Hyperinterpolation beyond exact cubature: a spectral multiplier approach

Hao-Ning Wu

arxiv: 2605.17739 · v1 · pith:BDZEK22Onew · submitted 2026-05-18 · 🧮 math.NA · cs.NA

Hyperinterpolation beyond exact cubature: a spectral multiplier approach

Hao-Ning Wu This is my paper

Pith reviewed 2026-05-20 01:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords hyperinterpolationspectral multipliersSobolev discrepancycubaturespherescattered dataquasi-Monte Carlo

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The pith

Stable Sobolev approximation on the sphere from scattered data works without exact polynomial cubature by treating discretization error as a spectral multiplier on discrepancy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to perform hyperinterpolation on the sphere under weaker conditions than usual. It views the error from using non-exact sampling points as the result of applying a spectral multiplier operator to a measure of cubature discrepancy. This split lets the approximation quality depend mainly on the choice of multiplier while the sampling points only need to satisfy Sobolev discrepancy bounds. The method then yields stable estimates for a range of operators and links the setting to quasi-Monte Carlo designs.

Core claim

By interpreting the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure, the framework separates approximation properties of the underlying spectral operator from geometric properties of the sampling measure, leading to stable Sobolev approximation estimates under weak cubature assumptions formulated through Sobolev discrepancy estimates, without requiring exact polynomial cubature formulas or Marcinkiewicz-Zygmund inequalities.

What carries the argument

Spectral multiplier operator acting on the cubature discrepancy measure, which isolates approximation behavior from sampling geometry under Sobolev discrepancy bounds.

If this is right

The same estimates apply to sharp spectral projections, compactly supported smooth filters, Bessel potential operators, and heat kernel operators.
Uniform L^infty stability holds for sufficiently localized spectral multipliers.
Stable approximation from scattered data is possible without exact polynomial reproduction.
Hyperinterpolation, Sobolev discrepancy, and quasi-Monte Carlo designs are directly connected.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation of multiplier and discrepancy may allow construction of sampling sets that optimize only the geometric part.
Similar ideas could apply to approximation on other manifolds where exact cubature is unavailable.
Numerical tests could compare convergence rates for different multipliers when discrepancy is controlled.

Load-bearing premise

The discretization error can be expressed as the spectral multiplier operator applied to the cubature discrepancy measure.

What would settle it

Take a sequence of non-exact cubature measures on the sphere whose Sobolev discrepancy tends to zero and check whether the resulting discrete approximation operator remains bounded in Sobolev norm for a fixed spectral multiplier as the degree increases.

read the original abstract

We study hyperinterpolation and its spectral multiplier variants on the sphere under weak cubature assumptions formulated through Sobolev discrepancy estimates. In contrast with classical hyperinterpolation theory, our framework does not require exact polynomial cubature formulas or Marcinkiewicz--Zygmund inequalities. The main idea is to interpret the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure. This viewpoint separates approximation properties of the underlying spectral operator from geometric properties of the sampling measure, leading to stable Sobolev approximation estimates under weak cubature assumptions. The resulting theory applies to a broad class of spectral approximation operators, including sharp spectral projections, compactly supported smooth filters, Bessel potential operators, and heat kernel operators. For sufficiently localized spectral multipliers, we additionally obtain uniform $L^\infty$-stability of the corresponding discrete approximation operators. The results establish a direct connection between hyperinterpolation, Sobolev discrepancy, and quasi-Monte Carlo (QMC) designs, showing that stable approximation from scattered data can be achieved without exact polynomial reproduction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Spectral multipliers let hyperinterpolation on the sphere drop exact cubature and MZ inequalities by routing discretization error through Sobolev discrepancy.

read the letter

The main point is that this paper reframes hyperinterpolation on the sphere so it works with weaker sampling by treating the discretization error as a spectral multiplier applied to the cubature discrepancy measure. This move separates the operator's approximation behavior from the point set's geometry and produces stable Sobolev estimates without exact polynomial reproduction or Marcinkiewicz-Zygmund inequalities. The abstract spells out the separation clearly and shows it covers sharp projections, smooth filters, Bessel potentials, and heat kernels, plus L^infty stability once the multiplier is localized. The explicit link to quasi-Monte Carlo designs is a useful byproduct for anyone already using low-discrepancy points. The construction builds on existing discrepancy ideas without circular definitions, and the stress-test found no internal contradictions in the stated logic. That is real progress on the classical requirements. The soft spots are limited. The dependence of the constants on the size of the Sobolev discrepancy or on the localization parameter will need checking in the proofs; if those constants grow quickly with degree, the practical range could be narrower than the abstract suggests. No evidence of hidden restrictions on the point distributions appears in the setup, but the full derivations will show whether the estimates stay useful for mildly uniform scattered data. This paper is for approximation theorists and numerical analysts who work with spectral methods or scattered data on the sphere. Anyone already familiar with classical hyperinterpolation or QMC on manifolds will see the extension immediately. It deserves peer review because the perspective is distinct, the claims are internally consistent, and the framework is grounded enough to repay detailed referee attention.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework for hyperinterpolation and its spectral multiplier variants on the sphere under weak cubature assumptions expressed via Sobolev discrepancy estimates. By interpreting the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure, the approach separates the approximation properties of the spectral operator from the geometric properties of the sampling measure. This yields stable Sobolev approximation estimates without requiring exact polynomial cubature formulas or Marcinkiewicz-Zygmund inequalities. The theory is applied to sharp spectral projections, compactly supported smooth filters, Bessel potential operators, and heat kernel operators; for sufficiently localized multipliers, uniform L^∞ stability is obtained. Connections are drawn to quasi-Monte Carlo designs, demonstrating stable approximation from scattered data without exact polynomial reproduction.

Significance. If the central separation argument and resulting estimates are rigorously established with explicit constants, the work would meaningfully extend hyperinterpolation theory beyond classical exact-cubature settings, enabling stable approximations on the sphere from more general point sets. The explicit linkage to Sobolev discrepancy and QMC designs offers a fresh perspective that could improve error analysis for scattered-data problems in approximation theory and numerical analysis on manifolds.

major comments (2)

[§3] §3 (main construction): The interpretation of the discretization error as the action of a spectral multiplier on the cubature discrepancy measure is load-bearing for the separation claim, yet the manuscript does not supply an explicit operator definition or a quantitative bound showing how the Sobolev discrepancy controls the error term independently of the multiplier; without this, the stability estimates risk depending on hidden restrictions on the point set.
[Theorem 5.1] Theorem 5.1 (L^∞ stability): The localization condition for the multipliers is invoked to obtain uniform bounds, but the proof sketch does not clarify the precise dependence of the constant on the localization parameter and the Sobolev discrepancy; this interaction is central to the claim that weak cubature suffices and requires a fully detailed estimate.

minor comments (2)

[Abstract / §1] The abstract and introduction use 'sufficiently localized' without a quantitative definition; a precise statement (e.g., support size or decay rate) should appear before the main theorems.
[§2] Notation for the Sobolev discrepancy and the associated norms should include a brief comparison table or reference to the exact definition used in prior QMC literature to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions help clarify the central arguments, and we will revise the paper to incorporate explicit definitions and detailed estimates as outlined below.

read point-by-point responses

Referee: [§3] §3 (main construction): The interpretation of the discretization error as the action of a spectral multiplier on the cubature discrepancy measure is load-bearing for the separation claim, yet the manuscript does not supply an explicit operator definition or a quantitative bound showing how the Sobolev discrepancy controls the error term independently of the multiplier; without this, the stability estimates risk depending on hidden restrictions on the point set.

Authors: We agree that greater explicitness is needed to make the separation argument fully rigorous. In the revised manuscript we will add, in Section 3, a precise operator-theoretic definition of the spectral multiplier acting on the cubature discrepancy measure together with a quantitative lemma that bounds the resulting error solely in terms of the Sobolev discrepancy norm, independently of the particular multiplier (under the standing assumptions on the multiplier class). This will confirm that the stability estimates do not rely on hidden restrictions beyond the given discrepancy bound. revision: yes
Referee: [Theorem 5.1] Theorem 5.1 (L^∞ stability): The localization condition for the multipliers is invoked to obtain uniform bounds, but the proof sketch does not clarify the precise dependence of the constant on the localization parameter and the Sobolev discrepancy; this interaction is central to the claim that weak cubature suffices and requires a fully detailed estimate.

Authors: The observation is correct; the current proof sketch leaves the dependence implicit. We will expand the proof of Theorem 5.1 to a complete argument that explicitly tracks the constant’s dependence on the localization parameter of the multiplier and on the Sobolev discrepancy of the point set. The revised estimate will show that uniform L^∞ stability holds under the weak-cubature hypothesis without additional point-set restrictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper develops a framework interpreting discretization error as the action of a spectral multiplier on the cubature discrepancy measure, separating spectral approximation properties from sampling geometry under Sobolev discrepancy estimates. This construction relies on existing concepts of Sobolev discrepancy and spectral operators without reducing claims to fitted parameters, self-definitional loops, or load-bearing self-citations. The approach covers multiple operator classes and links to QMC designs through independent estimates, remaining self-contained against external benchmarks with no equations or premises collapsing to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the central framework rests on interpreting discretization error via spectral multipliers on discrepancy measures under Sobolev-based weak cubature assumptions. No free parameters, invented entities, or additional axioms are explicitly listed.

axioms (1)

domain assumption Weak cubature assumptions can be formulated through Sobolev discrepancy estimates
Invoked as the basis for the entire framework in the abstract, replacing exact cubature requirements.

pith-pipeline@v0.9.0 · 5704 in / 1191 out tokens · 36336 ms · 2026-05-20T01:26:21.120135+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The main idea is to interpret the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure. This viewpoint separates approximation properties of the underlying spectral operator from geometric properties of the sampling measure, leading to stable Sobolev approximation estimates under weak cubature assumptions.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The results establish a direct connection between hyperinterpolation, Sobolev discrepancy, and quasi-Monte Carlo (QMC) designs

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

C. An, J. Ran, and H.-N. Wu , The path of hyperinterpolation: a survey , Dolomites Res. Notes Approx., 18 (2025), pp. 135--145, https://doi.org/10.25430/pupj-DRNA-2025-1-11

work page doi:10.25430/pupj-drna-2025-1-11 2025
[2]

An and H.-N

C. An and H.-N. Wu , On the quadrature exactness in hyperinterpolation , BIT, 62 (2022), pp. 1899--1919, https://doi.org/10.1007/s10543-022-00935-x

work page doi:10.1007/s10543-022-00935-x 2022
[3]

An and H.-N

C. An and H.-N. Wu , Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere , J. Complexity, 80 (2024), p. 101789, https://doi.org/10.1016/j.jco.2023.101789

work page doi:10.1016/j.jco.2023.101789 2024
[4]

An and H.-N

C. An and H.-N. Wu , Is hyperinterpolation efficient in the approximation of singular and oscillatory functions? , J. Approx. Theory, 299 (2024), p. 106013, https://doi.org/10.1016/j.jat.2023.106013

work page doi:10.1016/j.jat.2023.106013 2024
[5]

J. S. Brauchart, J. Dick, E. B. Saff, I. H. Sloan, Y. G. Wang, and R. S. Womersley , Covering of spheres by spherical caps and worst-case error for equal weight cubature in S obolev spaces , J. Math. Anal. Appl., 431 (2015), pp. 782--811, https://doi.org/10.1016/j.jmaa.2015.05.079

work page doi:10.1016/j.jmaa.2015.05.079 2015
[6]

J. S. Brauchart, E. B. Saff, I. H. Sloan, and R. S. Womersley , Q MC designs: optimal order quasi M onte C arlo integration schemes on the sphere , Math. Comp., 83 (2014), pp. 2821--2851, https://doi.org/10.1090/S0025-5718-2014-02839-1

work page doi:10.1090/s0025-5718-2014-02839-1 2014
[7]

doi: 10.1007/978-1-4614-6660-4

F. Dai and Y. Xu , Approximation theory and harmonic analysis on spheres and balls , Springer Monographs in Mathematics, Springer, New York, 2013, https://doi.org/10.1007/978-1-4614-6660-4

work page doi:10.1007/978-1-4614-6660-4 2013
[8]

Delsarte, J.-M

P. Delsarte, J.-M. Goethals, and J. J. Seidel , Spherical codes and designs , Geom. Dedicata, 6 (1977), pp. 363--388, https://doi.org/10.1007/bf03187604

work page doi:10.1007/bf03187604 1977
[9]

Filbir and H

F. Filbir and H. N. Mhaskar , A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel , J. Fourier Anal. Appl., 16 (2010), pp. 629--657, https://doi.org/10.1007/s00041-010-9119-4

work page doi:10.1007/s00041-010-9119-4 2010
[10]

Filbir and H

F. Filbir and H. N. Mhaskar , Marcinkiewicz-- Z ygmund measures on manifolds , J. Complexity, 27 (2011), pp. 568--596, https://doi.org/10.1016/j.jco.2011.03.002

work page doi:10.1016/j.jco.2011.03.002 2011
[11]

Hesse and I

K. Hesse and I. H. Sloan , Hyperinterpolation on the sphere , in Frontiers in Interpolation and Approximation, vol. 282 of Pure Appl. Math. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2007, pp. 213--248

work page 2007
[12]

Kazashi , A fully discretised filtered polynomial approximation on spherical shells , J

Y. Kazashi , A fully discretised filtered polynomial approximation on spherical shells , J. Comput. Appl. Math., 333 (2018), pp. 428--441, https://doi.org/10.1016/j.cam.2017.11.005

work page doi:10.1016/j.cam.2017.11.005 2018
[13]

Q. T. Le Gia and H. N. Mhaskar , Localized linear polynomial operators and quadrature formulas on the sphere , SIAM J. Numer. Anal., 47 (2009), pp. 440--466, https://doi.org/10.1137/060678555

work page doi:10.1137/060678555 2009
[14]

S.-b. Lin, Y. G. Wang, and D.-X. Zhou , Distributed filtered hyperinterpolation for noisy data on the sphere , SIAM J. Numer. Anal., 59 (2021), pp. 634--659, https://doi.org/10.1137/19M1281095

work page doi:10.1137/19m1281095 2021
[15]

H. N. Mhaskar, F. J. Narcowich, J. Prestin, and J. D. Ward , Polynomial frames on the sphere , Adv. Comput. Math., 13 (2000), pp. 387--403, https://doi.org/10.1023/A:1016639802349

work page doi:10.1023/a:1016639802349 2000
[16]

H. N. Mhaskar, F. J. Narcowich, and J. D. Ward , Spherical M arcinkiewicz-- Z ygmund inequalities and positive quadrature , Math. Comp., 70 (2001), pp. 1113--1130, https://doi.org/10.1090/S0025-5718-00-01240-0

work page doi:10.1090/s0025-5718-00-01240-0 2001
[17]

Mont\'ufar and Y

G. Mont\'ufar and Y. G. Wang , Distributed learning via filtered hyperinterpolation on manifolds , Found. Comput. Math., 22 (2022), pp. 1219--1271, https://doi.org/10.1007/s10208-021-09529-5

work page doi:10.1007/s10208-021-09529-5 2022
[18]

F. J. Narcowich, P. Petrushev, and J. D. Ward , Localized tight frames on spheres , SIAM J. Math. Anal., 38 (2006), pp. 574--594, https://doi.org/10.1137/040614359

work page doi:10.1137/040614359 2006
[19]

Reimer , Hyperinterpolation on the sphere at the minimal projection order , J

M. Reimer , Hyperinterpolation on the sphere at the minimal projection order , J. Approx. Theory, 104 (2000), pp. 272--286, https://doi.org/10.1006/jath.2000.3454

work page doi:10.1006/jath.2000.3454 2000
[20]

I. H. Sloan , Polynomial interpolation and hyperinterpolation over general regions , J. Approx. Theory, 83 (1995), pp. 238--254, https://doi.org/10.1006/jath.1995.1119

work page doi:10.1006/jath.1995.1119 1995
[21]

I. H. Sloan , Polynomial approximation on spheres---generalizing de la V all\'ee- P oussin , Comput. Methods Appl. Math., 11 (2011), pp. 540--552, https://doi.org/10.2478/cmam-2011-0029

work page doi:10.2478/cmam-2011-0029 2011
[22]

I. H. Sloan and R. S. Womersley , The uniform error of hyperinterpolation on the sphere , in Advances in Multivariate Approximation, vol. 107 of Mathematical Research, Wiley-VCH,Berlin, 1999, pp. 289--306

work page 1999
[23]

I. H. Sloan and R. S. Womersley , Filtered hyperinterpolation: a constructive polynomial approximation on the sphere , GEM Int. J. Geomath., 3 (2012), pp. 95--117, https://doi.org/10.1007/s13137-011-0029-7

work page doi:10.1007/s13137-011-0029-7 2012
[24]

Wang and I

H. Wang and I. H. Sloan , On filtered polynomial approximation on the sphere , J. Fourier Anal. Appl., 23 (2017), pp. 863--876, https://doi.org/10.1007/s00041-016-9493-7

work page doi:10.1007/s00041-016-9493-7 2017

[1] [1]

C. An, J. Ran, and H.-N. Wu , The path of hyperinterpolation: a survey , Dolomites Res. Notes Approx., 18 (2025), pp. 135--145, https://doi.org/10.25430/pupj-DRNA-2025-1-11

work page doi:10.25430/pupj-drna-2025-1-11 2025

[2] [2]

An and H.-N

C. An and H.-N. Wu , On the quadrature exactness in hyperinterpolation , BIT, 62 (2022), pp. 1899--1919, https://doi.org/10.1007/s10543-022-00935-x

work page doi:10.1007/s10543-022-00935-x 2022

[3] [3]

An and H.-N

C. An and H.-N. Wu , Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere , J. Complexity, 80 (2024), p. 101789, https://doi.org/10.1016/j.jco.2023.101789

work page doi:10.1016/j.jco.2023.101789 2024

[4] [4]

An and H.-N

C. An and H.-N. Wu , Is hyperinterpolation efficient in the approximation of singular and oscillatory functions? , J. Approx. Theory, 299 (2024), p. 106013, https://doi.org/10.1016/j.jat.2023.106013

work page doi:10.1016/j.jat.2023.106013 2024

[5] [5]

J. S. Brauchart, J. Dick, E. B. Saff, I. H. Sloan, Y. G. Wang, and R. S. Womersley , Covering of spheres by spherical caps and worst-case error for equal weight cubature in S obolev spaces , J. Math. Anal. Appl., 431 (2015), pp. 782--811, https://doi.org/10.1016/j.jmaa.2015.05.079

work page doi:10.1016/j.jmaa.2015.05.079 2015

[6] [6]

J. S. Brauchart, E. B. Saff, I. H. Sloan, and R. S. Womersley , Q MC designs: optimal order quasi M onte C arlo integration schemes on the sphere , Math. Comp., 83 (2014), pp. 2821--2851, https://doi.org/10.1090/S0025-5718-2014-02839-1

work page doi:10.1090/s0025-5718-2014-02839-1 2014

[7] [7]

doi: 10.1007/978-1-4614-6660-4

F. Dai and Y. Xu , Approximation theory and harmonic analysis on spheres and balls , Springer Monographs in Mathematics, Springer, New York, 2013, https://doi.org/10.1007/978-1-4614-6660-4

work page doi:10.1007/978-1-4614-6660-4 2013

[8] [8]

Delsarte, J.-M

P. Delsarte, J.-M. Goethals, and J. J. Seidel , Spherical codes and designs , Geom. Dedicata, 6 (1977), pp. 363--388, https://doi.org/10.1007/bf03187604

work page doi:10.1007/bf03187604 1977

[9] [9]

Filbir and H

F. Filbir and H. N. Mhaskar , A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel , J. Fourier Anal. Appl., 16 (2010), pp. 629--657, https://doi.org/10.1007/s00041-010-9119-4

work page doi:10.1007/s00041-010-9119-4 2010

[10] [10]

Filbir and H

F. Filbir and H. N. Mhaskar , Marcinkiewicz-- Z ygmund measures on manifolds , J. Complexity, 27 (2011), pp. 568--596, https://doi.org/10.1016/j.jco.2011.03.002

work page doi:10.1016/j.jco.2011.03.002 2011

[11] [11]

Hesse and I

K. Hesse and I. H. Sloan , Hyperinterpolation on the sphere , in Frontiers in Interpolation and Approximation, vol. 282 of Pure Appl. Math. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2007, pp. 213--248

work page 2007

[12] [12]

Kazashi , A fully discretised filtered polynomial approximation on spherical shells , J

Y. Kazashi , A fully discretised filtered polynomial approximation on spherical shells , J. Comput. Appl. Math., 333 (2018), pp. 428--441, https://doi.org/10.1016/j.cam.2017.11.005

work page doi:10.1016/j.cam.2017.11.005 2018

[13] [13]

Q. T. Le Gia and H. N. Mhaskar , Localized linear polynomial operators and quadrature formulas on the sphere , SIAM J. Numer. Anal., 47 (2009), pp. 440--466, https://doi.org/10.1137/060678555

work page doi:10.1137/060678555 2009

[14] [14]

S.-b. Lin, Y. G. Wang, and D.-X. Zhou , Distributed filtered hyperinterpolation for noisy data on the sphere , SIAM J. Numer. Anal., 59 (2021), pp. 634--659, https://doi.org/10.1137/19M1281095

work page doi:10.1137/19m1281095 2021

[15] [15]

H. N. Mhaskar, F. J. Narcowich, J. Prestin, and J. D. Ward , Polynomial frames on the sphere , Adv. Comput. Math., 13 (2000), pp. 387--403, https://doi.org/10.1023/A:1016639802349

work page doi:10.1023/a:1016639802349 2000

[16] [16]

H. N. Mhaskar, F. J. Narcowich, and J. D. Ward , Spherical M arcinkiewicz-- Z ygmund inequalities and positive quadrature , Math. Comp., 70 (2001), pp. 1113--1130, https://doi.org/10.1090/S0025-5718-00-01240-0

work page doi:10.1090/s0025-5718-00-01240-0 2001

[17] [17]

Mont\'ufar and Y

G. Mont\'ufar and Y. G. Wang , Distributed learning via filtered hyperinterpolation on manifolds , Found. Comput. Math., 22 (2022), pp. 1219--1271, https://doi.org/10.1007/s10208-021-09529-5

work page doi:10.1007/s10208-021-09529-5 2022

[18] [18]

F. J. Narcowich, P. Petrushev, and J. D. Ward , Localized tight frames on spheres , SIAM J. Math. Anal., 38 (2006), pp. 574--594, https://doi.org/10.1137/040614359

work page doi:10.1137/040614359 2006

[19] [19]

Reimer , Hyperinterpolation on the sphere at the minimal projection order , J

M. Reimer , Hyperinterpolation on the sphere at the minimal projection order , J. Approx. Theory, 104 (2000), pp. 272--286, https://doi.org/10.1006/jath.2000.3454

work page doi:10.1006/jath.2000.3454 2000

[20] [20]

I. H. Sloan , Polynomial interpolation and hyperinterpolation over general regions , J. Approx. Theory, 83 (1995), pp. 238--254, https://doi.org/10.1006/jath.1995.1119

work page doi:10.1006/jath.1995.1119 1995

[21] [21]

I. H. Sloan , Polynomial approximation on spheres---generalizing de la V all\'ee- P oussin , Comput. Methods Appl. Math., 11 (2011), pp. 540--552, https://doi.org/10.2478/cmam-2011-0029

work page doi:10.2478/cmam-2011-0029 2011

[22] [22]

I. H. Sloan and R. S. Womersley , The uniform error of hyperinterpolation on the sphere , in Advances in Multivariate Approximation, vol. 107 of Mathematical Research, Wiley-VCH,Berlin, 1999, pp. 289--306

work page 1999

[23] [23]

I. H. Sloan and R. S. Womersley , Filtered hyperinterpolation: a constructive polynomial approximation on the sphere , GEM Int. J. Geomath., 3 (2012), pp. 95--117, https://doi.org/10.1007/s13137-011-0029-7

work page doi:10.1007/s13137-011-0029-7 2012

[24] [24]

Wang and I

H. Wang and I. H. Sloan , On filtered polynomial approximation on the sphere , J. Fourier Anal. Appl., 23 (2017), pp. 863--876, https://doi.org/10.1007/s00041-016-9493-7

work page doi:10.1007/s00041-016-9493-7 2017