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I encountered the following claim (verbatim):

Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel

$$ K(x,y) = \sum_n e_n(x)e_n(y). $$

For any function $f \in V$,

$$ f(y) = \int f(x) K(x,y) dy. $$

Questions:

  • I am under the impression that a RKHS is a Hilbert space of functions, where pointwise evaluation is a bounded functional. $L^2$ is not even a space of functions, how does the claim make sense (e.g. take an ONB for the entire $L^2$)?
  • Also, are there no convergence requirements on the reproducing kernel $K$, or is it purely formal?
share|improve this question
    
A formula like that would hold if $V$ were a reproducing kernel Hilbert space (so in particular it would have to be a space of functions). As written it does not make sense. Where did you encounter this? –  Jonas Meyer Jan 30 at 3:55
    
p29 of Statistical Modeling by Wavelets, by Vidakovic. –  Michael Jan 30 at 23:29
1  
Thanks, I just checked it out on a Google Books preview. Whatever it is they're doing there, it is nonstandard from my experience. In an actual RKHS, the sum would be convergent, and the identity would be a consequence of Parseval's identity. I don't really know what to say in an answer though, because I don't know the context of what type of object the author does have in mind. –  Jonas Meyer Jan 30 at 23:34
    
Thanks. That helps. Struck me as a bit strange, hence the question. –  Michael Jan 31 at 0:59

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