ $f$,1372 2004 90-9681which is also a discrete analogue of the Cauchy-Riemann equation. This leads to the abovefunctional equation (E.I) which also has a simple geometric interpretation on the plane. Asa main result of this note we show that equation (E.I) for $f$ : $G\cross Garrow \mathbb{C}$ does not leadessentially beyond a quadratic function.In Q2 we determine the general (\S 2.1) and regular (\S 2.2) solutions (when $G$ is replaced by R)of equation (E. 1).We also show in \S 3 that similar results hold for certain related functional equations. In \S 3.1we consider the functional equation(E.I) $f(x+t,y+t)-f(x-t,y-t)=-i[f(x-t,y+t)-f(x+t, y-t)]$ .In \S 3.2 we study the partial difference differential functional equations$\frac{\partial f(x,y)}{\partial x}=-i$.$[ \frac{f(x,y+t)-f(x,y-t)}{2t}\rceil,$ $\frac{f(x+t,y)-f(x-t,y)}{2t}=-i$. $\frac{\partial f((x,y)}{y}$2. On the functional equation (E.I)In this section we determine the general and regular solutions of functional equation (E. 1).2.1. The general solutionA function $A^{1}$ : $Garrow \mathbb{C}$ is said to be additive if $A^{1}$ satisfies $A^{1}(x+y)=A^{1}(x)+A^{1}(y)$for all $c$ , $y\in G.$ A function $A_{2}$ : $G\cross Garrow \mathbb{C}$ is said to be $\mathrm{b}\mathrm{i}$-additive if $A_{2}$ satisfies bothequations $A_{2}(x+y, z)=$ A2 $(\mathrm{x}z)+$ Al (y) $z)$ and A2 $(x, y+z)=$ A2 $(x, y)+$ A2 $(\mathrm{x}, z)$ for all$x$ , $y$ , $z\in G.$ A function $A^{2}$ : $Garrow \mathbb{C}$ is a diagonalization of symmetric $\mathrm{b}\mathrm{i}$-additive function if$A^{2}(x)=$ A2 $(x, y)|_{y=x}$ , where $A_{2}$ : $G\cross Garrow \mathbb{C}$ is symmetric and additive in each argument.Our main result of this note is as follows.Theorem 2.1. A function $f$ : $G\cross Garrow \mathbb{C}$ satisfies equation (E.I) for all $x$ , $y$ , $z\in G$ if andonly if there exist(i) an arbitrary complex constant $A^{0}$ ,(ii) an arbitrary additive function $A^{1}$ : $Garrow \mathbb{C}$ ,(iii) an arbitrary symmetric $bi$-additive function $A_{2}$ : $G\cross$ $Garrow \mathbb{C}$ , and(iv) a diagonalization $A^{2}$ : $G\sim \mathbb{C}$ of the above function $A_{2}$such that(5.1) $f(x, y)=A^{0}+A^{1}(x)+iA^{1}(y)+A^{2}(x)-A^{2}(y)+2iA_{2}(x,y)$for all $x$ , $y\in G.$We impose the following notations. Define the shift operators $X^{t}$ and $\mathrm{Y}^{t}$ by $(X^{t}f)(x, y)=$$f(x+t,y)$ and $(\mathrm{Y}^{t}f)(x, y)=f(x, y+t)$ for all $x$ , $y,t\in G.$ In particular $1=X^{0}=\mathrm{Y}^{0}$ denotethe identity operator. Further, define the partial mean difference operators $\delta_{x,t}$ and $\delta_{y,t}$ by32$\delta_{x,t}=X^{t}-X^{-t}$ and $\delta_{y,t}=\mathrm{Y}^{t}-\mathrm{Y}^{-t}$ for all $x$ , $y$ , $t\in G.$ Notice that the ring of operatorsgenerated by this family of operators is commutative and distributive.In order to prove Theorem 2.1 we need the following two lemmas. One of them is:Lemma 2.1 If a function $f$ : $G\cross Garrow \mathbb{C}$ satisfies equation (E.I) for all $x$ , $y$ , $t\in G,$ then$f$ also satisfies each one of the following three functional equations(2.1) $(\delta_{x,t}^{3}f)(x, y)=0$ and $(\delta_{y,t}^{3}f)(x, y)=0$(2.2) $((\delta_{x,t}^{2}+ \mathrm{a}_{y,t}^{2})f)(x,y)=0$or as the expanded $form$ ($\mathit{2}t$ replaced by $t$)(2.3) $f(x+t,y)+f(x-t,y)+f(x,y+t)+f(x,y-t)=4f(x,y)$for all $x,$ $y\in G.$The above Lemma 2.1 shows that equation (E.V) yields equation (2.3). On the other hand,J. Aczel, H. Haruki, M. A. McKiernan and G. N. SakoviS 1968 [2, p.43, Lemma 3] proved thatequation (2.3) is equivalent to the Haruki functional equation (M. A. McKiernan [11], H. Swiatak[13], among others)(2.4) $f.(x+t,y+t)+f(x+t,y-t)+f(x-. t, y+t)+f(x-t,y-t)=4f(x, y)$.Hence, if we directly apply a general theorem of M. A. McKiernan 1970 [12, p.32, Theorem 2]to the equation (2.4), then we obtain(2.5) $(\delta_{x,t}^{k}f)(x,y)=0$ and $(\delta_{y,t}^{k}f)(x,y)=0$with $k=11.$ On the other hand, it is also known in [2, p.43, Lemma 3] that if an arbitrary$f$ satisfies (2.4), then $f$ ako satisfies difference equations (2.5) for $k=4.$ However, in orderto obtain the general solution of equation (E.I) our result (2.5) with $k=3,$ that is, (2.$\cdot$1), isbetter result and easier to prove the ‘only if ’ part of Theorem 2.1, since if $k>3$ in (2.5), then asolution of (2.1) contains more symmetric multiadditive functions in higher order (cf. S. Mazurand W. Orlicz 1934 [9], and M. A. McKiernan 1967 [10], among others). We emphasis that wedo not apply the above mensioned result for the case $k=4.$The other is the following lemma which is a particular case of Lemma 6 in [2, p.49-50]. Wenote that if we replace $f$ : $\mathbb{R}\cross$ $\mathbb{R}arrow \mathbb{R}$ by $f$ : $G\cross$ $Garrow \mathbb{C}$ in Lemma 6 of [2], then it followsfrom a general theorem of S. Mazur and W. Orlicz [9] that the result of Lemma 6 in [2] stillholds for the case $\delta_{x,t}=X^{t}-X^{-t}$ and $\delta_{y,t}=\mathrm{Y}^{t}-\mathrm{Y}^{-t}$ instead of $\triangle_{x}$ ,$t=X^{t/2}-X^{-t/2}$ and$\triangle_{y,t}=\mathrm{Y}^{t/2}-\mathrm{Y}^{-t/2}$ defined in [2, p.43].Lemma 2.2. A function $f:G\mathrm{x}$ $Garrow \mathbb{C}$ satisfies both equations (2.1) for all $x$ , $y,t\in G$ ifand only if $f$ is given by(2.6) $f(x,y)=A^{0}+A^{1}(x)+A^{2}.(x)+B^{1}(y)+B^{2}(y)$$+A^{\dot{1}}$ ’1 $(x;y)+$ $4^{2,1}$ $(x;y)+$ $4^{1,2}(x;y)+A^{2,2}(x;y$33for all $x$ , $y\in G,$ where $A^{0}$ , and $A^{1}$ , $A^{2}$ : $Garrow \mathbb{C}$ are defin$ed$ in Theorem 2.1, $B^{1}$ : $Garrow \mathbb{C}$ isan arbitrary additive function, and $B^{2}$ : $Garrow \mathbb{C}$ is a diagonalization of an arbitrary symmetric$bi$-additive function. Further, functions $A^{1,1}$ , $A^{2,1}$ , $A^{1,2}$ , and $A^{2,2}$ : $G\cross Garrow \mathbb{C}$ are definedas follows. $\mathit{4}1_{:}1$ $(x;y)$ is additive in $x$ and $y$, and not symmetric. $4^{2,1}(x;y):=A_{2,1}(x, x;y)$ isadditive in all variables, symmetric in the first two. $A^{1,2}(x;y):=A_{1,2}(x;y, y)$ is additive inall variables, symmetric in the last two places. $A^{2,2}(x;y):=A_{2,2}(x,x;y, y)$ is additive in allvariables, symmetric in the first two and in the last two places.2.2. Regular solutionsIn addition, as soon as some suitable regularity assumptions are imposed on $f$ for the case $G=\mathbb{R}$in the above Theorem 2.1, then it can be readily shown by the following lemma that $f$ is anordinary complex polynomial of degree at most two. The following lemma is a consequence ofTheorem 2.1.Lemma 2.3. Let $(F, +)$ be an additive group. If $f$ : $F\cross Farrow \mathbb{C}$ is given by (S.I) for all$x,y\in F,$ then all functions $A^{1}$ , $A^{2}$ : $Farrow \mathbb{C}$ and $A_{2}$ : $F\cross Farrow \mathbb{C}$ can be represented in termsof $f$ and an arbitrary constant $A^{0}$ by(2.26) $A^{1}(x)=[f(x, y)-f(-x, -y)-f(-x, y)+f(x, -y)]/4$,(2.27) $A^{1}(y)=$ $[f(x, y)-f(-x, -y) +f(-x, y)-f(x, -y)]/(4\mathrm{j})$ ,(2.28) A2(x, $y$ ) $=[f(x,y)+f(-x, -y)-f(-x,y)-f(x, -y)]/(8i)$ ,(2.29) $A^{2}(x)=f(x, 0)-A^{0}-[f(x,y)-f(-x, -y)-f(-x, y)+f(x, -y)]$ /4,(2.30) $A^{2}(y)=-f(0, y)+A^{0}+$ [$/(\mathrm{x},$ $y)-$ $/(-\mathrm{x},$ $-y)+f(-x$, $y)-$ /(x, $-y)$ ] $/4$for all $x$ , $y\in F.$If we assume that, for example, $f$ : $\mathbb{R}\mathrm{x}$ $\mathbb{R}arrow \mathbb{C}$ is continuous everywhere, then by applyingthe above Lemma 2.3 we have the following result.Theorem 2.2. A continuous function $f$ : $\mathbb{R}\cross \mathbb{R}$ $arrow \mathbb{C}$ satisfies (E.I) for all $x$ , $y$ , $t\in \mathbb{R}$ ifand only if $f$ is given by(2.38) $f(x, y)=a(x^{2}-y^{2}+2ixy)+b(x+iy)+c$for all $x$ , $y\in \mathbb{R}$ , where $a$ , $b$, and $c$ are arbitrary complex constants.Equation (E.I) can also be rewritten in the complex form(2.39) $f(z+t)-f(z-t)=-i$[f$(z+it)-f$ (z-it)]$\theta 4$for all $z\in \mathbb{C}$ and $t\in \mathbb{R}$, where $f(z):=$ f{x $y$ ) for all $x$ , $y\in \mathbb{R}$ and $f$ : $\mathbb{C}arrow$ C. In this case thecontinuous solution (2.38) is given by a complex polynomial of degree at most two such that$f(z)=az^{2}+bz+c$for all $z\in$ C.3. Certain related functional $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}_{\dot{1}}\mathrm{o}\mathrm{n}\mathrm{s}$3.1. Related equations I.In \S 3.1 we mainly consider the functional equation(E.3) $f(x+t, y+t)-f(x-t,y-t)=-i$[f$(x-t,y+t)-f(x+t,y-t)$]for all $x$ , $y$ , $t\in G,$ where $f$ : $G\cross Garrow \mathbb{C}$, and determine the general and regular solution of(E.3).There are various applications of functional equations. One of them is an application offunctional equations to a geometric characterization of complex polynomials ffom the stand-point of conformal-mapping properties. Prom this point of view H. Haruki 1971 [4] obtains thefunctional equation(3.1) $f(z+te^{\pi i/4})-f(z-te^{\pi i/4})=i[f(z+te^{-\pi i/4})-f(z+te^{-\pi i/4})]$for all $z\in \mathbb{C}$ and $t\in \mathbb{R}$ , where $f$ : $\mathbb{C}arrow \mathbb{C}$ , under two geometric properties on $f$ . Equation(3.1) yields the above equation (E.3) for all $x$ , $y$ , $t\in \mathbb{R}$ , where $f(x,y):=f(z)$ for $z=x- l$ $iy$and $f$ : $\mathbb{R}\cross \mathbb{R}arrow$ C. It is obtained in [4] that the continuous solution of equation (3.1) for$f$ : $\mathbb{C}arrow \mathbb{C}$ by applying the regularity of Haruki’s functional equation (2.4). We note that thecontinuity assumption in order to consider equation (3.1) is natural and meaningful with twogeometric properties in [4]. Further, it is possible to obtain the general solution of equation (E.3)and (3.1) for $f$ : $\mathbb{R}\cross \mathbb{R}arrow \mathbb{C}$ and $f$ : $\mathbb{C}arrow \mathbb{C}$ when no regularity assumptions are imposed on$f$ , since it is shown in [4] that (3.1) implies (2.4), and the general solution of (2.4) is obtained in[2, p.50-51, Theorem 5]. However, in this section we first show that equation (E.3) is equivalentto (E.1) when no regularity assumptions are imposed on $f$ so that by Theorem 2.1 in \S 2.1 weimmediately obtain the general solution of equation (E.3) under no regularity assumptions on$f$ . We emphasis that we do not apply the general solution of equation (2.4). We will be able togeneralize our results ffom $\mathbb{R}$ to $G$ .Theorem 3.1. If $f$ : $G\cross Garrow \mathbb{C}$ satisfies equation (E.I) for all $x$ , $y$ , $t\in G,$ then alsoequation (E.3) for all $x,y$ , $t\in G$ and conversely so that (E.I) and (E.3) are equivalent.The following result is an immediate consequence of Theorem 2.1 and Theorem 3.1.Corollary 3.1. A function $f$ : $G\cross Garrow \mathbb{C}$ satisfies equation (E.3) for all $x$ , $y$ , $t\in G$ ifand only if $f$ is given by expression (S.I) in \S 2.1 for all $x$ , $y\in G.$We also readily obtain ffom Theorem 3.1 and Theorem 2.2 in the case of $G=\mathbb{R}$ that acontinuous function $f$ : $\mathbb{R}\cross \mathbb{R}arrow \mathbb{C}$ satisfies (E.3) for all $x$ , $y,t\in \mathbb{R}$ if and only if $f$ is given by(2.38) for al $x,y\in$ R.$\mathrm{E}\mathrm{I}5$H. Haruki proved the following theorem in [4, p.37, Theorem] by applying a regularity offunctional equation (2.4).Theorem 3.2. A continuous function $f$ : $\mathbb{C}arrow \mathbb{C}$ satisfies equation (3.1) for all $z\in \mathbb{C}$ and$t\in \mathbb{R}$ if and only if $f$ is given by a quadratic polynomial of $z$ .Theorem 3.3. If $f$ : $G\cross Garrow \mathbb{C}$ satisfies each one of the following three functionalequations(3.1) $f(x+t,y)-f(x-t, y)=-\mathrm{i}[f(x,y+t)-f(x, y-t)]$(E.I) $f(x+t,y)-f(x,y)=-\mathrm{i}[f(x,y+t)-f(x,y)]$(E.3) $f(x+t,y+t)-f(x-t,y-t)=-i[f(x-t, y+t)-f(x+t,y-t)]$for all $x$ , $y$ , $t\in G,$ then each one of (E. $\mathrm{I}$), (E.2), or (E.3) also satisfies either the Harukifunctional equation(2.4) $f(x+t,y+t)+f(x-t,y+t)+f(x+t, y-t)+f(x-t,y-t)=4f(x,y)$or(E.3) $f(x+t, y)+f(x-t, y)+f(x,y+t)+f(x,y-t)=4f(x,y)$for all $x$ , $y$ , $t\in G.$3.2. Related equations $\mathrm{I}\mathrm{I}$ .If one side of the Cauchy-Riemann equation $\partial f/\partial x=-i\partial ff\partial y$ is replaced by the operators $!_{s,t}f$or $)_{y,t}f$ defined in \S 1, then we have the following two partial difference differential functionalequations(3.7) $\frac{\partial f(x,y)}{\partial x}=-i$. $\lceil\frac{f(x,y+t)-f(x,y-t)}{2t}\rceil$(3.8) $\frac{f(x+t,y)-f(x-t,y)}{2t}=-\cdot$ $\frac{\partial f((x,y)}{y}$ .In \S 3.2 we determine the general solutions of each one of the above two equations.Theorem 3.4. A function $f$ : $\mathbb{R}\cross \mathbb{R}arrow \mathbb{C}$ satisfies equation (3.7) for all $x$ , $y\in \mathbb{R}$ and$t\in \mathbb{R}$ ’ {0} if and only if $f$ is given by(3.9) $\mathrm{f}(\mathrm{x},\mathrm{y}=\frac{1}{2}a$ ($y2-x^{2}-$ 2ixy) $+b(y-ix)+c,$where $a$ , $b$ , and $c$ are arbitrary complex constants.In view of the forms (3.7) and (3.8) the following theorem readily follows from a proof similarto the above proof of Theorem 3.4.Theorem 3.5. A function $f$ : $\mathbb{R}\cross \mathbb{R}arrow \mathbb{C}$ satisfies equation (3.8) for all $x,y\in \mathbb{R}$ and$t\in \mathbb{R}3$ $\{0\}$ if and only if $f$ is given by$\mathrm{f}(\mathrm{x},\mathrm{y}=\frac{1}{2}a$ ($x2-y^{2}+$ 2ixy) $+b(x+iy)+c,$where $a$ , $b$, and $c$ are arbitrary complex constants.36References[1] J. Aczel, Lectures on functional equations and their applications, Academic Press, New Yorkand London, 1966.[2] J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic, General and regular solutionsof functional equations characterizing $ha$ monic polynomials, Aequationes Math. 1 (1968),37-53.[3] J. Aczel and S. Haruki, Partial difference equations analogous to the Cauchy-Riemannequations, Funkcial. Ekvac. 24 (1981), 95-102.[4] H. Haruki, On an application of the square functional equation to a geometric characteri-zation of quadratic functions from the standpoint of confomal-mapping properties, Aequa-tiones Math. 6 (1971), 36-38.[5] S. Haruki, A property of quadratic polynomials, Amer. Math. Monthly 86 (1979), 577-579.[6] S. Haruki, Partial difference equations analogous to the Cauchy-Riemann equations $II$,Funkcial. Ekvac. 29 (1986), 237-241.[7] S. Haruki and C. T. Ng, Partial difference equations analogous to the Cauchy-Riemannequations and related functional equations on rings and fields, Results in Math. 26 (1994),316-323.[8] J. H. B. Kemperman, A general functional equation, Trans. Amer. Math. Soc. 86 (1957),28-56.[9] S. Mazur and W. Orlicz, Grundlegende Eigenschaften der polynomischen Operationen, Stu-dia Mathematica 5 (1934), 50-68.[10] M. A. McKiernan, On vanishing $n$-th ordered differences and Hamel basis, Ann. Polon.Math. 19 (1967), 331-336.[11] M. A. McKiernan, On HarukVs functional equation, Aequationes Math. 1 (1968), 143.[12] M. A. McKiernan, Boundedness on a set of positive measure and the mean-value propertycharacterizes polynomials on a space $V^{n}$ , Aequationes Math. 4 (1970), 31-36.[13] H. Swiatak, A generalization of the Haruki functional equation, Ann. Polon. Math. 22(1970), $370rightarrow 376$ .

Partial difference functional equations arising from the Cauchy-Riemann equations (Mathematical models and dynamics of functional equations)

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/25496/1/1372-14.pdf

Partial difference functional equations arising from the Cauchy-Riemann equations (Mathematical models and dynamics of functional equations)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository