Gauss-Hermite quadratures for functions from Hilbert spaces with Gaussian reproducing kernels
Let \(H(K_{\gamma})\) be the Hilbert space with the Gaussian reproducing kernel \(K_{\gamma}=\exp(-\gamma^2(x-y)^2)\) for \(x,\,y\in \mathbb R\) and \(\gamma >0\). For \(f\in H(K_{\gamma})\), the authors study Gauss-Hermite quadratures for the integral \[ I(f) = \int_{-\infty}^{\infty} f(x)\, \exp(-x^2/(2\alpha))\,dx \] with \(\alpha>0\). The Gauss-Hermite quadratures approximate \(I(f)\) by \[ Q_n(f) = \sum_{j=1}^n w_j\, f(x_j)\,, \] where \(w_j\) are positive weights and \(x_j\) are the zeros of the generalized Hermite polynomial \[ H_n^{[\alpha]}(x) = (2\alpha)^{-n/2}\, H_n(x\,(2\alpha)^{-1/2})\,. \] Here \(H_n\) denotes the standard Hermite polynomial. The authors prove an explicit formula for the error of Gauss-Hermite quadrature. Thus, for \(2\alpha \gamma^2 < 1\) one has an exponential convergence rate of the Gauss-Hermite quadratures as \(n\to \infty\). In the case \(2\alpha \gamma^2 = 1\) one has no convergence. For \(2\alpha \gamma^2 > 1\) one has an exponential rate of divergence.
scientific article; zbMATH DE number 6062472
scientific article