Difference and $(\Delta )$ properties for some new classes

Abstract

\begin{abstract} {In this paper we study difference and $(\Delta )$ properties for the classes of the form $C_0(J,X)$, $\frak {g} \U$, $\U+\frak {g} \V$, where $\U, \V\in \{BUC(J,X), UC(J,X)\}$ and $\mathfrak{g}(t)=e^{i{t}^2}$, $t\in \mathbb{R}$. For functions whose differences belong to $ \F \in\{C_0(J,X)$, $\frak {g} \U\}$, we prove a new stronger $(\Delta )$ property (S$\Delta$P): If $f:J\to X$ and $Δ_h f \in \F$, $h>0$, then $f\in C(J,X)$ and $(f-M_hf)\in \F$, $h>0$. \noindent See (Lemma 2.5, Theorems 3.1, 3.2, Lemma 3.4, Theorem 3.7). These results enabled us to prove $(\Delta )$ for $\U+\frak {g} \V$ even when $\U, \V\in UC(J,X) $ (Theorem 4.2). We give a new proof of a theorem of De Bruijn [10] stating: if $J\in \{\Rdb_+, \Rdb\}$, $ϕ\in \Cdb^J$ and $Δ_s ϕ\in C(J,\Cdb)$ for each $s>0$, then $\varphi =G+H$, where $G \in C(J,\Cdb)$ and $H(t+s)=H(t)+H(s)$, $t,s\in J$, for functions $\varphi :X^J$.} \end{abstract}

10 pages