Let $iA_j(1\leq j\leq n)$ be generators of commuting bounded strongly continuous groups, $A\equiv(A_1,A_2,\ldots,A_n)$ . We show that, when $f$ has sufficiently many polynomially bounded derivatives, then there
exist $k,r > 0$ such that $f(A)$ has a $(1+|A|^2)^-r$ -regularized $BC^{k}(f(\mathbf{R}^n))$ functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when $f(\mathbf{R}^n)\subseteq\mathbf{R}$ , then, for appropriate $k,r$ , $t\mapsto(1-it)^{-k}e^{-itf(A)}(1+|A|^2)^{-r}$ is a Fourier-Stieltjes transform, and when $f(\mathbf{R}^n)\subseteq[0,\infty)$ , then $t\mapsto(1+t)^{-k}e^{-tf(A)}(1+|A|^2)^{-r}$ is a Laplace-Stieltjes transform. With $A\equiv i(D_1,\ldots,D_n),f(A)$ is a pseudodifferential operator on $L^{p}(\mathbf{R}^n) (1\leq p < \infty)$ or $BUC(\mathbf{R}^n)$ .