Landau kernel

The previous post was about the trick Lebesgue used to construct a sequence of polynomials converging to |x| on the interval [−1, 1]. This was the main step in his proof of the Weierstrass approximation theorem.

Before that, I wrote a post on Bernstein’s proof that used his eponymous polynomials to prove Weierstrass’ theorem. This is my favorite proof because it’s an example of using results from probability to prove a statement that has nothing to do with randomness.

This morning I’ll present one more way to prove the approximation theorem, this one due to Landau.

The Landau kernel is defined as

K_n(u) = (1 - u^2)^n

Denote its integral by

k_n = \int_{-1}^1 K_n(u)\, du

Let f(x) be any continuous function on [−1, 1]. Then the convolution of the normalized Landau kernel with f gives a sequence of polynomial approximations that converge uniformly to f. By “normalized” I mean dividing the kernel by its integral so that it integrates to 1.

For each n,

\frac{1}{k_n}\int_{-1}^1 K_n(t-x)\, f(t)\, dt

is a polynomial in x of degree 2n, and as n goes to infinity this converges uniformly to f(x).


There are a few connections I’d like to mention. First, the normalized Landau kernel is essentially a beta distribution density, just scaled to live on [−1, 1] rather than [0, 1].

And as with Bernstein’s proof of the Weierstrass approximation theorem, you could use probability to prove Landau’s result. Namely, you could use the fact that two independent random variables X and Y, the PDF of their sum is the convolution of their PDFs.

The normalizing constants kn have a simple closed form in terms of double factorials:

\frac{k_n}{2} = \frac{(2n)!!}{(2n+1)!!}

I don’t know which Landau is responsible for the Landau kernel. I’ve written before about the Edmund Landau and his Big O notation, and I wrote about Lev Landau and his license plate game. Edmund was a mathematician, so it makes sense that he might be the one to come up with another proof of Weierstrass’ theorem. Lev was a physicist, and I could imagine he would be interested in the Landau kernel as an approximation to the delta function.

If you know which of these Landaus, or maybe another, is behind the Landau kernel, please let me know.

Update: Someone sent me this paper which implies Edmund Landau is the one we’re looking for.