Correction to: Numerical Algorithms https://doi.org/10.1007/s11075-020-00973-y

  • An equation in Section 2.1 has been corrected to

    $$ \begin{array}{@{}rcl@{}} | f(\pmb{x}) - s_{f}(\pmb{x}) | &=& \left| \left\langle f , K(\cdot, \pmb{x}) - \sum\limits_{k=1}^{n} K(\cdot, \pmb{x}_{k}) u_{k}(\pmb{x}) \right\rangle_{\mathcal{H}_{K}({\Omega})} \right| \\ &\leq& \| f \|_{\mathcal{H}_{K}({\Omega})} \left\| K(\cdot, \pmb{x}) - \sum\limits_{k=1}^{n} K(\cdot, \pmb{x}_{k}) u_{k}(\pmb{x}) \right\|_{\mathcal{H}_{K}({\Omega})}\\ &=:& \| f \|_{\mathcal{H}_{K}({\Omega})} P_{\mathcal{X}_{n}}(\pmb{x}) \end{array} $$

    from

    $$ \begin{array}{@{}rcl@{}} | f(\pmb{x}) - s_{f}(\pmb{x}) | &=& \left| \left\langle f K(\cdot, \pmb{x}) - \sum\limits_{k=1}^{n} K(\cdot, \pmb{x}_{k}) u_{k}(\pmb{x}) \right\rangle_{\mathcal{H}_{K}({\Omega})} \right| \\ &\leq& \| f \|_{\mathcal{H}_{K}({\Omega})} \left\| K(\cdot, \pmb{x}) - {\sum}_{k=1}^{n} K(\cdot, \pmb{x}_{k}) u_{k}(\pmb{x}) \right\|_{\mathcal{H}_{K}({\Omega})}\\ &=:& \| f \|_{\mathcal{H}_{K}({\Omega})} P_{\mathcal{X}_{n}}(\pmb{x}). \end{array} $$

  • An inline equation in Section 2.2 has been corrected to \(f = {\sum }_{\ell =1}^{\infty } \langle f, \varphi _{\ell } \rangle _{{\mathscr{H}}_{K}({\Omega })} \varphi _{\ell }\) from \(f = {\sum }_{\ell =1}^{\infty } \langle \varphi _{\ell } \rangle _{{\mathscr{H}}_{K}({\Omega })} \varphi _{\ell }\).

  • An equation in Section 2.2 has been corrected to

    $$ \begin{array}{@{}rcl@{}} \langle f, K(\cdot, \pmb{x}) \rangle_{\mathcal{H}_{K}({\Omega})} &=& \sum\limits_{\ell,k=1}^{\infty} \langle \varphi_{\ell}, \varphi_{k} \rangle_{\mathcal{H}_{K}({\Omega})} \langle f, \varphi_{\ell} \rangle_{\mathcal{H}_{K}({\Omega})} \varphi_{k}(\pmb{x}) \\ &=& \sum\limits_{\ell=1}^{\infty} \langle f, \varphi_{\ell} \rangle_{\mathcal{H}_{K}({\Omega})} \varphi_{\ell}(\pmb{x}) \\ &=& f(\pmb{x}) \end{array} $$

    from

    $$ \begin{array}{@{}rcl@{}} \langle K(\cdot, \pmb{x}) \rangle_{\mathcal{H}_{K}({\Omega})} &=& \sum\limits_{\ell,k=1}^{\infty} \langle \varphi_{\ell} \varphi_{k} \rangle_{\mathcal{H}_{K}({\Omega})} \langle \varphi_{\ell} \rangle_{\mathcal{H}_{K}({\Omega})} \varphi_{k}(\pmb{x}) \\ &=& \sum\limits_{\ell=1}^{\infty} \langle \varphi_{\ell} \rangle_{\mathcal{H}_{K}({\Omega})} \varphi_{\ell}(\pmb{x}) \\ &=& f(\pmb{x}). \end{array} $$

  • An inline equation in Section 3.2 has been corrected to \(f_{\ell } = \langle f, \varphi _{\ell } \rangle _{{\mathscr{H}}_{K}({\Omega })}\) from \(f_{\ell } = \langle \varphi _{\ell } \rangle _{{\mathscr{H}}_{K}({\Omega })}\).

  • An equation in Section 3.3 has been corrected to

    $$ \begin{array}{@{}rcl@{}} \mathcal{H}_{K}({\Omega}) = \left\{ f \in L^{2}(\mu) \colon \| f \|_{\mathcal{H}_{K}({\Omega})}^{2} = \sum\limits_{\ell=1}^{\infty} \frac{ \langle f, \psi_{\ell} \rangle_{L^{2}(\mu)}^{2}}{\lambda_{\ell}} < \infty \right\} \end{array} $$

    from

    $$ \begin{array}{@{}rcl@{}} \mathcal{H}_{K}({\Omega}) = \left\{ f \in L^{2}(\mu) \colon \| f \|_{\mathcal{H}_{K}({\Omega})}^{2} = \sum\limits_{\ell=1}^{\infty} \frac{ \langle f \psi_{\ell} \rangle_{L^{2}(\mu)}^{2}}{\lambda_{\ell}} < \infty \right\}. \end{array} $$

  • An equation in Section 3.3 has been corrected to

    $$ \begin{array}{@{}rcl@{}} T(L^{2}(\mu)) = \left\{ f \in L^{2}(\mu) \colon \| f \|_{\mathcal{H}_{K}({\Omega})}^{2} = \sum\limits_{\ell=1}^{\infty} \frac{ \langle f ,\psi_{\ell} \rangle_{L^{2}(\mu)}^{2}}{\lambda_{\ell}^{2}} < \infty \right\} \subset \mathcal{H}_{K}({\Omega}) \end{array} $$

    from

    $$ \begin{array}{@{}rcl@{}} T(L^{2}(\mu)) = \left\{ f \in L^{2}(\mu) \colon \| f \|_{\mathcal{H}_{K}({\Omega})}^{2} = \sum\limits_{\ell=1}^{\infty} \frac{ \langle \psi_{\ell} \rangle_{L^{2}(\mu)}^{2}}{\lambda_{\ell}^{2}} < \infty \right\} \subset \mathcal{H}_{K}({\Omega}) \end{array} $$

  • An equation in Section 4.3 has been corrected to

    $$ \begin{array}{@{}rcl@{}} | g_{i}(x_{i}) | = \left| \langle g, K_{i}(\cdot, x_{i}) \rangle_{\mathcal{H}_{K_{i}}({\Omega}_{i})} \right| \leq \| g_{i} \|_{ \mathcal{H}_{K_{i}}({\Omega}_{i})} \text{ and } | s_{i,g_{i}}(x_{i}) | \leq \| g_{i} \|_{\mathcal{H}_{K_{i}}({\Omega}_{i})} \end{array} $$

    from

    $$ \begin{array}{@{}rcl@{}} | g_{i}(x_{i}) | = \left| \langle K_{i}(\cdot, x_{i}) \rangle_{\mathcal{H}_{K_{i}}({\Omega}_{i})} \right| \leq \| g_{i} \|_{ \mathcal{H}_{K_{i}}({\Omega}_{i})} \text{ and } | s_{i,g_{i}}(x_{i}) | \leq \| g_{i} \|_{\mathcal{H}_{K_{i}}({\Omega}_{i})}. \end{array} $$