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arxiv: 1906.12305 · v1 · pith:VGFKAGTInew · submitted 2019-06-25 · 🧮 math.CA · cs.NA· math.NA

Orthogonal polynomials in and on a quadratic surface of revolution

Sheehan Olver , Yuan Xu This is my paper

Pith reviewed 2026-05-25 15:57 UTC · model grok-4.3

classification 🧮 math.CA cs.NAmath.NA
keywords orthogonal polynomialsquadratic surfaces of revolutionconeshyperboloidsparaboloidscubaturespherical harmonics
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The pith

Explicit orthogonal polynomials are constructed inside quadratic bodies of revolution and on their surfaces.

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

The paper gives explicit formulas for orthogonal polynomials over the interiors of cones, hyperboloids, and paraboloids. It likewise supplies orthogonal polynomials on the surfaces of these same quadratic surfaces of revolution, extending the role played by spherical harmonics on the sphere. These constructions are then applied to produce cubature rules and fast approximation schemes. A reader would care because the explicit forms remove the need to solve large linear systems for each new geometry in this family.

Core claim

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

What carries the argument

Coordinate systems adapted to quadratic surfaces of revolution in which the orthogonality measure factors and produces explicit polynomial solutions.

If this is right

  • Cubature formulas for numerical integration follow directly from the orthogonal polynomials on each surface and interior.
  • Fast approximation algorithms become available for functions defined inside or on cones, hyperboloids, and paraboloids.
  • The surface constructions supply a direct analogue of spherical harmonics for these non-spherical quadratic surfaces.
  • The same separation technique yields polynomial bases that respect the rotational symmetry of each body.

Where Pith is reading between the lines

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

  • The explicit bases may simplify spectral discretizations of partial differential equations posed on these surfaces.
  • Similar coordinate separations could be tested on other surfaces of revolution that are not quadratic.
  • The cubature rules could be compared against Monte-Carlo methods for accuracy on representative test integrals over cones and paraboloids.

Load-bearing premise

The geometry of each quadratic surface of revolution permits a coordinate change that separates the orthogonality integral into independent factors whose solutions are ordinary polynomials.

What would settle it

An explicit check that any of the proposed polynomial families fails to satisfy the stated inner-product orthogonality relation on the given domain or surface.

Figures

Figures reproduced from arXiv: 1906.12305 by Sheehan Olver, Yuan Xu.

Figure 1
Figure 1. Figure 1: Cone and Double Cone 3.4. Ball. If φ is given by φ(t) = √ 1 − t 2 for t ∈ (0, 1), then V d+1 is a ball V d+1 =  (x, t) : kxk 2 ≤ 1 − t 2 , 0 ≤ t ≤ 1, x ∈ B d [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Paraboloid and Hyperboloid 3.6. Hyperboloid of revolution. If φ is given by φ(t) = p t 2 + ρ 2 for t ∈ (a, b), then V d+1 is a hyperboloid V d+1 =  (x, t) : kxk 2 ≤ t 2 + ρ 2 , a ≤ t ≤ b, x ∈ R d [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hyperboloid and Hyperboloid of two sheets 3.7. Hyperboloid of two sheets. If φ is given by φ(t) = p t 2 − ρ 2 for t ∈ {t ∈ R : |t| > ρ}, then V d+1 is a hyperboloid of two sheets, V d+1 =  (x, t) : kxk 2 ≤ t 2 − ρ 2 , |t| ≥ ρ > 0, x ∈ R d [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Decay of numerically calculated coefficients on (left) and in (right) the cone, for f(x, y, t) = ex cos(20y−t) , g(x, y, t) = 1 x2+y2+(t−0.02)2 , h(x, y, t) = cos 100x(y−1) 1+50t , and r(x, y, t) = 1 x2+y2+(t+0.02)2 . We plot the 2-norm of the degree n coefficients. for three choices of f. Mimicking the univariate case, functions holomorphic in x, y, t in a neighbourhood of the cone are observed to achieve… view at source ↗
read the original abstract

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution (cones, hyperboloids, paraboloids) and on their surfaces, generalizing spherical harmonics, obtained via separation of variables in orthogonal curvilinear coordinates; these are then applied to cubature formulas and fast approximation methods.

Significance. If the constructions are as described, the work offers a useful extension of classical orthogonal polynomial theory to non-spherical quadratic domains. The direct verification of three-term recurrences and orthogonality integrals with respect to standard volume/surface measures, without hidden fitting parameters, is a clear strength and supports potential applications in numerical analysis on these geometries.

minor comments (2)
  1. The abstract and introduction could more explicitly name the curvilinear coordinate systems (e.g., conical, parabolic) employed for each surface to aid readers in locating the constructions.
  2. Notation for the weight functions and measures on the surfaces should be introduced consistently in the first section where the orthogonality integrals appear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's explicit constructions of orthogonal polynomials rely on separation of variables in curvilinear coordinates adapted to each quadratic surface of revolution (cones, hyperboloids, paraboloids), followed by direct verification that the resulting families satisfy the orthogonality integrals and three-term recurrences with respect to the standard measures. These steps are self-contained algebraic and integral identities that do not reduce to fitted parameters, self-citations, or redefinitions of the target quantities. No load-bearing premise is justified solely by prior work of the same authors, and the derivations remain independent of the final claimed families.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5575 in / 999 out tokens · 21028 ms · 2026-05-25T15:57:27.163473+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

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