Linear span of poisson kernels dense in $L^1(\mathbb{T})$

Question

A paper I am reading ("Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in $L^2(\mathbb{T})$" by Sergei Khrushchev...really a great paper) repeatedly mentions the denseness of the linear span of Poisson kernels in $L^1(\mathbb{T})$.

I am aware of the delta-approximation of standard Poisson kernel convolution and the various senses of this convergence. Does this imply our denseness conclusion? I may be missing something here but to me this seems nontrivial.

Thanks for the help!

Answer 1

Let $M$ be the linear span of these kernels. If $M$ is not dense in $L^1,$ then by Hahn-Banach, there is an $f\in L^\infty,$ $f$ not the zero function, such that $\int fg = 0$ for all $g\in M.$ This gives $P_r*f(\theta) \equiv 0$ for $r\in [0,1), \theta \in (0,2\pi].$ But we know as $r\to 1,$ $P_r*f\to f$ in, say, $L^2,$ which contains $L^\infty.$ This tells us $f$ is the zero function after all, contradiction.