Sometimes one needs to switch from an approximation of the identity to another with some other property which is helpful to the problem in exam. In Stein’s Harmonic Analysis book there’s this nice little result which allows one to do this switch.
Notice that by we denote the dilated function .
Proposition 1 Let be such that , and let be another Schwartz function. Then there exist a sequence such that we have the following nice decomposition:
- the ‘s size decreases rapidly in , in particular
for every , where are the seminorms of defining its topology.
Proof: As usual we’ll localize in the frequency domain. Take then a bump function which is , identically on and identically zero on (it is therefore in ). Now from define bump functions localized at (big) frequency by
where . They have the further properties that
because and these two terms have disjoint supports and comparable upperbounds; the other property is trivially that
From this second property then
and the are then defined by
i.e. . Of course we’re cheating a little: we need some lowerbound on for the ‘s to be well defined. But this is implicit in the definition since , and therefore we can assume in the whole ball – if the ball were smaller we should just shift the indices , as will be apparent in the following. In our case, since is localized at frequency one has therefore in the support of . It is thus evident that, since the denominator is bounded away from zero and all the functions are in , the ‘s are in .
What’s left to prove is that the seminorms decay rapidly in . This property actually follows from the frequency localization. The proof is a bit long and boring, but here it is anyway: using Hausdorff-Young inequality
and by the properties of the Fourier transform , so that we can estimate the norm of the latter. Using Leibniz’s rule
and since the support of is (localization!) the terms are essentially bounded in size. We only care about a uniform bound, so we write
and again by boundedness of the support one has
We have therefore to estimate . Again, by Leibniz’s rule (discarding the constants) and writing ,
, therefore there’s a constant such that for every , and once again the localization of frequency to size allows us to write
Notice the constants here depend on and thus on , and thus on in the end (other than and ). Now, by the definition of we have
therefore we can see how these terms are all uniformly bounded by a constant depending only on and . Then we’ve shown that
for whatever we choose.
In the end we go back to the time domain applying the inverse Fourier transform to , which becomes
Notice, in case the ball where were smaller, say for some , it would suffice to define instead
so that the sum starts from instead of .
Notice we assumed very little about and , this last one in particular can be any Schwartz function.