Let $ \mathcal{X} $ be a compact space equipped with a strictly positive finite Borel measure $ \mu $ and $ K: \mathcal{X} \times \mathcal{X} \to \mathbb{R} $ a continuous, symmetric, and positive definite function. Define the integral operator $ T_K: L^2(\mathcal{X}) \rightarrow L^2(\mathcal{X}) $ as
$$ [T_K f](\cdot) =\int_\mathcal{X} K(\cdot,t) f(t)\, d\mu(t) $$where $ L^2(\mathcal{X}) $ is the space of square integrable functions with respect to $ \mu $.
Then $ K $ can be written in terms of the eigenvalues and continuous eigenfunctions of $T_k$ as
$$ K(x,y) = \sum_{j=1}^\infty \lambda_j \, \phi_j(x) \, \phi_j(y) $$Let $\mathcal{X}$ be a nonempty set and $k$ a positive-definite real-valued kernel on $\mathcal{X} \times \mathcal{X}$ with corresponding reproducing kernel Hilbert space $H_k$. Given a training sample $(x_1, y_1), \dotsc, (x_n, y_n) \in \mathcal{X} \times \mathbb{R}$, a strictly monotonically increasing real-valued function $g \colon [0, \infty) \to \mathbb{R}$, and an arbitrary empirical risk function $E \colon (\mathcal{X} \times \mathbb{R}^2)^n \to \mathbb{R} \cup \lbrace \infty \rbrace$, then for any $f^{*} \in H_k$ satisfying
$$ f^{*} = \operatorname{arg min}_{f \in H_k} \left\lbrace E\left( (x_1, y_1, f(x_1)), ..., (x_n, y_n, f(x_n)) \right) + g\left( \lVert f \rVert \right) \right \rbrace, $$$f^{*}$ admits a representation of the form:
$$ f^{*}(\cdot) = \sum_{i = 1}^n \alpha_i k(\cdot, x_i), $$where $\alpha_i \in \mathbb{R}$ for all $1 \le i \le n$.