Jaeyoung Chung

Characterizations of the Gelfand-Shilov spaces via Fourier transforms

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type S and type W. These results explain more clearly the invariance of these spaces under the Fourier transformations.

Stability of functional equations on restricted domains in a group and their asymptotic behaviors

In this paper we consider Hyers-Ulam stability problems for the Pexider equation, the Cauchy equation, and the Jensen equation in general restricted domains in a group. The main purpose of this paper is to find restricted domains such that the functional inequality satisfied in those domains extends to the inequality for the whole domain and such that the Hyers-Ulam stability theorem holds for the inequalities as it does when the inequality holds globally. We also consider a distributional version of the Hyers-Ulam stability of the Pexider equation in restricted domains and its asymptotic behaviors.

A Functional Equation of Aczél and Chung in Generalized Functions

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.

THE STABILITY OF THE SINE AND COSINE FUNCTIONAL EQUATIONS IN SCHWARTZ DISTRIBUTIONS

We prove the Hyers-Ulam stability of the sine and cosine functional equations in the spaces of generalized functions such as Schwar- tz distributions, Fourier hyperfunctions, and Gelfand generalized func- tions.

There exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable

We verify that there exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable in the sense that for given s > 1, there is a nowhere Gevrey differentiable function on R of order s that is Gevrey differentiable of order r for any r > s, which also gives a strong example that is Gevrey differentiable but nowhere analytic.

Solution of Several Functional Equations on Nonunital Semigroups Using Wilson's Functional Equations with Involution

Let S be a nonunital commutative semigroup, an involution, and the set of complex numbers. In this paper, first we determine the general solutions of Wilson’s generalizations of d’Alembert’s functional equations and on nonunital commutative semigroups, and then using the solutions of these equations we solve a number of other functional equations on more general domains.

HYERS-ULAM STABILITY OF TRIGONOMETRIC FUNCTIONAL EQUATIONS

In this article we prove the Hyers–Ulam stability of trigonometric functional equations.

A generalized Hyers-Ulam stability of a Pexiderized logarithmic functional equation in restricted domains

AbstractLet ℝ+ and B be the set of positive real numbers and a Banach space, respectively, f, g, h : ℝ+ → B and ψ:ℝ+2→ℝ be a nonnegative function of some special forms. Generalizing the stability theorem for a Jensen-type logarithmic functional equation, we prove the Hyers-Ulam stability of the Pexiderized logarithmic functional inequality ||f(xy)-g(x)-h(y)||≤ψ(x,y) in restricted domains of the form {(x, y) : xkys≥ d} for fixed k, s ∈ ℝ, d > 0. We also discuss an L∞-version of the Hyers-Ulam stability of the inequality. 2000 MSC: 39B22.

Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

We consider the Hyers-Ulam stability problems for the Jensen-type functional equations in general restricted domains. The main purpose of this paper is to find the restricted domains for which the functional inequality satisfied in those domains extends to the inequality for whole domain. As consequences of the results we obtain asymptotic behavior of the equations.

Stability of Trigonometric Functional Equations in Generalized Functions

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.