Multiscale Approximation and Reproducing Kernel Hilbert Space Methods

@article{Griebel2015MultiscaleAA,
  title={Multiscale Approximation and Reproducing Kernel Hilbert Space Methods},
  author={Michael Griebel and Christian Rieger and Barbara Zwicknagl},
  journal={SIAM J. Numer. Anal.},
  year={2015},
  volume={53},
  pages={852-873}
}
We consider reproducing kernels $K:\Omega\times \Omega \to \mathbb{R}$ in multiscale series expansion form, i.e., kernels of the form $K\left(\boldsymbol{x},\boldsymbol{y}\right)=\sum_{\ell\in\mathbb{N}}\lambda_\ell\sum_{j\in I_\ell}\phi_{\ell,j}\left(\boldsymbol{x}\right)\phi_{\ell,j}\left(\boldsymbol{y}\right)$ with weights $\lambda_\ell$ and structurally simple basis functions $\left\{\phi_{\ell,i}\right\}$. Here, we deal with basis functions such as polynomials or frame systems, where, for… 

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