Reproducing kernel vs. Riesz kernel.

Question

I'm attending an Analysis course and we are studying the Hardy space $H^2$ in the unit disk, from where the concept of RKHS (Reproducing Kernel Hilbert Space) came out. Acording to my notes:

By the Riesz representation theorem for every evaluation functional $\phi_a:H^2\to\mathbb{C}$, $\phi_af=f(a)$, there is exactly one $k_a\in H^2$ such that $$ \phi_af=\langle f,k_a\rangle_{H^2},\ \ f\in H^2. $$ This equation is called the reproducing property, for which $k_a$ is called a reproducing kernel of $H^2$. In fact we can calculate that $k_a(z)=1/(1-\bar az)$ since $$ \phi_af=\sum_{n=0}^\infty \hat f(n)a^n=\sum_{n=0}^\infty \hat f(n)\overline{\bar a^n}=\langle f\,, \,\frac{1}{1-\bar az}\rangle_{H^2}. $$ The function $k_a(z)=1/(1-\bar a z)$ receives the name of Riesz kernel.

I have no problem with the math, but with the terminology; why is $k_a$ called both reproducing kernel and Riesz kernel? Is a reproducing kernel exactly the same as a Riesz kernel? If not, how are they different?

Thank you in advance for your help!

Answer 1

"Reproducing": I am not sure what was originally intended by the person who coined that term, but there are two plausible ways to think of the kernel function as "reproducing" (as in producing again, not as in giving offspring):

1. Given $f\in H^2$, if you know the kernel you can "reproduce" all function values of $f$ using $k$, because $f(z)=\langle f,k_z\rangle$.
2. Given the domain, in this case $\mathbb D$, and the kernel function $k:\mathbb D\times \mathbb D\to \mathbb C$ you can "reproduce" the function space as the unique Hilbert space of functions containing the functions $k_a:z\to k(a,z)$ whose evaluations are given by inner products with the functions $k_a$. This is called the Moore–Aronszajn theorem and given much more precision and elaboration on Wikipedia.

Many sources including your notes refer to 1. as the reason for the term "reproducing".

"Riesz": I have never seen that name for it before, but it might be in reference to the Riesz lemma guaranteeing its existence whenever you have a Hilbert space of functions with continuous evaluation maps. Or it mean something more specific that this kernel is an example of. I can only speculate, and Googling hasn't helped quickly.

Some special spaces' kernels have special names; yours is called the Szegő kernel, and there are other Szegő kernels for generalizations of $H^2$ on other domains. There are also the Bergman kernels for spaces of holomorphic functions with inner product given by integration with respect to area measure. Perhaps there is some class of kernels called "Riesz kernels" of which the Szegő kernel is an example.