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High-order quasi-interpolation with generalized Gaussian kernels restricted over tori

Wenwu Gao, Zhengjie Sun, Changwei Wang

TL;DR

The paper tackles high-order periodic function approximation on $\mathbb{T}^d$ via quasi-interpolation using radial kernels. It develops a tensor-product kernel by restricting generalized Gaussian kernels to circles, proving the restricted kernel preserves the periodic Strang-Fix order and yields $O(N^{-s})$ convergence in Wiener spaces; it then extends to high dimensions with a sparse-grid variant to mitigate the curse of dimensionality. The main contributions are the construction of $\psi_{2m+2}$ with PSF order $2m+2$, the rigorous error bounds for the torus quasi-interpolant, and numerical demonstrations of accuracy and efficiency. This approach provides a simple, scalable alternative for high-accuracy periodic approximation on tori with potential broad applications in multidimensional signal processing and approximation theory.

Abstract

The paper proposes a novel and efficient quasi-interpolation scheme with high approximation order for periodic function approximation over tori. The resulting quasi-interpolation takes the form of Schoenberg's tensor-product generalized Gaussian kernels restricted over circles. Notably, theoretical analysis shows that it achieves the highest approximation order equal to the order of the generalized Strang-Fix condition satisfied by the generalized Gaussian kernels. This is in sharp contrast to classical quasi-interpolation counterparts, which often provide much lower approximation orders than those dictated by the generalized Strang-Fix conditions satisfied by the kernels. Furthermore, we construct a sparse grid counterpart for high-dimensional periodic function approximation to alleviate the curse of dimensionality. Numerical simulations provided at the end of the paper demonstrate that our quasi-interpolation scheme is simple and computationally efficient.

High-order quasi-interpolation with generalized Gaussian kernels restricted over tori

TL;DR

The paper tackles high-order periodic function approximation on via quasi-interpolation using radial kernels. It develops a tensor-product kernel by restricting generalized Gaussian kernels to circles, proving the restricted kernel preserves the periodic Strang-Fix order and yields convergence in Wiener spaces; it then extends to high dimensions with a sparse-grid variant to mitigate the curse of dimensionality. The main contributions are the construction of with PSF order , the rigorous error bounds for the torus quasi-interpolant, and numerical demonstrations of accuracy and efficiency. This approach provides a simple, scalable alternative for high-accuracy periodic approximation on tori with potential broad applications in multidimensional signal processing and approximation theory.

Abstract

The paper proposes a novel and efficient quasi-interpolation scheme with high approximation order for periodic function approximation over tori. The resulting quasi-interpolation takes the form of Schoenberg's tensor-product generalized Gaussian kernels restricted over circles. Notably, theoretical analysis shows that it achieves the highest approximation order equal to the order of the generalized Strang-Fix condition satisfied by the generalized Gaussian kernels. This is in sharp contrast to classical quasi-interpolation counterparts, which often provide much lower approximation orders than those dictated by the generalized Strang-Fix conditions satisfied by the kernels. Furthermore, we construct a sparse grid counterpart for high-dimensional periodic function approximation to alleviate the curse of dimensionality. Numerical simulations provided at the end of the paper demonstrate that our quasi-interpolation scheme is simple and computationally efficient.
Paper Structure (9 sections, 10 theorems, 88 equations, 4 figures, 1 table)

This paper contains 9 sections, 10 theorems, 88 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wiener spaces
  4. Periodic Strang-Fix conditions
  5. Our main results
  6. Generalized Gaussian kernels restricted over tori $\mathbb{T}^d$
  7. Quasi-interpolation with generalized Gaussian kernels restricted over tori $\mathbb{T}^d$
  8. Numerical examples
  9. Conclusions and discussions

Key Result

Lemma 2.1

Let $f\in A_q^{\mu}$ with $q\geq 1$, $\mu\geq \alpha\geq 0$, and $\mu> d(1-1/q)$. Suppose that the kernel $\chi_N$ of the above cardinal interpolation $I_Nf$ satisfies periodic Strang-Fix conditions of order $m\geq d/2$. Then for $\sigma=\min\{\mu,m+\alpha\}$, there exists a constant $C_{\sigma}$ su holds true for all $f\in A_q^{\mu}(\mathbb{T}^d)$.

Figures (4)

  • Figure 1: Approximation errors of the two-dimensional function $G_6(\mathbb{T}^2)$ by using quasi-interpolation with generalized Gaussian kernels satisfying different orders of periodic Strang-Fix conditions.
  • Figure 2: Relative $L_{\infty}$-norm approximation errors for approximating $G_6(\bm{\alpha})$ in different dimensions via sparse grid quasi-interpolation with restricted generalized Gaussian kernel $\psi_{2m+2}$.
  • Figure 3: Relative $L_2$-norm approximation errors for approximating $G_6(\bm{\alpha})$ in different dimensions via sparse grid quasi-interpolation with restricted generalized Gaussian kernel $\psi_{2m+2}$.
  • Figure 4: Relative $L_{\infty}$-norm approximation errors of sparse grid quasi-interpolation with restricted Gaussian kernels satisfying different orders of periodic Strang-Fix conditions for approximating $G_6(\bm{\alpha})$ in dimension three and dimension five.

Theorems & Definitions (17)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.1
  • ...and 7 more