Abstract
We consider an -dimensional version of the functional equations of Aczél and Chung in the spaces of generalized functions such as the Schwartz distributions and Gelfand generalized functions. As a result, we prove that the solutions of the distributional version of the equation coincide with those of classical functional equation.
Highlights
In 1, Aczel and Chung introduced the following functional equation:l m fj αj x βj y gk x hk y, j1k1 where fj, gk, hk : R → C and αj, βj ∈ R for j 1, . . . , l, k 1, . . . , m
We consider an n-dimensional version of the functional equations of Aczel and Chung in the spaces of generalized functions such as the Schwartz distributions and Gelfand generalized functions
We prove that the solutions of the distributional version of the equation coincide with those of classical functional equation
Summary
In 1 , Aczel and Chung introduced the following functional equation:. k1 where fj , gk, hk : R → C and αj , βj ∈ R for j 1, . . . , l, k 1, . . . , m. In 1 , Aczel and Chung introduced the following functional equation:. Under the natural assumptions that {g1, . Hm} are linearly independent, and αj βj / 0, αiβj / αj βi for all i / j, i, j 1, . L, it was shown that the locally integrable solutions of 1.1 are exponential polynomials, that is, the functions of the form q erkxpk x , k1 where rk ∈ C and pk’s are polynomials for all k 1, 2, . Under the natural assumptions that {g1, . . . , gm} and {h1, . . . , hm} are linearly independent, and αj βj / 0, αiβj / αj βi for all i / j, i, j 1, . . . , l, it was shown that the locally integrable solutions of 1.1 are exponential polynomials, that is, the functions of the form q erkxpk x , k1 where rk ∈ C and pk’s are polynomials for all k 1, 2, . . . , q
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