Quick description
Suppose one has an object, such as a function , that one wants to somehow represent as combinations of various modifications (e.g. translations, rescalings, etc.) of another object, such as
. If the first object
exhibits some invariance properties with respect to some sort of symmetry, and is in some sense a "canonical" example of an object with that symmetry, then often one can obtain
(or something similar to
) by starting with
and averaging it with respect to the symmetries that
enjoys. This will often create a new object which is closer to
, and which is thus presumably an easier starting point to build
out of.
Prerequisites
Real analysis; calculus
Example 1
To represent the Riesz potential in
for some
in terms of Gaussians
one can use the scale invariance
for
. More specifically, averaging the rescaled Gaussians
against the multiplicative Haar measure
soon yields the useful identity

thus allowing one to represent as the average of Gaussians. This is useful in many ways, for instance it allows one to compute the Fourier transform of the Riesz potential
; see this blog post for further discussion.
Example 2
The construction of Haar measure on compact groups by repeatedly averaging non-Haar measures with respect to rotations is another example: see this blog post for further discussion.
Example 3
One can often represent singular integrals with a dilation symmetry as an average of dyadic counterparts of those singular integrals. This trick is occasionally used in the harmonic analysis literature, e.g. in this paper of Lacey. (Need more examples here...)
General discussion
One key danger to avoid when averaging is to ensure that the average doesn't cancel itself out to zero. This may require an appropriate renormalization of the averaging process, or by ensuring that all the various transformed versions of are somehow "uniformly non-degenerate", for instance their value when tested against some dual object might be uniformly bounded away from zero.