.I am not a fan of spreading out the delta. I like concentrating it on a node. The main reason is: you don't want to introduce an extraneous limit (besides taking the lattice spacing to zero). If you concentrate it on a node, it adjusts 'automatically' with the lattice spacing. For example with uniform spacing of a one-factor probem, my method has lattice Dirac delta with amplitude [$]1/\Delta x[$].
Alan
I have tried some of the other suspects but this one looks good (its integral is 1 and has compact support in a small interval). However, it is a rectangle and will that give nasty oscillations?