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

Hyers-Ulam stability problems of functional equations go back to 1940 when Ulam proposed the following question [

Let

Then does there exist a group homomorphism

This problem was solved affirmatively by Hyers [

For the first step, convolving

For the second step, using similar idea of Székelyhidi [

We first introduce the spaces

One denotes by

One denotes by

It is well known that the following topological inclusions hold:

It is known that the space

By virtue of Theorem 6.12 of [

Let

In particular, let

Let

For more details of pullback and tensor product of distributions we refer the reader to Chapter V-VI of [

Let

Let

In view of (

We call

A function

For the proof of stabilities of (

Let

Suppose that

In view of (

Let

We discuss the solutions of the corresponding trigonometric functional equations

The solutions

The continuous solutions

Also, the continuous solutions

For the proof of the stability of (

Let

Suppose that inequality (

Also by (

From (

Since

So by our assumption, the left hand side of (

Now by the definition of

In the following lemma we assume that

Let

Then

for all

for all

for all

for all

for all

If

From (

Fixing

Replacing

In the following we assume that

Let

for all

for all

for all

Convolving in (

Similarly we have

By Lemma

Let

Let

for all

for all

Now we prove the stability of (

Let

As in Lemma

Since we may assume that

By equating (

In (

Here, we have

Considering (

In the following lemma we assume that

Let

Then

for all

for all

If

From Lemma

Putting (

Similarly as in the proof of Lemma

Putting

From (

Let

for all

Similarly as in the proof of Theorem

Firstly we assume that (

Putting (

Secondly we assume that (

By Lemma

Every infraexponential function

Let

for all

for all

Suppose that

there exists

for almost every

Taking the growth of

Let

for all

for all

for all

Let

for all

The first author was supported by Basic Science Research Program through the National Foundation of Korea (NRF) funded by the Ministry of Education Science and Technology (MEST) (no. 2012008507), and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2011-0005235).