We obtain the generalized codimension-$p$Cauchy–Kovalevsky extension of the exponential function $e^{i\langle\underline{y},\underline{t}\rangle}$ in R^{$m$}=R^{$p$}⊕R^{$q$}, where$p$>1, $\underline{y},\underline{t}\in\mathbf{R}^{q}$ , and prove the corresponding codimension-$p$Paley–Wiener theorems.