Convolution $(*)$ is an operation defined on two functions as
$$(a*b)(t) = \int_{-\infty}^\infty a(\tau)b(t-\tau)d\tau$$ The other day I came to think about an equation regarding it:
$$(f_i * f_j) (t) = g(t)$$ And for which sets of functions $\mathcal F = \{f_1,f_2,\cdots\}$ we have for some $\{c_k\}$ $$g(t)= \sum_{\forall i} c_if_i(t)$$
Own work The only thing I can think of are the truncated monomials
$$f_k = H(t)t^k = \cases{0,\,\,\,\ t< 0\\t^k,\,\, t\geq 0\\}$$
One reason for investigating this is that a family of interpolating splines, including nearest neighbor, linear spline, cubic splines can be generated like so: $$f_0(t) = H(t)\cdot H(1-t)\\f_k(t) = (f_0 *f_{k-1})(t)$$