A simple proof of Stolarsky's invariance principle (Q2838965)

In order to analyze the integration error by approximation by an equal weight integration rule for a set of \(N\) points \(\{\mathbf{z}_{k}\}_{k=0}^{N-1}\) on the \(d\)-dimensional sphere \(\mathbb{S}^{d}\), \(Q_{N}(f)=\frac{1}{N} \sum _{k=0}^{ N-1} f(\mathbf{z}_{k}),\) the worst-case error is defined by \(e(\mathcal{H}, Q_{N}) = \sup _{f\in\mathcal{H}, \|f\|\leq 1} | \int_{\mathbb{S}^{d}} f(\mathbf{x}) d\sigma_{d}(\mathbf{x}) - Q_{N}(f)|\), where \(\mathcal{H}\) is a \(\|.\|\)-normed function space and \(\sigma_{d}\) is the normalized Lebesgue surface area measure on \(\mathbb{S}^{d}\). Notice that the rate of decay of the worst-case error depends on the function space and the integration nodes. For a fixed space, this error can serve as a quality criterion for different sets of integration nodes, in the sense that the performance of a set of \(N\) quadrature points on the sphere can be compared to another set of \(N\) quadrature points on the sphere taking into account the corresponding worst-case errors.NEWLINENEWLINEA second quality criterion for points on the sphere is based on the Riesz \(s\)-energy of point configurations modeling unit charges which interact through such a kind of Riesz potential. It is well known from potential theory (see [\textit{G. Björck}, Ark. Mat. 3, 255--269 (1956; Zbl 0071.10105)]) that the normalized sum of distances of optimal \(N\)-point configurations \(\frac{1}{N^{2}} \sum _{j,k=0}^{N-1} \|\mathbf{z}_{j}- \mathbf{z}_{k}\|\), \(\mathbf{z}_{k}\in \mathbb{S}^{d}\), \(k= 0,1, \dotsc , N-1,\) approaches the distance integral \(\int_{\mathbb{S}^{d}} \int_ {\mathbb{S}^{d}} \|\mathbf{z}- \mathbf{x}\| d \sigma_{d} (\mathbf{z}) d \sigma_{d} (\mathbf{x})\), when \(N \rightarrow\infty\). The corresponding error also quantifies the quality of points on the sphere.NEWLINENEWLINEAnother quality criterion is the so-called spherical cap \(L_{2}\) discrepancy given by NEWLINE\[NEWLINEL_{2}(\mathbf{z}_{0}, \dotsc ,\mathbf{z}_{N-1}) = (\int_{-1}^{1} \int_ {\mathbb{S}^{d}} | \sigma_{d} (C(\mathbf{z};t)) - \frac{1}{N} \sum _{k=0}^{ N-1} 1_{C(\mathbf{z};t)} (\mathbf{z}_{k})|^{2} d\sigma_{d} (\mathbf{z}) dt) ^{1/2},NEWLINE\]NEWLINE where \(C(\mathbf{x};t) = \{\mathbf{z}\in \mathbb{S}^{d}, \langle \mathbf{x}, \mathbf{z} \rangle \geq t \}\) denotes the spherical cap centered at \(\mathbf{z}\in \mathbb{S}^{d}\) with height \(t\in [-1,1]\) and \(1_{J}\) is the indicator function for a set \(J\in\mathbb{S}^{d}\).NEWLINENEWLINEIn [\textit{K. B. Stolarsky}, Proc. Am. Math. Soc. 41, 575--582 (1973; Zbl 0274.52012)], it was proved that the deviation of the sum of distances of points on the sphere with respect to the (\(-1\))-energy is, up to a constant factor, the square of the spherical cap discrepancy. For some choice of function space, such a deviation also coincides, up to a constant factor, with the squared worst-case error as announced in a work in progress by \textit{J. S. Brauchart} and \textit{R. S. Womersley} [``Numerical integration over the unit sphere, \(L_{2}\) discrepancy and sum of distances''].NEWLINENEWLINEIn the paper under review, a simple and elegant proof of the above results based on the theory of reproducing kernel Hilbert spaces is deduced. Such Hilbert spaces are associated with a symmetric and positive reproducing kernel function \(K_{C}(\mathbf{x},\mathbf{y}) = \int_{-1}^{1} \int_ {\mathbb{S}^{d}} 1_{C(\mathbf{x};t)} (\mathbf{z}) 1_{C(\mathbf{y};t)} (\mathbf{z}) d\sigma_{d} (\mathbf{z}) dt\), \(\mathbf{x}, \mathbf{y} \in \mathbb{S}^{d}\). Finally, an extension of the above results to weighted reproducing kernel Hilbert spaces is given.