Stability of quadratic functional equations in tempered distributions

Abstract

We reformulate the following quadratic functional equation:

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

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.

1 Introduction

In 1940, Ulam [31] raised a question concerning the stability of group homomorphisms as follows: The case of approximately additive mappings was solved by Hyers [16] under the assumption that $G_2$ is a Banach space. In 1978, Rassias [25] 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 $d(\cdot ,\cdot )$. Given $ϵ>0$, does there exist a $\delta >0$ such that if a function $h:G_1\to G_2$ satisfies the inequality $d(h(xy),h(x)h(y))<\delta$ for all $x,y\in G_1$, then there exists a homomorphism $H:G_1\to G_2$ with $d(h(x),H(x))<ϵ$ 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

$$f(x+y)+f(x-y)=2f(x)+2f(y)$$

(1.1)

was proved by Skof [29]. 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 $f(x)=B(x,x)$ (see [1, 13, 17, 19, 27]). The biadditive function B is given by

$$B(x,y)=\frac{1}{4}(f(x+y)-f(x-y)).$$

Recently, Lee et al. [21] introduced the following quadratic functional equation which is equivalent to (1.1):

$$f(kx+y)+f(kx-y)=2k^{2}f(x)+2f(y),$$

(1.2)

where k is a fixed positive integer. They proved the Hyers-Ulam-Rassias stability of this equation in Banach spaces. Wang [32] considered the intuitionistic fuzzy stability of (1.2) by using the fixed-point alternative. Saadati and Park [26] 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 [8] 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 [7–11, 22], we reformulate (1.2) and the related inequality in the spaces of generalized functions as follows:

(1.3)

(1.4)

where A, B, P, and Q are the functions defined by

$$A(x,y)=kx+y,B(x,y)=kx-y,P(x,y)=x,Q(x,y)=y.$$

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 [15] for pullbacks and to [2, 7–11] 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

$$u=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j+\mu (x),$$

where μ is a bounded measurable function such that

$${\parallel \mu \parallel }_{L^{\mathrm{\infty }}}\le \{\begin{array}{cc}\frac{ϵ}{2},\hfill & k=1,\hfill \\ \frac{(k^2+1)ϵ}{2k^2(k^2-1)},\hfill & k\ge 2.\hfill \end{array}$$

2 Preliminaries

In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the n-dimensional notations. If $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}_0^n$, 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 $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$, we denote $x^{\alpha }=x_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}$ and ${\partial }^{\alpha }={(\partial /\partial x_1)}^{{\alpha }_1}\cdots {(\partial /\partial x_n)}^{{\alpha }_n}$.

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

$${\parallel \phi \parallel }_{\alpha ,\beta }=\underset{x\in {\mathbb{R}}^n}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|<\mathrm{\infty }$$

(2.1)

for all $\alpha ,\beta \in {\mathbb{N}}_0^n$. The vector space of such functions is denoted by $\mathcal{S}({\mathbb{R}}^n)$. A linear functional u on $\mathcal{S}({\mathbb{R}}^n)$ is said to be a tempered distribution if there exists the constant $C\ge 0$ and the nonnegative integer N such that

$$|〈u,\phi 〉|\le C\sum _{|\alpha |,|\beta |\le N}\underset{x\in {\mathbb{R}}^n}{sup}\left|x^{\alpha }{\partial }^{\beta }\phi \right|$$

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

We note that, if $\phi \in \mathcal{S}({\mathbb{R}}^n)$, 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 $\phi (x)=exp(-a{|x|}^2)$, where $a>0$ belongs to $\mathcal{S}({\mathbb{R}}^n)$, but $\psi (x)={(1+{|x|}^2)}^{-1}$ is not a member of $\mathcal{S}({\mathbb{R}}^n)$. It is known from [5] that (2.1) is equivalent to

$$\underset{x\in {\mathbb{R}}^n}{sup}|x^{\alpha }\phi (x)|<\mathrm{\infty },\underset{\xi \in {\mathbb{R}}^n}{sup}|{\xi }^{\beta }\hat{\phi }(\xi )|<\mathrm{\infty }$$

for all $\alpha ,\beta \in {\mathbb{N}}_0^n$, where $\hat{\phi }$ is the Fourier transform of φ.

For example, every polynomial $p(x)={\sum }_{|\alpha |\le m}{a}_{\alpha }{x}^{\alpha }$, where $a_{\alpha }\in \mathbb{C}$, defines a tempered distribution by

$$〈p(x),\phi 〉={\int }_{{\mathbb{R}}^n}p(x)\phi (x)dx,\phi \in \mathcal{S}\left({\mathbb{R}}^n\right).$$

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 ([6])

We denote by $\mathcal{F}({\mathbb{R}}^n)$ the set of all infinitely differentiable functions φ in ${\mathbb{R}}^n$ such that

$${\parallel \phi \parallel }_{A,B}=\underset{x,\alpha ,\beta }{sup}\frac{|x^{\alpha }{\partial }^{\beta }\phi (x)|}{A^{|\alpha |}{B}^{|\beta |}\alpha !\beta !}<\mathrm{\infty }$$

(2.2)

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

It can be verified that the seminorm (2.2) is equivalent to

$${\parallel \phi \parallel }_{h,k}=\underset{x,\alpha }{sup}\frac{|{\partial }^{\alpha }\phi (x)|expk|x|}{h^{|\alpha |}\alpha !}<\mathrm{\infty }$$

for some constants $h,k>0$. Furthermore, it is shown in [6] that (2.2) is equivalent to

$$\underset{x\in {\mathbb{R}}^n}{sup}|\phi (x)|expk|x|<\mathrm{\infty },\underset{\xi \in {\mathbb{R}}^n}{sup}|\hat{\phi }(\xi )|exph|\xi |<\mathrm{\infty }$$

for some $h,k>0$.

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

$$\mathcal{F}\left({\mathbb{R}}^n\right)\hookrightarrow \mathcal{S}\left({\mathbb{R}}^n\right),{\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right)\hookrightarrow {\mathcal{F}}^{\prime }\left({\mathbb{R}}^n\right).$$

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,

$$E_t(x)=\{\begin{array}{cc}{(4\pi t)}^{-n/2}exp(-{|x|}^2/4t),\hfill & x\in {\mathbb{R}}^n,t>0,\hfill \\ 0,\hfill & x\in {\mathbb{R}}^n,t\le 0.\hfill \end{array}$$

Since for each $t>0$, $E_t(\cdot )$ belongs to the space $\mathcal{F}({\mathbb{R}}^n)$, the convolution

$$\tilde{u}(x,t)=(u\ast E_t)(x)=〈u_y,E_t(x-y)〉$$

is well defined for all u in ${\mathcal{F}}^{\prime }({\mathbb{R}}^n)$, which is called the Gauss transform of u. Subsequently, the semigroup property

$$(E_t\ast E_s)(x)=E_{t+s}(x)$$

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 ([23])

Let $u\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n)$. Then its Gauss transform $\tilde{u}$ is a $C^{\mathrm{\infty }}$-solution of the heat equation

$$(\partial /\partial t-\mathrm{\Delta })\tilde{u}(x,t)=0$$

satisfying

• (i) There exist positive constants C, M, and N such that

  $$|\tilde{u}(x,t)|\le C{t}^{-M}{(1+|x|)}^N\mathit{\text{in}}{\mathbb{R}}^n\times (0,\delta ).$$

  (3.1)

• (ii) $\tilde{u}(x,t)\to u$ as $t\to 0^{+}$ in the sense that for every $\phi \in \mathcal{S}({\mathbb{R}}^n)$,

  $$〈u,\phi 〉=\underset{t\to 0^{+}}{lim}\int \tilde{u}(x,t)\phi (x)dx.$$

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

Similarly, we can represent Fourier hyperfunctions as a special case of the results as in [20]. 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

$$|\tilde{u}(x,t)|\le C_{ϵ}exp\left(ϵ(|x|+1/t)\right)\text{in }{\mathbb{R}}^n\times (0,\delta ).$$

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

Lemma 3.2 Suppose that $f:{\mathbb{R}}^n\times (0,\mathrm{\infty })\to \mathbb{C}$ is a continuous function satisfying the equation

$$f(kx+y,k^{2}t+s)+f(kx-y,k^{2}t+s)=2k^{2}f(x,t)+2f(y,s)$$

(3.2)

for all $x,y\in {\mathbb{R}}^n$, $t,s>0$. Then the solution f is the quadratic-additive function

$$f(x,t)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j+bt$$

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

Proof Define a function $h:{\mathbb{R}}^n\times (0,\mathrm{\infty })\to \mathbb{C}$ as $h(x,t):=f(x,t)-f(0,t)$. We immediately have $h(0,t)=0$ and

$$h(kx+y,k^{2}t+s)+h(kx-y,k^{2}t+s)=2k^{2}h(x,t)+2h(y,s)$$

(3.3)

for all $x,y\in {\mathbb{R}}^n$, $t,s>0$. Putting $y=0$ in (3.3) yields

$$h(kx,k^{2}t+s)=k^{2}h(x,t)$$

(3.4)

for all $x\in {\mathbb{R}}^n$, $t,s>0$. Letting $s\to 0^{+}$ in (3.4) gives

$$h(kx,k^{2}t)=k^{2}h(x,t)$$

(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

$$h(x,t+s)=h(x,t)$$

for all $x\in {\mathbb{R}}^n$, $t,s>0$. This shows that $h(x,t)$ is independent with respect to the second variable. Thus, we see that $H(x):=h(x,t)$ 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

$$H(x)=h(x,t)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j,$$

where $a_{ij}\in \mathbb{C}$.

On the other hand, putting $x=y=0$ in (3.2) yields

$$f(0,k^{2}t+s)=k^{2}f(0,t)+f(0,s)$$

(3.6)

for all $t,s>0$. In view of (3.6), we verify that ${lim}_{s\to 0^{+}}f(0,s)=0$ and

$$f(0,k^{2}t)=k^{2}f(0,t)$$

(3.7)

for all $t>0$. It follows from (3.6) and (3.7) that we see that $f(0,t)$ satisfies the Cauchy functional equation

$$f(0,t+s)=f(0,t)+f(0,s)$$

for all $t,s>0$. Given the continuity, we have

$$f(0,t)=bt$$

for some $b\in \mathbb{C}$. Therefore, we finally obtain

$$f(x,t)=h(x,t)+f(0,t)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j+bt$$

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 ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ (or ${\mathcal{F}}^{\prime }({\mathbb{R}}^n)$, 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 $E_t(\xi )E_s(\eta )$ of n-dimensional heat kernels in both sides of (1.3), we have

$$\begin{array}{rcl}[(u\circ A)\ast (E_t(\xi )E_s(\eta ))](x,y)& =& 〈u\circ A,E_t(x-\xi )E_s(y-\eta )〉\\ =& 〈u_{\xi },k^{-n}\int E_t(x-\frac{\xi -\eta }{k})E_s(y-\eta )d\eta 〉\\ =& 〈u_{\xi },k^{-n}\int E_t\left(\frac{kx+y-\xi -\eta }{k}\right)E_s(\eta )d\eta 〉\\ =& 〈u_{\xi },\int E_{k^{2}t}(kx+y-\xi -\eta )E_s(\eta )d\eta 〉\\ =& 〈u_{\xi },(E_{k^{2}t}\ast E_s)(kx+y-\xi )〉\\ =& 〈u_{\xi },E_{k^{2}t+s}(kx+y-\xi )〉\\ =& \tilde{u}(kx+y,k^{2}t+s)\end{array}$$

and similarly we get

Thus, (1.3) is converted into the classical functional equation

$$\tilde{u}(kx+y,k^{2}t+s)+\tilde{u}(kx-y,k^{2}t+s)=2k^2\tilde{u}(x,t)+2\tilde{u}(y,s)$$

for all $x,y\in {\mathbb{R}}^n$, $t,s>0$. We note that the Gauss transform $\tilde{u}$ is a $C^{\mathrm{\infty }}$ function and so, by Lemma 3.2, the solution $\tilde{u}$ is of the form

$$\tilde{u}(x,t)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j+bt$$

(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 ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ (or ${\mathcal{F}}^{\prime }({\mathbb{R}}^n)$, 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

$$T(x)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j$$

such that

$$\parallel u-T(x)\parallel \le \frac{ϵ}{2}.$$

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 $f:{\mathbb{R}}^n\times (0,\mathrm{\infty })\to \mathbb{C}$ is a continuous function satisfying the inequality

$${\parallel f(kx+y,k^{2}t+s)+f(kx-y,k^{2}t+s)-2k^{2}f(x,t)-2f(y,s)\parallel }_{L^{\mathrm{\infty }}}\le ϵ.$$

(4.1)

Then there exist a unique function $g(x,t)$ satisfying the quadratic-additive functional equation

$$g(kx+y,k^{2}t+s)+g(kx-y,k^{2}t+s)=2k^{2}g(x,t)+2g(y,s)$$

such that

$${\parallel f(x,t)-g(x,t)\parallel }_{L^{\mathrm{\infty }}}\le \frac{k^2+1}{2k^2(k^2-1)}ϵ.$$

Proof Putting $x=y=0$ in (4.1) yields

$$|f(0,k^{2}t+s)-k^{2}f(0,t)-f(0,s)|\le \frac{ϵ}{2}$$

(4.2)

for all $t,s>0$. In view of (4.2), we see that

$$c:=\underset{t\to 0^{+}}{lim\hspace{0.17em}sup}f(0,t)$$

exists. Letting $t=t_n\to 0^{+}$ so that $f(0,t_n)\to c$ in (4.2) gives

$$|c|\le \frac{ϵ}{2k^2}.$$

(4.3)

Putting $y=0$ and letting $s=s_n\to 0^{+}$ so that $f(0,s_n)\to c$ in (4.1) we have

$$|f(kx,k^{2}t)-k^{2}f(x,t)-c|\le \frac{ϵ}{2}$$

(4.4)

for all $x\in {\mathbb{R}}^n$, $t>0$. Using (4.3), we can rewrite (4.4) as

$$|\frac{f(kx,k^{2}t)}{k^2}-f(x,t)|\le \frac{k^2+1}{2k^4}ϵ$$

for all $x\in {\mathbb{R}}^n$, $t>0$. By the induction argument yields

$$|\frac{f(k^{n}x,k^{2n}t)}{k^{2n}}-f(x,t)|\le \frac{k^2+1}{2k^2(k^2-1)}ϵ$$

(4.5)

for all $n\in \mathbb{N}$, $x\in {\mathbb{R}}^n$, $t>0$. We claim that the sequence $\{k^{-2n}f(k^{n}x,k^{2n}t)\}$ converges. Replacing x by $k^{m}x$ and t by $k^{2m}t$ in (4.5), respectively, where $m\ge n$, we get

$$|\frac{f(k^{m+n}x,k^{2(m+n)}t)}{k^{2(m+n)}}-\frac{f(k^{m}x,k^{2m}t)}{k^{2m}}|\le \frac{k^2+1}{2k^{2(m+1)}(k^2-1)}ϵ.$$

Letting $n\to \mathrm{\infty }$, by Cauchy convergence criterion, we see that the sequence $\{k^{-2n}f(k^{n}x,k^{2n}t)\}$ is a Cauchy sequence. We can now define a function $h:{\mathbb{R}}^n\times (0,\mathrm{\infty })\to \mathbb{C}$ by

$$g(x,t):=\underset{n\to \mathrm{\infty }}{lim}\frac{f(k^{n}x,k^{2n}t)}{k^{2n}}.$$

Letting $n\to \mathrm{\infty }$ in (4.5) we obtain

$${\parallel f(x,t)-g(x,t)\parallel }_{L^{\mathrm{\infty }}}\le \frac{k^2+1}{2k^2(k^2-1)}ϵ.$$

(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

$$g(kx+y,k^{2}t+s)+g(kx-y,k^{2}t+s)=2k^{2}g(x,t)+2g(y,s)$$

(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 $h:{\mathbb{R}}^n\times (0,\mathrm{\infty })\to \mathbb{C}$ such that h satisfies (4.6) and (4.7). Since g and h satisfy (4.7), we see from Lemma 3.2 that

$$g(k^{n}x,k^{2n}t)=k^{2n}g(x,t),h(k^{n}x,k^{2n}t)=k^{2n}h(x,t)$$

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 $g(x,t)=h(x,t)$ 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 ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ (or ${\mathcal{F}}^{\prime }({\mathbb{R}}^n)$, resp.) satisfies the inequality (1.4). Then there exists a unique quadratic form

$$T(x)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j$$

such that

$$\parallel u-T(x)\parallel \le \{\begin{array}{cc}\frac{ϵ}{2},\hfill & k=1,\hfill \\ \frac{(k^2+1)ϵ}{2k^2(k^2-1)},\hfill & k\ge 2.\hfill \end{array}$$

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 $E_t(\xi )E_s(\eta )$ of n-dimensional heat kernels in both sides of (1.4), we have

$${\parallel \tilde{u}(kx+y,k^{2}t+s)+\tilde{u}(kx-y,k^{2}t+s)-2k^2\tilde{u}(x,t)-2\tilde{u}(y,s)\parallel }_{L^{\mathrm{\infty }}}\le ϵ.$$

By Lemma 4.2, there exists a unique function $g(x,t)$ satisfying the quadratic-additive functional equation

$$g(kx+y,k^{2}t+s)+g(kx-y,k^{2}t+s)=2k^{2}g(x,t)+2g(y,s)$$

such that

$${\parallel \tilde{u}(x,t)-g(x,t)\parallel }_{L^{\mathrm{\infty }}}\le \frac{k^2+1}{2k^2(k^2-1)}ϵ.$$

(4.8)

It follows from Lemma 3.2 that $g(x,t)$ is of the form

$$g(x,t)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j+bt$$

for some $a_{ij},b\in \mathbb{C}$. Letting $t\to 0^{+}$ in (4.8), we have

$$\parallel u-\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j\parallel \le \frac{k^2+1}{2k^2(k^2-1)}ϵ.$$

This completes the proof. □

Remark 4.4 The resulting inequality in Theorem 4.3 implies that $u-T(x)$ is a measurable function. Thus, all of the solution u in ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ (or ${\mathcal{F}}^{\prime }({\mathbb{R}}^n)$, resp.) can be written uniquely in the form

$$u=T(x)+\mu (x),$$

where

$${\parallel \mu \parallel }_{L^{\mathrm{\infty }}}\le \{\begin{array}{cc}\frac{ϵ}{2},\hfill & k=1,\hfill \\ \frac{(k^2+1)ϵ}{2k^2(k^2-1)},\hfill & k\ge 2.\hfill \end{array}$$