Technology and Art
This article lays the ground for taking a second perspective to Kernel Functions using Mercer’s Theorem. We discussed this theorem in Functional Analysis: Norms, Operators, and Some Theorems briefly. We will see that Mercer’s Theorem applies somewhat more directly to the characterisation of Kernel Functions, and there is no need for an elaborate construction, like we do for Reproducing Kernel Hilbert Spaces. Before we do that, this post will lay out the mathematical concepts necessary for understanding the proof behind Mercer’s Theorem.
The specific posts discussing the background are:
It is also advisable (though not necessary) to review Kernel Functions with Reproducing Kernel Hilbert Spaces to contrast and compare that approach with the one shown here.
Here is the roadmap for understanding the concepts relating to Mercer’s Theorem.
Recall what Mercer’s Theorem states:
\[\kappa(x,y)=\sum_{i=1}^\infty \lambda_i \psi_i(x)\psi_i(y)\]where \(\kappa(x,y)\) is a positive semi-definite function and \(\psi_i(\bullet)\) is the \(i\)th eigenfunction. Note that this implies that there are an infinite number of eigenfunctions.
The Evaluation Functional is an interesting function: it takes another function as an input, and applies a specific argument to that function. As an example, if we have a function, like so:
\[f(x)=2x+3\]We can define an evaluation functional called \(\delta_3(f)\) such that:
\[\delta_3(f)=f(3)=2.3+3=9\]Here we will treat the Evaluation Functional in its functional form (the “formula view”, if you like). Is the graph of the Evaluation Functional continuous. We can prove that if a linear functional is bounded, then it is also continuous. In this case, we will prove that the Evaluation Functional is bounded in the function space \(\mathcal{H}\).