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The path of hyperinterpolation: A survey

Congpei An, Jiashu Ran, Hao-Ning Wu

TL;DR

The paper surveys hyperinterpolation, a discrete analogue of $L^2$ projection for multivariate polynomial approximation, built from quadrature rules with exactness for polynomials of degree $2n$. It discusses domain-specific constructions across intervals, circles, squares, disks, and spheres, and then introduces relaxing exactness via Marcinkiewicz--Zygmund inequalities to reduce quadrature requirements. Variants such as generalized, filtered, regularized, and efficient hyperinterpolation are presented, highlighting theoretical guarantees, practical performance, and connections to discrete Galerkin and Nyström methods. The survey also covers applications to partial differential and integral equations, illustrating how hyperinterpolation integrates quadrature into numerical analysis and study of singular or oscillatory kernels. Overall, hyperinterpolation has evolved into a flexible toolkit for high-dimensional approximation and numerical analysis, with ongoing challenges in constructing minimal or relaxed rules on complex domains and in adapting to nonlinear and singular problems.

Abstract

This paper surveys hyperinterpolation, a quadrature-based approximation scheme. We cover classical results, provide examples on several domains, review recent progress on relaxed quadrature exactness, introduce methodological variants, and discuss applications to differential and integral equations.

The path of hyperinterpolation: A survey

TL;DR

The paper surveys hyperinterpolation, a discrete analogue of projection for multivariate polynomial approximation, built from quadrature rules with exactness for polynomials of degree . It discusses domain-specific constructions across intervals, circles, squares, disks, and spheres, and then introduces relaxing exactness via Marcinkiewicz--Zygmund inequalities to reduce quadrature requirements. Variants such as generalized, filtered, regularized, and efficient hyperinterpolation are presented, highlighting theoretical guarantees, practical performance, and connections to discrete Galerkin and Nyström methods. The survey also covers applications to partial differential and integral equations, illustrating how hyperinterpolation integrates quadrature into numerical analysis and study of singular or oscillatory kernels. Overall, hyperinterpolation has evolved into a flexible toolkit for high-dimensional approximation and numerical analysis, with ongoing challenges in constructing minimal or relaxed rules on complex domains and in adapting to nonlinear and singular problems.

Abstract

This paper surveys hyperinterpolation, a quadrature-based approximation scheme. We cover classical results, provide examples on several domains, review recent progress on relaxed quadrature exactness, introduce methodological variants, and discuss applications to differential and integral equations.

Paper Structure

This paper contains 21 sections, 1 theorem, 41 equations.

Table of Contents

  1. Introduction
  2. Hyperinterpolation on various domains
  3. The interval
  4. The circle
  5. The square
  6. On the disk
  7. On the sphere
  8. Recent advances in relaxed quadrature exactness
  9. Marcinkiewicz--Zygmund (MZ) inequalities
  10. Hyperinterpolation using exactness-relaxed quadrature rules
  11. Significance of such relaxation
  12. Variants of hyperinterpolation
  13. Generalized hyperinterpolation
  14. Filtered hyperinterpolation
  15. Regularized hyperinterpolation
  16. ...and 6 more sections

Key Result

Theorem 4.1

Given a positive-weight quadrature rule equ:quad with exactness $2n$ on $\mathbb{S}^d$, the exact integration of both hyperinterpolant and filtered hyperinterpolant recovers the quadrature rule used in its construction in the sense of

Theorems & Definitions (1)

  • Theorem 4.1