nLab approximation of the identity

“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.

Examples

We illustrate the technique with two examples.

Example

Consider first the problem of showing that restrictions of polynomials to $[-1, 1]$ are dense in $L^p([-1, 1])$ (under Lebesgue measure). The idea is to take the formula

$$f(y) = (f * \delta)(y) = \int_{-1}^1 f(x) \delta(y - x) d x$$

(which literally makes no sense because $\delta$ is not an actual integrable function) and then replace $\delta$ by polynomial functions $p_n$ which “approximate” to it (so each $p_n$ has “mass” 1 and is vanishingly small outside a given neighborhood of 0, if $n$ is sufficiently large).

Put for example $f_n(x) = (1 - x^2)^n$ and “normalize” it, putting

$$p_n = \frac{f_n}{\|f_n\|_1}$$

where $\|g\|_1$ indicates $L^1$ norm. By “differentiating under the integral sign”, we have

$$D^j (f * p_n) = (-1)^j f * D^j(p_n)$$

so that for each $n$, the $j^{th}$ derivative of $f * p_n$ is identically zero for $j$ sufficiently large. Hence $f * p_n$ is polynomial. Next, the claim is that for $f \in L^p$, we have

$$lim_{n \to \infty} \|f - (f * p_n)\|_p = 0$$

Intuitively, the idea is that $f - (f * p_n) = f * (\delta - p_n)$ and that (because $L^p$ is a module over the Banach algebra $L^1$)

$$\|f * (\delta - p_n)\|_p \leq \|f\|_p \|\delta - p_n\|_1 \to 0$$

as $n \to \infty$. For a more careful proof, see theorem 9.6 in Wheeden and Zygmund (referenced below).

Example

For a second example, consider how to prove that the functions $z^n$, with $n$ ranging over integers, forms an orthonormal basis of the Hilbert space $L^2(S^1)$ where $S^1$ is the unit circle in the complex plane, where the inner product is given by

$$\langle f, g \rangle = \frac1{2\pi i} \int_{S^1} \overline{f(z)} g(z) \frac{d z}{z}$$

The monomials $z^n$ are clearly orthonormal, so again the idea is to use appropriate linear combinations of the $z^n$ (i.e., Laurent polynomials) to approximate a Dirac mass concentrated at the identity $z = 1$ in $S^1$. There are various ways of doing that; one of the most useful is by taking the Féjer kernel

$$F_n(z) = \frac1{n+1} (\sum_{-n \leq k \leq n} z^{k/2})^2 = \frac1{n+1}(z^{-n} + 2z^{-n+1} + \ldots + (n+1)z^0 + \ldots + 2z^{n-1} + z^n)$$

Each Laurent polynomial $F_n(z)$ is real-valued, nonnegative, and its $L^1$ norm is 1. Putting $z = e^{i x}$, we have

$$F_n(e^{i x}) = \frac1{n+1} \frac{\sin^2((n+1)x/2)}{\sin^2(x/2)}$$

which makes it clear that $F_n(e^{i x})$ becomes very small outside a neighborhood of 0 (in $\mathbb{R}/2\pi\mathbb{Z}$) as $n$ grows large. Thus $F_n$ approximates the identity; therefore for any $L^2$ function $f$ on $S^1$, we have

$$\lim_{n \to \infty} \|(F_n * f) - f\|_2 = 0$$

Finally, $F_n * f$ is itself a Laurent polynomial; this follows from the fact that for the function $e_n(z) = z^n$, one has

$$(e_n * f)(w) = \frac1{2\pi i}\int_{S^1} (w/z)^n f(z) \frac{d z}{z} = e_n(w)\langle e_n, f \rangle$$

It follows from all this that the Laurent polynomials on $S^1$ are dense in $L^2(S^1)$.

A similar technique applies to any compact Hausdorff abelian group $G$ equipped with its normalized Haar measure $d\mu$, in place of the measure space $(S^1, \frac1{2\pi i}\frac{d z}{z})$, and shows that the characters on the group span a dense subspace in $L^2$ norm. In other words, the characters form an orthonormal basis of $L^2(G, d\mu)$.