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 1Let 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.