  We consider the classical Cauchy problem for the linear heat equation and
integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case
$N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1
\leq p < \infty$ and a fast growing weight $w$, there is a Schauder basis
$(e_n)_{n=1}^\infty$ in $L_ w^p(\mathbb{R})$ with the following property: given
a positive integer $m $ there exists $n_m > 0$ such that, if the initial data
$f$ belongs to the closed linear space of $e_n$ with $n \geq n_m$, then the
decay rate of the solution of the heat equation is at least $t^{-m}$. The
result is also generalized to the case $N >1$ with a slightly weaker
formulation. The proof is based on a construction of a Schauder basis of
$L_w^p( \mathbb{R}^N)$, which annihilates an infinite sequence of bounded
functionals.
