Hilbert Space With Reproducing Kernel and Uniform Distribution Preserving Maps, Ii

For Hilbert space \(H\) with reproducing kernel \(K(\mathbf{x}, \mathbf{y})\),
we express the mean square worse-case error
\[
\int_{[0,1]^s} \sup_{\substack{f \in H \\ \|f\| \le 1}}
\left|
\frac{1}{N} \sum_{n=0}^{N-1} \Phi(\{\mathbf{x}_n + \boldsymbol{\sigma}\})
- \int_{[0,1]^s} f(\mathbf{x})\, d\mathbf{x}
\right|^2 d\boldsymbol{\sigma}
\]
as
\[
\frac{1}{N^2} \sum_{n,m=0}^{N-1} \int_{[0,1]^s}
K(\Phi(\mathbf{x}), \Phi(\mathbf{y}))\, d\mathbf{x}\, d\mathbf{y}\,
g_{m,n}(\mathbf{x}, \mathbf{y})
- \int_{[0,1]^{2s}} K(\mathbf{x}, \mathbf{y})\, d\mathbf{x}\, d\mathbf{y},
\]
where \(\Phi(\mathbf{x})\) is a uniform distribution preserving map,
\(\mathbf{x}_0, \ldots, \mathbf{x}_{N-1} \in [0,1]^s\), and
\(g_{m,n}(\mathbf{x}, \mathbf{y})\) are copulas associated with points
\(\mathbf{x}_m\) and \(\mathbf{x}_n\). Applying this, for dimension \(s = 1\),
we find that the minimum of the mean square worse-case error is attained in
the sequence \(x_n = \dfrac{n}{N}\), for the kernel
\(K(x, y) = 1 - \max(x, y)\), and \(\Phi(x) = x\).