I'm trying to prove that, given the heat equation $u_t = \Delta u$ with boundary values $u(x,0) = f(x)$, the solution given by
$$u(x,t) = f \star H_t^{(d)}(x)$$
is continuous up to the boundary $t=0$, where $H_t^{(d)} = \frac{1}{(4 \pi^2 t)^{d/2}} e^{-\frac{|x|^2}{4t}}$ and $f \in \mathbb{S}(\mathbb{R})$
We know that $H_t^{(d)}$ is a family of good kernels/approx. to the identity as $t \to 0$
Question: Can we say $ f \star H_t^{(d)}(x)$ converges to $f(x)$ uniformly as $t \to 0$ as in the one dimensional case? Stein and Shakarchi say that, for a good family of kernels $K_{\delta}$ in $\mathbb{R}^n$, $f \star K_{\delta} \to f(0)$ for $f$ of Schwartz class, which confuses me.