What are non-tagential limits?

Question

I'm reading this article where they use a set of functions, $H^{\infty}$, defined like this "Let $H^{\infty }$ be the closed subalgebra of $L^{\infty }({\mathbb R})$ that consists of all functions being non-tangential limits on ${\mathbb R}$ of bounded analytic functions on the upper half plane." The thing is I don't know what is reffered as a non tangential limit. Please help.

Answer 1

See Fatou's Theorem for an example of why this matters. In that context, a tangential limit is a limit along a path that is tangent to the boundary of a disk. A function may have "easy" limits along paths running radially to the edge of the disk, and paths that curve outward (but always with some radial component), but at the edge of the disk the function may be so badly behaved that limits along the edge are impossible.

In your case, tangential limits again run along the boundary of the "good space" -- i.e. tangential limits are along the real line, being the boundary of the upper half plane. This is also the boundary of a "disc" since the upper half-plane is a hemisphere of the Riemann sphere.

Answer 2

If $x_n\to x$, we say that $(x_n)$ is a non-tangential sequence in the upper half plane if $\inf_{n > 0}\text{Im}(x_n-x)/\text{Re}(x_n-x) > 0$. So $f(x_n)$ converges non-tangentially to $y$ if $f(x_n)$ converges to $y$, and $(x_n)$ is a non-tangential sequence.

So $x_n = (a+ib)/n$ for $b>0$ is non-tangential, but $x_n = \frac1n + i\frac1{n^2}$ is not non-tangential.

http://demonstrations.wolfram.com/StolzAngle/ illustrates the concept (because non-tangential convergence is equivalent to convergence in so called Stolz domains).