“Approximation of the identity” is a rubric for a general technique in functional analysis for proving that certain inclusions of topological vector spaces are dense. It refers to the fact that an identity for a convolution product (aka “Dirac distribution”) may not literally exist in a particular TVS, but is virtually there in the sense that it can be approximated by elements in the subspace which is being included.
We illustrate the technique with two examples.
Consider first the problem of showing that restrictions of polynomials to are dense in (under Lebesgue measure). The idea is to take the formula
(which literally makes no sense because is not an actual integrable function) and then replace by polynomial functions which “approximate” to it (so each has “mass” 1 and is vanishingly small outside a given neighborhood of 0, if is sufficiently large).
Put for example and “normalize” it, putting
where indicates norm. By “differentiating under the integral sign”, we have
so that for each , the derivative of is identically zero for sufficiently large. Hence is polynomial. Next, the claim is that for , we have
Intuitively, the idea is that and that (because is a module over the Banach algebra )
as . For a more careful proof, see theorem 9.6 in Wheeden and Zygmund (referenced below).
For a second example, consider how to prove that the functions , with ranging over integers, forms an orthonormal basis of the Hilbert space where is the unit circle in the complex plane, where the inner product is given by
The monomials are clearly orthonormal, so again the idea is to use appropriate linear combinations of the (i.e., Laurent polynomials) to approximate a Dirac mass concentrated at the identity in . There are various ways of doing that; one of the most useful is by taking the Féjer kernel
Each Laurent polynomial is real-valued, nonnegative, and its norm is 1. Putting , we have
which makes it clear that becomes very small outside a neighborhood of 0 (in ) as grows large. Thus approximates the identity; therefore for any function on , we have
Finally, is itself a Laurent polynomial; this follows from the fact that for the function , one has
It follows from all this that the Laurent polynomials on are dense in .
A similar technique applies to any compact Hausdorff abelian group equipped with its normalized Haar measure , in place of the measure space , and shows that the characters on the group span a dense subspace in norm. In other words, the characters form an orthonormal basis of .