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PreprintCC BY-NC-SA 4.0arXiv

From Pólya's Conditions to a Complete Characterization of the L2 Convergence of Hyperinterpolation

  • Congpei An
  • Xiannan Hu
  • Xiaoming Yuan
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Abstract

It has remained open to identify the necessary and sufficient conditions for the L2 convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the L1-L2 Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal L1-L2 MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya's classical conditions for quadrature convergence are not sufficient for the L2 convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.

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Cite

BibTeXRIS
BibTeX
@misc{an2026from,
  title = {{From Pólya's Conditions to a Complete Characterization of the $L^2$ Convergence of Hyperinterpolation}},
  author = {Congpei An and Xiannan Hu and Xiaoming Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.00525},
  eprint = {2608.00525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.00525},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - From Pólya's Conditions to a Complete Characterization of the L^2 Convergence of Hyperinterpolation
AU  - Congpei An
AU  - Xiannan Hu
AU  - Xiaoming Yuan
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.00525
UR  - https://arxiv.org/abs/2608.00525
AB  - It has remained open to identify the necessary and sufficient conditions for the L^2 convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the L^1-L^2 Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal L^1-L^2 MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya's classical conditions for quadrature convergence are not sufficient for the L^2 convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.
ER  -