Hello MathOverFlow

I have some questions about frames and reproducing kernels and here they are:

For a Hilbert space $H$ spanned by a frame $\lbrace f_n \rbrace$ there exists a reproducing kernel $K(y,y')$ such that

$f(y) = \int f(y') K(y,y') dy'$

where

$K(y,y') = \sum_n f_n(y) g_n(y')$

and $\lbrace g_n\rbrace$ is the dual frame to $\lbrace f_n \rbrace$.

I assume that in general the kernel will not be a convolution kernel (ie $K(y,y′)=K′(y−y′)$). Is this true? Under what conditions is this frame-based kernel a convolution kernel?

Thank you