To represent one object as combinations of another, try averaging the latter object using the symmetries of the former

Quick description

Suppose one has an object, such as a function $f(x)$ , that one wants to somehow represent as combinations of various modifications (e.g. translations, rescalings, etc.) of another object, such as $g(x)$ . If the first object $f$ 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 $f$ (or something similar to $f$ ) by starting with $g$ and averaging it with respect to the symmetries that $f$ enjoys. This will often create a new object which is closer to $f$ , and which is thus presumably an easier starting point to build $f$ out of.

Prerequisites

Real analysis; calculus

Example 1

To represent the Riesz potential $= \frac{1}{|x|^\alpha}$ in $\R^n$ for some $\alpha > 0$ in terms of Gaussians $e^{-\pi t^2 |x|^2}$ one can use the scale invariance $t^\alpha F(tx) = F(x)$ for $t > 0$ . More specifically, averaging the rescaled Gaussians $t^\alpha e^{-\pi t^2 |x|^2}$ against the multiplicative Haar measure $\frac{dt}{t}$ soon yields the useful identity

$\int_0^\infty t^{\alpha} e^{-\pi t^2 |x|^2} \frac{dt}{t} = \frac{1}{2} \pi^{-\alpha/2} |x|^{-\alpha} \Gamma(\alpha/2)$

thus allowing one to represent $F(x)$ 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 $F(x)$ ; 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 $g$ are somehow "uniformly non-degenerate", for instance their value when tested against some dual object might be uniformly bounded away from zero.