Behavior in $ L^\infty $ of convolution transforms with dilated kernels

Assuming that $ K(x) $ is in $ L^1( {\mathbb R}) $, $ K_t(x) = t^{-1} K(x/t) $, and $ f(x) $ is in $ L^\infty( {\mathbb R}) $, we study the behavior of the convolution $ K_t*f(x) $ as the parameter $ t $ tends to $ \infty $. It turns out that the limit need not exist and, if it does exist, the limit is a constant independent of $ x $. Situations where the limit exists and those where it fails to exist are identified. Several issues related to this are addressed, including the multivariate case. As one application, these results provide an accessible description of the behavior of bounded solutions to the initial value problem for the heat equation.