QUESTION : Formulate a variant of Proposition 6.3 in which f is not required to have compact support.

Please use the proposition below to formulate the variant proposition

Proposition 6.3. Let f be an integrable function on R" with compact support. Let a E R" and set fe (x ) = of(* (x -a)) for E > 0. Furthermore, we define, for each multi-index a, the coefficient ca E C by Ca = xa f(x) dx, and we introduce, for every j E Zzo, the distribution u; e D'(R") by Ca gaga. uj = Land Jal= j We then have, for all k E Zzo, k lime* (fe - >(-E)' uj) = 0 in D'(R"). E40 1=0 Proof. For every o E Coo(R") we have fc ($ ) = 1 " f (2 ( x - a) ) $ ( )dx - [ f(h ) (a+ ch )ah . The assertion now follows by substituting the Taylor expansion of o at the point a and of order k. 0 The coefficient ca is known as the ath moment of the function f. A consequence of the proposition is that e fe converges distributionally to (-1)Kuk as e 4 0 if Ca = 0 for all multi-indices a with la|