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# Stability of quadratic functional equations in tempered distributions

Journal of Inequalities and Applications20122012:177

https://doi.org/10.1186/1029-242X-2012-177

• Received: 12 September 2011
• Accepted: 6 August 2012
• Published:

## Abstract

We reformulate the following quadratic functional equation:

$f\left(kx+y\right)+f\left(kx-y\right)=2{k}^{2}f\left(x\right)+2f\left(y\right)$

as the equation for generalized functions. Using the fundamental solution of the heat equation, we solve the general solution of this equation and prove the Hyers-Ulam stability in the spaces of tempered distributions and Fourier hyperfunctions.

## Keywords

• quadratic functional equation
• stability
• tempered distribution
• heat kernel
• Gauss transform

## 1 Introduction

In 1940, Ulam  raised a question concerning the stability of group homomorphisms as follows: The case of approximately additive mappings was solved by Hyers  under the assumption that ${G}_{2}$ is a Banach space. In 1978, Rassias  generalized Hyers’ result to the unbounded Cauchy difference.

Let ${G}_{1}$ be a group and let ${G}_{2}$ be a metric group with the metric $d\left(\cdot ,\cdot \right)$. Given $ϵ>0$, does there exist a $\delta >0$ such that if a function $h:{G}_{1}\to {G}_{2}$ satisfies the inequality $d\left(h\left(xy\right),h\left(x\right)h\left(y\right)\right)<\delta$ for all $x,y\in {G}_{1}$, then there exists a homomorphism $H:{G}_{1}\to {G}_{2}$ with $d\left(h\left(x\right),H\left(x\right)\right)<ϵ$ for all $x\in {G}_{1}$?

During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [13, 14, 17, 19, 24, 27, 30]). In particular, the stability problem of the following quadratic functional equation
$f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$
(1.1)
was proved by Skof . Thereafter, many authors studied the stability problems of (1.1) in various settings (see [3, 4, 12, 18]). Usually, quadratic functional equations are used to characterize the inner product spaces. Note that a square norm on an inner product space satisfies the parallelogram equality
${\parallel x+y\parallel }^{2}+{\parallel x-y\parallel }^{2}=2{\parallel x\parallel }^{2}+2{\parallel y\parallel }^{2}$
for all vectors x, y. By virtue of this equality, the quadratic functional equation (1.1) is induced. It is well known that a function f between real vector spaces satisfies (1.1) if and only if there exists a unique symmetric biadditive function B such that $f\left(x\right)=B\left(x,x\right)$ (see [1, 13, 17, 19, 27]). The biadditive function B is given by
$B\left(x,y\right)=\frac{1}{4}\left(f\left(x+y\right)-f\left(x-y\right)\right).$
Recently, Lee et al.  introduced the following quadratic functional equation which is equivalent to (1.1):
$f\left(kx+y\right)+f\left(kx-y\right)=2{k}^{2}f\left(x\right)+2f\left(y\right),$
(1.2)

where k is a fixed positive integer. They proved the Hyers-Ulam-Rassias stability of this equation in Banach spaces. Wang  considered the intuitionistic fuzzy stability of (1.2) by using the fixed-point alternative. Saadati and Park  proved the Hyers-Ulam-Rassias stability of (1.2) in non-Archimedean $\mathcal{L}$-fuzzy normed spaces.

In this paper, we solve the general solution and the stability problem of (1.2) in the spaces of generalized functions such as ${\mathcal{S}}^{\prime }$ of tempered distributions and ${\mathcal{F}}^{\prime }$ of Fourier hyperfunctions. Using pullbacks, Chung and Lee  reformulated (1.1) as the equation for generalized functions and proved that every solution of (1.1) in ${\mathcal{S}}^{\prime }$ (or ${\mathcal{F}}^{\prime }$, resp.) is a quadratic form. Also, Chung [7, 11] proved the stability problem of (1.1) in the spaces ${\mathcal{S}}^{\prime }$ and ${\mathcal{F}}^{\prime }$. Making use of the similar methods as in [711, 22], we reformulate (1.2) and the related inequality in the spaces of generalized functions as follows:
where A, B, P, and Q are the functions defined by
$A\left(x,y\right)=kx+y,\phantom{\rule{2em}{0ex}}B\left(x,y\right)=kx-y,\phantom{\rule{2em}{0ex}}P\left(x,y\right)=x,\phantom{\rule{2em}{0ex}}Q\left(x,y\right)=y.$

Here, denotes the pullback of generalized functions and the inequality $\parallel v\parallel \le ϵ$ in (1.4) means that $|〈v,\phi 〉|\le ϵ{\parallel \phi \parallel }_{{L}^{1}}$ for all test functions φ. We refer to  for pullbacks and to [2, 711] for more details of the spaces of generalized functions.

As results, we shall prove that every solution u in ${\mathcal{S}}^{\prime }$ (or ${\mathcal{F}}^{\prime }$, resp.) of Eq. (1.3) is a quadratic form
$u=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j},$
where ${a}_{ij}\in \mathbb{C}$. Also, we shall prove that every solution u in ${\mathcal{S}}^{\prime }$ (or ${\mathcal{F}}^{\prime }$, resp.) of the inequality (1.4) can be written uniquely in the form
$u=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}+\mu \left(x\right),$
where μ is a bounded measurable function such that
${\parallel \mu \parallel }_{{L}^{\mathrm{\infty }}}\le \left\{\begin{array}{cc}\frac{ϵ}{2},\hfill & k=1,\hfill \\ \frac{\left({k}^{2}+1\right)ϵ}{2{k}^{2}\left({k}^{2}-1\right)},\hfill & k\ge 2.\hfill \end{array}$

## 2 Preliminaries

In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the n-dimensional notations. If $\alpha =\left({\alpha }_{1},\dots ,{\alpha }_{n}\right)\in {\mathbb{N}}_{0}^{n}$, where ${\mathbb{N}}_{0}^{n}$ is the set of nonnegative integers, then $|\alpha |={\alpha }_{1}+\cdots +{\alpha }_{n}$, $\alpha !={\alpha }_{1}!\cdots {\alpha }_{n}!$. For $x=\left({x}_{1},\dots ,{x}_{n}\right)\in {\mathbb{R}}^{n}$, we denote ${x}^{\alpha }={x}_{1}^{{\alpha }_{1}}\cdots {x}_{n}^{{\alpha }_{n}}$ and ${\partial }^{\alpha }={\left(\partial /\partial {x}_{1}\right)}^{{\alpha }_{1}}\cdots {\left(\partial /\partial {x}_{n}\right)}^{{\alpha }_{n}}$.

### 2.1 Tempered distributions

We present a very useful space of test functions for the tempered distributions as follows.

Definition 2.1 ([15, 28])

An infinitely differentiable function φ in ${\mathbb{R}}^{n}$ is called rapidly decreasing if
${\parallel \phi \parallel }_{\alpha ,\beta }=\underset{x\in {\mathbb{R}}^{n}}{sup}|{x}^{\alpha }{\partial }^{\beta }\phi \left(x\right)|<\mathrm{\infty }$
(2.1)
for all $\alpha ,\beta \in {\mathbb{N}}_{0}^{n}$. The vector space of such functions is denoted by $\mathcal{S}\left({\mathbb{R}}^{n}\right)$. A linear functional u on $\mathcal{S}\left({\mathbb{R}}^{n}\right)$ is said to be a tempered distribution if there exists the constant $C\ge 0$ and the nonnegative integer N such that
$|〈u,\phi 〉|\le C\sum _{|\alpha |,|\beta |\le N}\underset{x\in {\mathbb{R}}^{n}}{sup}|{x}^{\alpha }{\partial }^{\beta }\phi |$

for all $\phi \in \mathcal{S}\left({\mathbb{R}}^{n}\right)$. The set of all tempered distributions is denoted by ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$.

We note that, if $\phi \in \mathcal{S}\left({\mathbb{R}}^{n}\right)$, then each derivative of φ decreases faster than ${|x|}^{-N}$ for all $N>0$ as $|x|\to \mathrm{\infty }$. It is easy to see that the function $\phi \left(x\right)=exp\left(-a{|x|}^{2}\right)$, where $a>0$ belongs to $\mathcal{S}\left({\mathbb{R}}^{n}\right)$, but $\psi \left(x\right)={\left(1+{|x|}^{2}\right)}^{-1}$ is not a member of $\mathcal{S}\left({\mathbb{R}}^{n}\right)$. It is known from  that (2.1) is equivalent to
$\underset{x\in {\mathbb{R}}^{n}}{sup}|{x}^{\alpha }\phi \left(x\right)|<\mathrm{\infty },\phantom{\rule{2em}{0ex}}\underset{\xi \in {\mathbb{R}}^{n}}{sup}|{\xi }^{\beta }\stackrel{ˆ}{\phi }\left(\xi \right)|<\mathrm{\infty }$

for all $\alpha ,\beta \in {\mathbb{N}}_{0}^{n}$, where $\stackrel{ˆ}{\phi }$ is the Fourier transform of φ.

For example, every polynomial $p\left(x\right)={\sum }_{|\alpha |\le m}{a}_{\alpha }{x}^{\alpha }$, where ${a}_{\alpha }\in \mathbb{C}$, defines a tempered distribution by
$〈p\left(x\right),\phi 〉={\int }_{{\mathbb{R}}^{n}}p\left(x\right)\phi \left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}\phi \in \mathcal{S}\left({\mathbb{R}}^{n}\right).$

Note that tempered distributions are generalizations of ${L}^{p}$-functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform.

### 2.2 Fourier hyperfunctions

Imposing the growth condition on ${\parallel \cdot \parallel }_{\alpha ,\beta }$ in (2.1) Sato and Kawai introduced the new space of test functions for the Fourier hyperfunctions as follows.

Definition 2.2 ()

We denote by $\mathcal{F}\left({\mathbb{R}}^{n}\right)$ the set of all infinitely differentiable functions φ in ${\mathbb{R}}^{n}$ such that
${\parallel \phi \parallel }_{A,B}=\underset{x,\alpha ,\beta }{sup}\frac{|{x}^{\alpha }{\partial }^{\beta }\phi \left(x\right)|}{{A}^{|\alpha |}{B}^{|\beta |}\alpha !\beta !}<\mathrm{\infty }$
(2.2)

for some positive constants A, B depending only on φ. The strong dual of $\mathcal{F}\left({\mathbb{R}}^{n}\right)$, denoted by ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$, is called the Fourier hyperfunction.

It can be verified that the seminorm (2.2) is equivalent to
${\parallel \phi \parallel }_{h,k}=\underset{x,\alpha }{sup}\frac{|{\partial }^{\alpha }\phi \left(x\right)|expk|x|}{{h}^{|\alpha |}\alpha !}<\mathrm{\infty }$
for some constants $h,k>0$. Furthermore, it is shown in  that (2.2) is equivalent to
$\underset{x\in {\mathbb{R}}^{n}}{sup}|\phi \left(x\right)|expk|x|<\mathrm{\infty },\phantom{\rule{2em}{0ex}}\underset{\xi \in {\mathbb{R}}^{n}}{sup}|\stackrel{ˆ}{\phi }\left(\xi \right)|exph|\xi |<\mathrm{\infty }$

for some $h,k>0$.

Fourier hyperfunctions were introduced by Sato in 1958. The space ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$ is a natural generalization of the space ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$ and can be thought informally as distributions of a infinite order. Observing the above growth conditions, we can easily see the following topological inclusions:
$\mathcal{F}\left({\mathbb{R}}^{n}\right)↪\mathcal{S}\left({\mathbb{R}}^{n}\right),\phantom{\rule{2em}{0ex}}{\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)↪{\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right).$

## 3 General solution in generalized functions

In order to solve the general solution of (1.3), we employ the n-dimensional heat kernel, fundamental solution of the heat equation,
${E}_{t}\left(x\right)=\left\{\begin{array}{cc}{\left(4\pi t\right)}^{-n/2}exp\left(-{|x|}^{2}/4t\right),\hfill & x\in {\mathbb{R}}^{n},t>0,\hfill \\ 0,\hfill & x\in {\mathbb{R}}^{n},t\le 0.\hfill \end{array}$
Since for each $t>0$, ${E}_{t}\left(\cdot \right)$ belongs to the space $\mathcal{F}\left({\mathbb{R}}^{n}\right)$, the convolution
$\stackrel{˜}{u}\left(x,t\right)=\left(u\ast {E}_{t}\right)\left(x\right)=〈{u}_{y},{E}_{t}\left(x-y\right)〉$
is well defined for all u in ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$, which is called the Gauss transform of u. Subsequently, the semigroup property
$\left({E}_{t}\ast {E}_{s}\right)\left(x\right)={E}_{t+s}\left(x\right)$

of the heat kernel is very useful to convert Eq. (1.3) into the classical functional equation defined on upper-half plane. We also use the following famous result, the so-called heat kernel method, which is stated as follows.

Theorem 3.1 ()

Let $u\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$. Then its Gauss transform $\stackrel{˜}{u}$ is a ${C}^{\mathrm{\infty }}$-solution of the heat equation
$\left(\partial /\partial t-\mathrm{\Delta }\right)\stackrel{˜}{u}\left(x,t\right)=0$
satisfying
1. (i)
There exist positive constants C, M, and N such that
$|\stackrel{˜}{u}\left(x,t\right)|\le C{t}^{-M}{\left(1+|x|\right)}^{N}\phantom{\rule{1em}{0ex}}\mathit{\text{in}}\phantom{\rule{1em}{0ex}}{\mathbb{R}}^{n}×\left(0,\delta \right).$
(3.1)

2. (ii)
$\stackrel{˜}{u}\left(x,t\right)\to u$ as $t\to {0}^{+}$ in the sense that for every $\phi \in \mathcal{S}\left({\mathbb{R}}^{n}\right)$,
$〈u,\phi 〉=\underset{t\to {0}^{+}}{lim}\int \stackrel{˜}{u}\left(x,t\right)\phi \left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$

Conversely, every ${C}^{\mathrm{\infty }}$-solution $U\left(x,t\right)$ of the heat equation satisfying the growth condition (3.1) can be uniquely expressed as $U\left(x,t\right)=\stackrel{˜}{u}\left(x,t\right)$ for some $u\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$.

Similarly, we can represent Fourier hyperfunctions as a special case of the results as in . In this case, the estimate (3.1) is replaced by the following:

For every $ϵ>0$, there exists a positive constant ${C}_{ϵ}$ such that

Here, we need the following lemma to solve the general solution of (1.3).

Lemma 3.2 Suppose that $f:{\mathbb{R}}^{n}×\left(0,\mathrm{\infty }\right)\to \mathbb{C}$ is a continuous function satisfying the equation
$f\left(kx+y,{k}^{2}t+s\right)+f\left(kx-y,{k}^{2}t+s\right)=2{k}^{2}f\left(x,t\right)+2f\left(y,s\right)$
(3.2)
for all $x,y\in {\mathbb{R}}^{n}$, $t,s>0$. Then the solution f is the quadratic-additive function
$f\left(x,t\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}+bt$

for some ${a}_{ij},b\in \mathbb{C}$.

Proof Define a function $h:{\mathbb{R}}^{n}×\left(0,\mathrm{\infty }\right)\to \mathbb{C}$ as $h\left(x,t\right):=f\left(x,t\right)-f\left(0,t\right)$. We immediately have $h\left(0,t\right)=0$ and
$h\left(kx+y,{k}^{2}t+s\right)+h\left(kx-y,{k}^{2}t+s\right)=2{k}^{2}h\left(x,t\right)+2h\left(y,s\right)$
(3.3)
for all $x,y\in {\mathbb{R}}^{n}$, $t,s>0$. Putting $y=0$ in (3.3) yields
$h\left(kx,{k}^{2}t+s\right)={k}^{2}h\left(x,t\right)$
(3.4)
for all $x\in {\mathbb{R}}^{n}$, $t,s>0$. Letting $s\to {0}^{+}$ in (3.4) gives
$h\left(kx,{k}^{2}t\right)={k}^{2}h\left(x,t\right)$
(3.5)
for all $x\in {\mathbb{R}}^{n}$, $t>0$. Replacing s by ${k}^{2}s$ in (3.4) and then using (3.5), we obtain
$h\left(x,t+s\right)=h\left(x,t\right)$
for all $x\in {\mathbb{R}}^{n}$, $t,s>0$. This shows that $h\left(x,t\right)$ is independent with respect to the second variable. Thus, we see that $H\left(x\right):=h\left(x,t\right)$ satisfies (1.2). Using the induction argument on the dimension n, we verify that every continuous solution of (1.2) in ${\mathbb{R}}^{n}$ is a quadratic form
$H\left(x\right)=h\left(x,t\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j},$

where ${a}_{ij}\in \mathbb{C}$.

On the other hand, putting $x=y=0$ in (3.2) yields
$f\left(0,{k}^{2}t+s\right)={k}^{2}f\left(0,t\right)+f\left(0,s\right)$
(3.6)
for all $t,s>0$. In view of (3.6), we verify that ${lim}_{s\to {0}^{+}}f\left(0,s\right)=0$ and
$f\left(0,{k}^{2}t\right)={k}^{2}f\left(0,t\right)$
(3.7)
for all $t>0$. It follows from (3.6) and (3.7) that we see that $f\left(0,t\right)$ satisfies the Cauchy functional equation
$f\left(0,t+s\right)=f\left(0,t\right)+f\left(0,s\right)$
for all $t,s>0$. Given the continuity, we have
$f\left(0,t\right)=bt$
for some $b\in \mathbb{C}$. Therefore, we finally obtain
$f\left(x,t\right)=h\left(x,t\right)+f\left(0,t\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}+bt$

for all $x\in {\mathbb{R}}^{n}$, $t>0$. □

As a direct consequence of the above lemma, we present the general solution of the quadratic functional equation (1.3) in the spaces of generalized functions.

Theorem 3.3 Every solution u in ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$ (or ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$, resp.) of Eq. (1.3) is the quadratic form
$u=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}$

for some ${a}_{ij}\in \mathbb{C}$.

Proof Convolving the tensor product ${E}_{t}\left(\xi \right){E}_{s}\left(\eta \right)$ of n-dimensional heat kernels in both sides of (1.3), we have
$\begin{array}{rcl}\left[\left(u\circ A\right)\ast \left({E}_{t}\left(\xi \right){E}_{s}\left(\eta \right)\right)\right]\left(x,y\right)& =& 〈u\circ A,{E}_{t}\left(x-\xi \right){E}_{s}\left(y-\eta \right)〉\\ =& 〈{u}_{\xi },{k}^{-n}\int {E}_{t}\left(x-\frac{\xi -\eta }{k}\right){E}_{s}\left(y-\eta \right)\phantom{\rule{0.2em}{0ex}}d\eta 〉\\ =& 〈{u}_{\xi },{k}^{-n}\int {E}_{t}\left(\frac{kx+y-\xi -\eta }{k}\right){E}_{s}\left(\eta \right)\phantom{\rule{0.2em}{0ex}}d\eta 〉\\ =& 〈{u}_{\xi },\int {E}_{{k}^{2}t}\left(kx+y-\xi -\eta \right){E}_{s}\left(\eta \right)\phantom{\rule{0.2em}{0ex}}d\eta 〉\\ =& 〈{u}_{\xi },\left({E}_{{k}^{2}t}\ast {E}_{s}\right)\left(kx+y-\xi \right)〉\\ =& 〈{u}_{\xi },{E}_{{k}^{2}t+s}\left(kx+y-\xi \right)〉\\ =& \stackrel{˜}{u}\left(kx+y,{k}^{2}t+s\right)\end{array}$
Thus, (1.3) is converted into the classical functional equation
$\stackrel{˜}{u}\left(kx+y,{k}^{2}t+s\right)+\stackrel{˜}{u}\left(kx-y,{k}^{2}t+s\right)=2{k}^{2}\stackrel{˜}{u}\left(x,t\right)+2\stackrel{˜}{u}\left(y,s\right)$
for all $x,y\in {\mathbb{R}}^{n}$, $t,s>0$. We note that the Gauss transform $\stackrel{˜}{u}$ is a ${C}^{\mathrm{\infty }}$ function and so, by Lemma 3.2, the solution $\stackrel{˜}{u}$ is of the form
$\stackrel{˜}{u}\left(x,t\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}+bt$
(3.8)
for some ${a}_{ij},b\in \mathbb{C}$. By the heat kernel method, we obtain
$u=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}$

as $t\to {0}^{+}$ in (3.8). □

## 4 Stability in generalized functions

In this section, we are going to solve the stability problem of (1.4). For the case of $k=1$ in (1.4), the result is known as follows.

Theorem 4.1 ([7, 10])

Suppose that u in ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$ (or ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$, resp.) satisfies the inequality
$\parallel u\circ A+u\circ B-2u\circ P-2u\circ Q\parallel \le ϵ.$
Then there exists a unique quadratic form
$T\left(x\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}$
such that
$\parallel u-T\left(x\right)\parallel \le \frac{ϵ}{2}.$

We here need the following lemma to solve the stability problem of (1.4).

Lemma 4.2 Let k be a fixed positive integer with $k\ge 2$. Suppose that $f:{\mathbb{R}}^{n}×\left(0,\mathrm{\infty }\right)\to \mathbb{C}$ is a continuous function satisfying the inequality
${\parallel f\left(kx+y,{k}^{2}t+s\right)+f\left(kx-y,{k}^{2}t+s\right)-2{k}^{2}f\left(x,t\right)-2f\left(y,s\right)\parallel }_{{L}^{\mathrm{\infty }}}\le ϵ.$
(4.1)
Then there exist a unique function $g\left(x,t\right)$ satisfying the quadratic-additive functional equation
$g\left(kx+y,{k}^{2}t+s\right)+g\left(kx-y,{k}^{2}t+s\right)=2{k}^{2}g\left(x,t\right)+2g\left(y,s\right)$
such that
${\parallel f\left(x,t\right)-g\left(x,t\right)\parallel }_{{L}^{\mathrm{\infty }}}\le \frac{{k}^{2}+1}{2{k}^{2}\left({k}^{2}-1\right)}ϵ.$
Proof Putting $x=y=0$ in (4.1) yields
$|f\left(0,{k}^{2}t+s\right)-{k}^{2}f\left(0,t\right)-f\left(0,s\right)|\le \frac{ϵ}{2}$
(4.2)
for all $t,s>0$. In view of (4.2), we see that
$c:=\underset{t\to {0}^{+}}{lim sup}f\left(0,t\right)$
exists. Letting $t={t}_{n}\to {0}^{+}$ so that $f\left(0,{t}_{n}\right)\to c$ in (4.2) gives
$|c|\le \frac{ϵ}{2{k}^{2}}.$
(4.3)
Putting $y=0$ and letting $s={s}_{n}\to {0}^{+}$ so that $f\left(0,{s}_{n}\right)\to c$ in (4.1) we have
$|f\left(kx,{k}^{2}t\right)-{k}^{2}f\left(x,t\right)-c|\le \frac{ϵ}{2}$
(4.4)
for all $x\in {\mathbb{R}}^{n}$, $t>0$. Using (4.3), we can rewrite (4.4) as
$|\frac{f\left(kx,{k}^{2}t\right)}{{k}^{2}}-f\left(x,t\right)|\le \frac{{k}^{2}+1}{2{k}^{4}}ϵ$
for all $x\in {\mathbb{R}}^{n}$, $t>0$. By the induction argument yields
$|\frac{f\left({k}^{n}x,{k}^{2n}t\right)}{{k}^{2n}}-f\left(x,t\right)|\le \frac{{k}^{2}+1}{2{k}^{2}\left({k}^{2}-1\right)}ϵ$
(4.5)
for all $n\in \mathbb{N}$, $x\in {\mathbb{R}}^{n}$, $t>0$. We claim that the sequence $\left\{{k}^{-2n}f\left({k}^{n}x,{k}^{2n}t\right)\right\}$ converges. Replacing x by ${k}^{m}x$ and t by ${k}^{2m}t$ in (4.5), respectively, where $m\ge n$, we get
$|\frac{f\left({k}^{m+n}x,{k}^{2\left(m+n\right)}t\right)}{{k}^{2\left(m+n\right)}}-\frac{f\left({k}^{m}x,{k}^{2m}t\right)}{{k}^{2m}}|\le \frac{{k}^{2}+1}{2{k}^{2\left(m+1\right)}\left({k}^{2}-1\right)}ϵ.$
Letting $n\to \mathrm{\infty }$, by Cauchy convergence criterion, we see that the sequence $\left\{{k}^{-2n}f\left({k}^{n}x,{k}^{2n}t\right)\right\}$ is a Cauchy sequence. We can now define a function $h:{\mathbb{R}}^{n}×\left(0,\mathrm{\infty }\right)\to \mathbb{C}$ by
$g\left(x,t\right):=\underset{n\to \mathrm{\infty }}{lim}\frac{f\left({k}^{n}x,{k}^{2n}t\right)}{{k}^{2n}}.$
Letting $n\to \mathrm{\infty }$ in (4.5) we obtain
${\parallel f\left(x,t\right)-g\left(x,t\right)\parallel }_{{L}^{\mathrm{\infty }}}\le \frac{{k}^{2}+1}{2{k}^{2}\left({k}^{2}-1\right)}ϵ.$
(4.6)
Replacing x, y, t, s by ${k}^{n}x$, ${k}^{n}y$, ${k}^{2n}t$, ${k}^{2n}s$ in (4.1), dividing both sides by ${k}^{2n}$ and letting $n\to \mathrm{\infty }$ we have
$g\left(kx+y,{k}^{2}t+s\right)+g\left(kx-y,{k}^{2}t+s\right)=2{k}^{2}g\left(x,t\right)+2g\left(y,s\right)$
(4.7)
for all $x,y\in {\mathbb{R}}^{n}$, $t,s>0$. Next, we shall prove that g is unique. Suppose that there exists another function $h:{\mathbb{R}}^{n}×\left(0,\mathrm{\infty }\right)\to \mathbb{C}$ such that h satisfies (4.6) and (4.7). Since g and h satisfy (4.7), we see from Lemma 3.2 that
$g\left({k}^{n}x,{k}^{2n}t\right)={k}^{2n}g\left(x,t\right),\phantom{\rule{2em}{0ex}}h\left({k}^{n}x,{k}^{2n}t\right)={k}^{2n}h\left(x,t\right)$
for all $n\in \mathbb{N}$, $x\in {\mathbb{R}}^{n}$, $t>0$. One gets from (4.6) that

for all $n\in \mathbb{N}$, $x\in {\mathbb{R}}^{n}$, $t>0$. Taking the limit as $n\to \mathrm{\infty }$, we conclude that $g\left(x,t\right)=h\left(x,t\right)$ for all $x\in {\mathbb{R}}^{n}$, $t>0$. □

We now state and prove the main theorem of this paper.

Theorem 4.3 Suppose that u in ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$ (or ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$, resp.) satisfies the inequality (1.4). Then there exists a unique quadratic form
$T\left(x\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}$
such that
$\parallel u-T\left(x\right)\parallel \le \left\{\begin{array}{cc}\frac{ϵ}{2},\hfill & k=1,\hfill \\ \frac{\left({k}^{2}+1\right)ϵ}{2{k}^{2}\left({k}^{2}-1\right)},\hfill & k\ge 2.\hfill \end{array}$
Proof As discussed above, it is done for the case of $k=1$. We assume that k is a fixed-positive integer with $k\ge 2$. Convolving the tensor product ${E}_{t}\left(\xi \right){E}_{s}\left(\eta \right)$ of n-dimensional heat kernels in both sides of (1.4), we have
${\parallel \stackrel{˜}{u}\left(kx+y,{k}^{2}t+s\right)+\stackrel{˜}{u}\left(kx-y,{k}^{2}t+s\right)-2{k}^{2}\stackrel{˜}{u}\left(x,t\right)-2\stackrel{˜}{u}\left(y,s\right)\parallel }_{{L}^{\mathrm{\infty }}}\le ϵ.$
By Lemma 4.2, there exists a unique function $g\left(x,t\right)$ satisfying the quadratic-additive functional equation
$g\left(kx+y,{k}^{2}t+s\right)+g\left(kx-y,{k}^{2}t+s\right)=2{k}^{2}g\left(x,t\right)+2g\left(y,s\right)$
such that
${\parallel \stackrel{˜}{u}\left(x,t\right)-g\left(x,t\right)\parallel }_{{L}^{\mathrm{\infty }}}\le \frac{{k}^{2}+1}{2{k}^{2}\left({k}^{2}-1\right)}ϵ.$
(4.8)
It follows from Lemma 3.2 that $g\left(x,t\right)$ is of the form
$g\left(x,t\right)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}+bt$
for some ${a}_{ij},b\in \mathbb{C}$. Letting $t\to {0}^{+}$ in (4.8), we have
$\parallel u-\sum _{1\le i\le j\le n}{a}_{ij}{x}_{i}{x}_{j}\parallel \le \frac{{k}^{2}+1}{2{k}^{2}\left({k}^{2}-1\right)}ϵ.$

This completes the proof. □

Remark 4.4 The resulting inequality in Theorem 4.3 implies that $u-T\left(x\right)$ is a measurable function. Thus, all of the solution u in ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^{n}\right)$ (or ${\mathcal{F}}^{\prime }\left({\mathbb{R}}^{n}\right)$, resp.) can be written uniquely in the form
$u=T\left(x\right)+\mu \left(x\right),$
where
${\parallel \mu \parallel }_{{L}^{\mathrm{\infty }}}\le \left\{\begin{array}{cc}\frac{ϵ}{2},\hfill & k=1,\hfill \\ \frac{\left({k}^{2}+1\right)ϵ}{2{k}^{2}\left({k}^{2}-1\right)},\hfill & k\ge 2.\hfill \end{array}$

## Authors’ Affiliations

(1)
Department of Mathematics, Sogang University, Seoul, 121-741, Republic of Korea

## References 