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Soon-Mo Jung

Publications and source records attributed to Soon-Mo Jung.

8 recordsLinked to original sources
arXiv2022

The conjecture of Ulam on the invariance of measure on Hilbert cube

A conjecture of Ulam states that the standard probability measure $π$ on the Hilbert cube $I^ω$ is invariant under the induced metric $d_a$ when the sequence $a = \{ a_i \}$ of positive numbers satisfies the condition $\sum\limits_{i=1}^\infty a_i^2 < \infty$. This conjecture was proved in \cite{jung1} when $E_1$ is a non-degenerate subset of $M_a$. In this paper, we prove the conjecture of Ulam completely by classifying cylinders as non-degenerate and degenerate cylinders and by treating the degenerate case that was overlooked in the previous paper.

math.FA↗
arXiv2021

Extension of isometries in real Hilbert spaces

In this paper, the notions of first-order and second-order generalized linear spans and index set are defined. Moreover, their properties are investigated and applied to the studies of extension of isometries. We develop the theory of extending the domain of local isometries to the generalized linear spans, where we call an isometry defined in a subset of a Hilbert space a local isometry. In addition, we prove that the domain of local isometry can be extended to any real Hilbert space, where the domain of local isometry does not have to be a convex body or an open set. This indicates that the main results of this paper are superior to those previously published.

math.FA↗
arXiv2021

Hyers-Ulam stability of isometries on bounded domains

More than 20 years after Fickett attempted to prove the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ in 1981, Väisälä improved Fickett's result significantly. In this paper, we will improve Fickett's theorem by proving the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ using a more intuitive method different from that used by Väisälä.

math.FA↗
arXiv2005

Stability of Generalized Jensen Equation on Restricted Domains

In this paper, we establish the conditional Hyers-Ulam-Rassias stability of the generalized Jensen functional equation $r f \left (\frac{sx+ty}{r}) = s g(x) + t h(y)$ on various restricted domains such as inside balls, outside balls, and punctured spaces. In addition, we prove the orthogonal stability of this equation and study orthogonally generalized Jensen mappings on Balls in inner product spaces.

math.FA↗
arXiv2005

Superstability of the generalized orthogonality equation on restricted domains

Chmieliński has proved in the paper [4] the superstability of the generalized orthogonality equation $|< f(x), f(y) >| = |< x, y >|$. In this paper, we will extend the result of Chmieliński by proving a theorem: Let $D_{n}$ be a suitable subset of $ \R^n$. If a function $f\hbox{:} D_{n} \to \R^n$ satisfies the inequality $||< f(x), f(y) >| - |< x, y >|| \leq ϕ(x,y)$ for an appropriate control function $ϕ(x,y)$ and for all $x, y \in D_{n}$, then $f$ satisfies the generalized orthogonality equation for any $x, y \in D_{n}$.

math.FA↗
arXiv2004

On some congruence with application to exponential sums

We will study the solution of a congruence, $x \equiv g^{(1/2)ω_g(2^n)} \bmod 2^n$, depending on the integers $g$ and $n$, where $ω_g(2^n)$ denotes the order of $g$ modulo $2^n$. Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.

math.NT↗
arXiv2003

Some functional equations originating from number theory

We will introduce new functional equations (\ref{eq:hana}) and (\ref{eq:hana2}) which are strongly related to well-known formulae (\ref{eq:young}) and (\ref{eq:young2}) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.

math.NT↗