Quasi-interpolation using generalized Gaussian kernels

This paper focuses on developing a framework for constructing quasi-interpolation with the highest achievable approximation order from generalized Gaussian kernels with the help of kernel restriction trick and periodization technique. We first demonstrate that when we restrict generalized Gaussian kernels satisfying generalized Strang-Fix conditions of order s over a torus, the corresponding restricted kernels in tensor-product forms fulfill periodic Strang-Fix conditions of the same order s. Then, based on these restricted kernels, we construct a periodic quasi-interpolant in Schoenberg’s form and derive its error estimates for periodic function approximation over a torus, which reveals that our quasiinterpolant attains the highest approximation order s. Finally, using the periodization technique, we extend the periodic quasi-interpolant to its nonperiodic counterpart with the highest approximation order s for approximating a general function defined over a cube via a torus-to-cube transformation. This result stands in stark contrast to classical quasi-interpolation counterparts, which often yield much lower approximation orders than those dictated by the generalized Strang-Fix conditions of generalized Gaussian kernels. Furthermore, we propose a sparse grid counterpart for high-dimensional function approximation to alleviate the curse of dimensionality. Numerical simulations confirm that our quasi-interpolation scheme is simple and computationally efficient.

💡 Research Summary

This paper presents a comprehensive framework for constructing quasi‑interpolation schemes that achieve the highest possible approximation order when using generalized Gaussian kernels. The authors begin by recalling the generalized Strang‑Fix conditions, which characterize the polynomial reproduction capability of a kernel and directly determine the asymptotic convergence rate of a quasi‑interpolant. While many radial basis functions satisfy these conditions, classical quasi‑interpolation formulas based on them often fall short of the theoretical order, especially in high dimensions.

To overcome this limitation, the authors focus on a two‑dimensional generalized Gaussian kernel
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