  If $f:{\bf R}^d\to{\bf C}$ is bounded and $f$'s H\"older $\alpha$-modulus of
continuity grows no faster than $(1+\vert x\vert)^M$ ($M\geq0$) then, for every
$\epsilon>0$, there is a $\beta>0$ such that $f$'s H\"older $\beta$-modulus
grows no faster than $(1+\vert x\vert)^{\epsilon}$. We use this easy fact to
show that, if $\vert f\vert$ decays as fast as $(1+\vert x\vert)^{-R}$ (for
$R>0$) and $f$'s $\alpha$-H\"older modulus grows no faster than $(1+\vert
x\vert)^M$, then, for every $0\leq R'< R$, there is a $\beta>0$ such that $f$'s
$\beta$-H\"older modulus decays as fast as $(1+\vert x\vert)^{-R'}$. We apply
this to strengthen a result of Coifman and Meyer on almost-orthogonality of
vaguelet families and to derive other useful facts about vaguelets and
vaguelet-like functions.
