Approximating compactly supported L2L^2 functions with Schwartz functions “from within”?

It is well known that the class of Schwartz functions S in dense in all Lp spaces therefore for each fL2 there exists a sequence of Schwartz functions (fk) such that ffkL20 as k.

If we suppose further that f has compact support can we find a sequence of Schwartz functions (fk) such that ffkL20 as k (as above) and additionally supp(fk)supp(f) for all k?

Answer

No, this is not possible, even if you define supp(f) only as an equivalence class (up to a null set). For a counterexample take the characteristic function f of a closed set E of positive measure but without interior points (like that fat Cantor set which Mateusz mentioned).

Indeed, if fn is a continuous function with supp(fn)supp(f)N=EN for some null set N, then fn=0, because otherwise EN would have to contain some interval (a,b) which must contain a point from the complement of E and which after shrinking (since E is closed) can thus be assumed to lie completely in the complement of E, hence N(a,b), a contradiction.

Attribution
Source : Link , Question Author : Dominic Shillingford , Answer Author : Martin Väth

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