Waring's Problem, conjecturing that every integer can be respresented as sum of a sufficiently large number of powers of other integers, is investigated subject to so-called digital restrictions. That is, the indeterminates , ..., all obey a congruence of the type , where denotes the -adic sum of digits function. Given , , , , and , we provide a Hardy-Littlewood like asymptotic formula for the number of such representations of , from which the fact that the set forms an asymptotic basis can be easily derived. This is done for a single digital restriction (Thuswaldner and Tichy [3]) as well as for systems of sum of digits congruences (Pfeiffer and Thuswaldner [2]) using methods introduced by Kim [1].