Stability of Trigonometric Functional Equations in Generalized Functions
© J. Chang and J. Chung. 2010
Received: 30 June 2009
Accepted: 10 January 2010
Published: 19 January 2010
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.
The Hyers-Ulam stability of functional equations go back to 1940 when Ulam proposed the following problem :
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 .
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 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
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 ,
are equal, respectively, to the smooth solution of corresponding classical functional equations (1.3) and (1.4).
For the proof of the stability of (1.5), we need the following lemma.
Also the inequality (3.5) together with (3.4) implies one of the followings;
Now we prove the stability of (1.6).
which gives (iv). This completes the 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.
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 . This work was supported by the Korea Research Foundation Grant (KRF) Grant funded by the Korea Government (MEST) (no. 2009-0063887).
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