# Stability of Trigonometric Functional Equations in Generalized Functions

- Jeongwook Chang
^{1}Email author and - Jaeyoung Chung
^{2}

**2010**:801502

https://doi.org/10.1155/2010/801502

© J. Chang and J. Chung. 2010

**Received: **30 June 2009

**Accepted: **10 January 2010

**Published: **19 January 2010

## Abstract

We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

## Keywords

## 1. Introduction

The Hyers-Ulam stability of functional equations go back to 1940 when Ulam proposed the following problem [1]:

*Let*
*be a mapping from a group*
*to a metric group*
*with metric*
*such that*

*Then does there exist a group homomorphism*
*and*
*such that*

where . Following the formulations as in [6, 20–22], we generalize the classical stability problems of above functional equations to the spaces of generalized functions as

where , and denote the pullback and the tensor product of generalized functions, respectively, and denotes the space of bounded measurable functions on .

We prove as results that if generalized function satisfies (1.5), then satisfies one of the followings

(ii) and are bounded measurable functions;

for some , and a bounded measurable function .

Also if generalized function satisfies (1.6), then satisfies one of the followings:

## 2. Generalized Functions

For the spaces of tempered distributions , we refer the reader to [23–25]. Here we briefly introduce the spaces of Gelfand generalized functions and Fourier hyperfunctions. Here we use the following notations: , , , , and , for , , where is the set of nonnegative integers and .

Definition 2.1.

The topology on the space
is defined by the seminorms
in the left-hand side of (2.1) and the elements of the dual space
of
are called *Gelfand-Shilov generalized functions*. In particular, we denote
by
and call its elements *Fourier hyperfunctions*.

It is known that if and , the space consists of all infinitely differentiable functions on that can be continued to an entire function on satisfying

It is well known that the following topological inclusions hold:

We refer the reader to [24, chapter V-VI], for tensor products and pullbacks of generalized functions.

## 3. Stability of the Equations

In view of (2.2), it is easy to see that the -dimensional heat kernel

belongs to the Gelfand-Shilov space for each . Thus the convolution is well defined for all . It is well known that is a smooth solution of the heat equation in and in the sense of generalized functions, that is, for every ,

We call
*the Gauss transform* of
. Let
be a semigroup and
be the field of complex numbers. A function
is said to be *additive* provided
, and
is said to be *exponential* provided
.

We first discuss the solutions of the corresponding trigonometric functional equations in the space of Gelfand generalized functions. As a consequence of the result [6, 26], we have the following.

Lemma 3.1.

are equal, respectively, to the smooth solution of corresponding classical functional equations (1.3) and (1.4).

Remark 3.2.

We refer the reader to Aczél [27, page 180] and Aczél and Dhombres [28, pages 209–217] for the general solutions and measurable solutions of (1.3) and (1.4).

For the proof of the stability of (1.5), we need the following lemma.

Lemma 3.3.

Also the inequality (3.5) together with (3.4) implies one of the followings;

(ii) and are bounded functions;

(iii) and where , , with and is a bounded function.

Proof.

Again considering (3.11) as a function of and for all fixed , we have .

for all . From (3.12) and (3.23), we have (iii). This completes the proof.

Remark 3.4.

for some and bounded solution of the heat equation.

Theorem 3.5.

Let satisfy (1.5). Then and satisfy one of the followings

(ii) and are bounded measurable functions;

for some , and a bounded measurable function .

Proof.

The nontrivial solutions of (3.27) are given by (iv) or which is included in case (iii). This completes the proof.

Now we prove the stability of (1.6).

Lemma 3.6.

Also the inequality (3.29) together with (3.28) implies one of the followings

(i) and are bounded functions;

(ii) is bounded; is an exponential;

(iii) for some bounded exponential ;

(iv) , , where , is a bounded exponential and is a bounded function.

Proof.

which gives (iv). This completes the proof.

Theorem 3.7.

Let satisfy (1.6). Then and satisfy one of the followings

(i) and are bounded measurable functions;

Proof.

for all , where is a bounded exponential. Using the continuity of , it follows from (3.43) that is bounded in and so is , which implies that both and are bounded measurable functions. For the case (iv) since , are unbounded continuous, is unbounded continuous, and . On the other hand, it follows from (3.28) that , which occurs only when . Thus both and are bounded in and are bounded measurable functions.

By Lemma 3.1, the nonconstant solution of (3.44) is given by , for some . This completes the proof.

Remark 3.8.

Taking the growth of as into account, or only when for some . Thus the Theorems 3.5 and 3.7 are reduced to the followings.

Corollary 3.9.

Let or satisfy (1.5). Then and satisfy one of the followings

(ii) and are bounded measurable functions;

for some , , and a bounded measurable function .

Corollary 3.10.

Let or satisfy (1.6). Then and are bounded measurable functions.

## Declarations

### Acknowledgments

The authors express their deep thanks to the referee for valuable comments on the paper, in particular, introducing the earlier results of Bourgin [3, 4] and Aoki [5]. This work was supported by the Korea Research Foundation Grant (KRF) Grant funded by the Korea Government (MEST) (no. 2009-0063887).

## Authors’ Affiliations

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