Thanks for reading, I'm glad you enjoyed it.

You are correct that there are few texts that discuss real-world examples of reproducing kernel Hilbert spaces without a significant amount of overhead to get there. In reality, I'm not sure there is an easy way to talk about reproducing kernel Hilbert spaces without all that functional analysis, which is a tough topic. I would argue that's probably the greatest thing holding back kernels - their high barrier to entry.

I'll spend one line here to mention my book (kernel-based approximation methods in Matlab) and also say that it is pretty heavy on the theory; the second half of the book deals with reproducing kernels as used in a variety of fields, but getting there requires a lot of very tough math.

Lemme take a moment to list places (some more real-world than others) where I know that reproducing kernels pop up - and anyone else reading this please feel free to add to this list:

1) Approximation theory - My bread and butter and the most important role for kernels as far as I'm concerned. When you have scattered data (especially in higher dimensions) and you want to interpolate and predict unobserved values, reproducing kernels give you several very nice guarantees. Through this, they also pop up in computer vision or surface reconstruction from point cloud data.

2) Spatial statistics - They actually play an identical role in kriging as they do in approximation theory, albeit for totally different reasons. People who are doing remote sensing or topological mapping may use reproducing kernels (or software that uses these kernels.

3) Machine learning - Both RBF networks and support vector machines use reproducing kernels, although in two different ways. The mechanism by which they are used in support vector machines (feature maps) is difficult for me to understand which is why I don't talk about them much. They are still a very important field that uses kernels.

4) Differential equations - The same theory that makes reproducing kernels work for approximation theory also works for numerically solving differential equations. You could lump stuff regarding Green's functions in here too, although those may not be reproducing kernels.

5) Numerical integration - Especially for integration in higher dimensions, basically all the theory of convergence for quasi Monte Carlo methods relies on reproducing kernel Hilbert spaces.

Maybe the best thing I can point you to is the same place where I first learned about this stuff: math.iit.edu/~fass/590. That's an old (recently updated) class website from a class I took at the Illinois Institute of Technology when I was an undergrad there. Dr. Fasshauer does an outstanding job putting notes and slides for his class together.

If you're feeling adventurous, I would recommend our book, but a good starting point would be his class notes, which I still turn back to often and are written to introduce graduate students (who know nothing about kernels) to the topic. I'll work on putting together a list of good application-oriented sources as well.

Thanks for your interest! Did you have a specific example/topic of kernels from data analysis that piqued your interest? I could look into more targeted content.