I try to derive the formula for the discrete convolution from the continuous convolution using piecewise constant interpolation.
Whereby the zero-order spline basis function is given as:
$$\phi^0(x) = \begin{cases} 1; \quad \text{ if } x \in [\frac{1}{2}, \frac{1}{2}) \\ 0; \quad \text{ otherwise}\end{cases}$$
The piecewise constant interpolation as:
$$u(x) = \sum_{i=1}^N \phi^0(x-i) U_i$$
At some point I arrive at the equation:
$$(u*h)(k) = \sum_{l \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} U_IH_m \int_{\mathbb{R}} \phi^0(x) \phi^0(k-l - m-x) dx $$
To get the correct result the following equivalence is needed: $$\int_{\mathbb{R}} \phi^0(x) \phi^0(k-l - m-x) dx = \begin{cases} 1; \quad \text{ if } m=k-l\\ 0; \quad \text{ otherwise} \end{cases}$$
However I don't totally get why this holds, can maybe someone help me with an intuitive explanation?