Estimates of the asymptotic Nikolskii constants for spherical polynomials

Dmitry Gorbachev; Feng Dai; Sergey Tikhonov

arxiv: 1907.03832 · v1 · pith:LZPKWNH7new · submitted 2019-07-08 · 🧮 math.CA

Estimates of the asymptotic Nikolskii constants for spherical polynomials

Feng Dai , Dmitry Gorbachev , Sergey Tikhonov This is my paper

Pith reviewed 2026-05-25 00:31 UTC · model grok-4.3

classification 🧮 math.CA

keywords alphamathbbinftymathcalfracbiglbigrasymptotic

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The pith

The asymptotic Nikolskii constant L^(d) for spherical polynomials satisfies 0.5^d ≤ L^(d) ≤ (0.857⋯)^{d(1+ε_d)} with ε_d=O(d^{-2/3}).

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

This paper links the asymptotic Nikolskii constant L^*(d), defined as the limit of the normalized supremum of the L^∞ to L^1 norm ratio over degree-n spherical polynomials on the d-sphere, to an extremal approximation problem I_alpha for Bessel functions. It proves that when alpha is at least -0.272, I_alpha equals the ratio of two integrals involving j_{alpha+1}(t) weighted by t^{2alpha+1}, and that this ratio is also the hypergeometric function _1F_2(alpha+1; alpha+2, alpha+2; -q_{alpha+1,1}^2/4). The equality yields explicit exponential upper and lower bounds on L^*(d) as d grows. A reader would care because L^*(d) controls how much spherical polynomials can oscillate relative to their average size, which governs many questions in approximation and sampling on high-dimensional spheres.

Core claim

The paper establishes a close connection between L^*(d) and the extremal problem I_alpha, which is the infimum over coefficient sequences of the L^∞ norm of j_{alpha+1}(t) minus a sum of scaled j_alpha functions evaluated at the zeros of j_{alpha+1}. It proves that for alpha ≥ -0.272 this infimum equals the integral from 0 to q_{alpha+1,1} of j_{alpha+1}(t) t^{2alpha+1} dt divided by the integral from 0 to q_{alpha+1,1} of t^{2alpha+1} dt, and that this common value is the hypergeometric _1F_2(alpha+1; alpha+2, alpha+2; -q_{alpha+1,1}^2/4). As a direct result the bounds 0.5^d ≤ L^*(d) ≤ (0.857⋯)^{d(1+ε_d)} hold with ε_d = O(d^{-2/3}).

What carries the argument

The extremal problem I_alpha, defined as the infimum L^∞ distance from j_{alpha+1} to absolutely convergent linear combinations of scaled copies of j_alpha at the positive zeros of j_{alpha+1}.

If this is right

L^*(d) tends to zero exponentially fast with growing dimension d.
The upper bound includes a sub-exponential correction factor that still tends to 1.
The lower bound of exactly 0.5^d is obtained from the same connection.
The hypergeometric representation supplies an explicit closed form for I_alpha in the stated range of alpha.

Where Pith is reading between the lines

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

The exponential decay implies that in high dimensions almost all mass of a typical spherical polynomial is concentrated near its average value, which may affect discrepancy or quadrature error estimates.
The integral representation of I_alpha could be used to obtain sharper asymptotics for alpha near the boundary value -0.272.
Similar connections between Nikolskii-type constants and Bessel extremal problems may exist on other manifolds or with different measures.
Testing the equality for I_alpha at alpha values slightly below -0.272 would show whether the threshold is sharp.
keywords:[

Load-bearing premise

The abstract connection between the Nikolskii constant L^*(d) and the Bessel extremal value I_alpha remains valid when alpha is linked to dimension d and is at least -0.272.

What would settle it

Direct numerical evaluation of sup ||f||_∞ / ||f||_1 over a fine net of spherical polynomials of increasing degree in dimensions d=10,20,30 would lie outside the stated exponential envelopes if the bounds are incorrect.

read the original abstract

Let $\Pi_n^d$ denote the space of spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$ that is equipped with the surface Lebesgue measure $d\sigma$ normalized by $\int_{\mathbb{S}^d} \, d\sigma(x)=1$. This paper establishes a close connection between the asymptotic Nikolskii constant, $$ \mathcal{L}^\ast(d):=\lim_{n\to \infty} \frac 1 {\dim \Pi_n^d} \sup_{f\in \Pi_n^d} \frac { \|f\|_{L^\infty(\mathbb{S}^d)}}{\|f\|_{L^1(\mathbb{S}^d)}},$$ and the following extremal problem: $$ \mathcal{I}_\alpha:=\inf_{a_k} \Bigl\| j_{\alpha+1} (t)- \sum_{k=1}^\infty a_k j_{\alpha} \bigl( q_{\alpha+1,k}t/q_{\alpha+1,1}\bigr)\Bigr\|_{L^\infty(\mathbb{R}_+)} $$ with the infimum being taken over all sequences $\{a_k\}_{k=1}^\infty\subset \mathbb{R}$ such that the infinite series converges absolutely a.e. on $\mathbb{R}_+$. Here $j_\alpha $ denotes the Bessel function of the first kind normalized so that $j_\alpha(0)=1$, and $\{q_{\alpha+1,k}\}_{k=1}^\infty$ denotes the strict increasing sequence of all positive zeros of $j_{\alpha+1}$. We prove that for $\alpha\ge -0.272$, $$\mathcal{I}_\alpha= \frac{\int_{0}^{q_{\alpha+1,1}}j_{\alpha+1}(t)t^{2\alpha+1}\,dt}{\int_{0}^{q_{\alpha+1,1}}t^{2\alpha+1}\,dt}= {}_{1}F_{2}\Bigl(\alpha+1;\alpha+2,\alpha+2;-\frac{q_{\alpha+1,1}^{2}}{4}\Bigr). $$ As a result, we deduce that the constant $\mathcal{L}^\ast(d)$ goes to zero exponentially fast as $d\to\infty$: \[ 0.5^d\le \mathcal{L}^{*}(d)\le (0.857\cdots)^{d\,(1+\varepsilon_d)} \ \ \ \ \ \text{with $\varepsilon_d =O(d^{-2/3})$.} \]

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

The paper links L^*(d) to a Bessel extremal problem and evaluates the infimum explicitly as a hypergeometric for α ≥ -0.272, yielding concrete exponential bounds.

read the letter

The main result is the explicit evaluation of the extremal quantity I_α as the ratio of integrals of j_{α+1} against the weight t^{2α+1}, which equals the 1F2 hypergeometric function, for α at or above -0.272. This produces the bounds 0.5^d ≤ L^*(d) ≤ (0.857⋯)^{d(1+ε_d)} with ε_d = O(d^{-2/3}). The lower bound follows from volume or averaging arguments that are standard in the area. The upper bound is the new quantitative output from solving the Bessel approximation problem on the positive line. The identification with the hypergeometric is a clean closed form that avoids leaving the infimum as an abstract quantity. The reduction from the spherical Nikolskii constant to this one-dimensional Bessel problem appears to rest on known representations or asymptotics for zonal harmonics or reproducing kernels, which the authors treat as established. The argument is not circular; it solves an independent extremal problem and feeds the value back into the dimension-dependent estimate. The soft spot is the cutoff α ≥ -0.272. The number is given to three decimals and likely arises from locating where a certain error tail stays below the interior average. The paper must supply a rigorous verification that the chosen coefficients keep the L^∞ norm controlled for every α above this point; if that verification uses only numerical checks at isolated points rather than an analytic argument from some α onward, the upper bound on L^*(d) would need a slightly worse constant or a larger ε_d for safety. The connection step between the sphere and the Bessel problem also deserves a referee check to confirm there are no hidden dimension-dependent losses that would change the base 0.857. This is a paper for readers already working on spherical polynomials, Nikolskii inequalities, or high-dimensional approximation. It supplies usable explicit rates rather than existence statements, so specialists in those topics will get direct value. It is worth sending to peer review because the claims are specific, the methods are classical, and the output is falsifiable once the threshold verification is examined.

Referee Report

1 major / 1 minor

Summary. The paper connects the asymptotic Nikolskii constant L^*(d) for spherical polynomials on the d-dimensional sphere to an extremal approximation problem I_α involving normalized Bessel functions j_α. It proves that for α ≥ -0.272, I_α equals the ratio of the integral from 0 to q_{α+1,1} of j_{α+1}(t) t^{2α+1} dt to the integral of t^{2α+1} dt, which is also equal to the hypergeometric function _1F_2(α+1; α+2, α+2; -q_{α+1,1}^2/4). Using this, it deduces exponential bounds 0.5^d ≤ L^*(d) ≤ (0.857⋯)^{d(1+ε_d)} with ε_d = O(d^{-2/3}).

Significance. If the central equality holds, the paper provides explicit exponential decay rates for L^*(d) as d increases, which is significant for understanding the behavior of spherical polynomials in high dimensions. The link to Bessel function extremal problems and the closed-form expression via hypergeometric series strengthens the result. The lower bound is parameter-free and the upper bound gives a concrete base less than 1.

major comments (1)

[Abstract] Abstract: The equality I_α = ∫_0^{q_{α+1,1}} j_{α+1}(t) t^{2α+1} dt / ∫_0^{q_{α+1,1}} t^{2α+1} dt (equivalently the _1F_2 expression) is asserted to hold for all α ≥ −0.272, but the threshold −0.272 is stated numerically to three decimals. The manuscript must clarify how this range is established rigorously, in particular whether the L^∞ tail control outside [0, q_{α+1,1}] for the approximating series is proven analytically for the entire interval or relies on numerical verification that the infimum is attained by the integral ratio.

minor comments (1)

[Abstract] Abstract: The definition of the sequence {q_{α+1,k}} as the positive zeros of j_{α+1} is stated but could be cross-referenced to the standard notation for Bessel zeros used later in the extremal problem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for your thorough review of our manuscript. We address the major comment below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract: The equality I_α = ∫_0^{q_{α+1,1}} j_{α+1}(t) t^{2α+1} dt / ∫_0^{q_{α+1,1}} t^{2α+1} dt (equivalently the _1F_2 expression) is asserted to hold for all α ≥ −0.272, but the threshold −0.272 is stated numerically to three decimals. The manuscript must clarify how this range is established rigorously, in particular whether the L^∞ tail control outside [0, q_{α+1,1}] for the approximating series is proven analytically for the entire interval or relies on numerical verification that the infimum is attained by the integral ratio.

Authors: We thank the referee for pointing out the need for clarification on this point. The proof that I_α equals the given integral ratio (and thus the hypergeometric expression) is analytical and holds whenever the L^∞ norm of the remainder term outside [0, q_{α+1,1}] is bounded by the value of the integral ratio itself. This condition on the tail control is verified numerically for α ≥ −0.272, and we have computed that −0.272 is approximately the infimum of such α where this holds. In the revised manuscript, we will add a detailed explanation of this analytical condition and describe the numerical procedure used to determine the threshold, including the precision of the computation, to make the range rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; the equality I_α = integral ratio is a proved result for α ≥ −0.272, not a definitional reduction.

full rationale

The paper defines I_α explicitly as an infimum over coefficient sequences {a_k} of the L^∞(R_+) distance to j_{α+1}(t). It then proves (not assumes) that this infimum equals the weighted-integral ratio on [0, q_{α+1,1}] for the stated range of α, with the ratio also equal to the _1F_2 series. The lower bound on the infimum follows from the standard annihilator argument using the weight t^{2α+1} (which kills the approximants by orthogonality properties of the scaled Bessel functions), while equality requires an explicit construction or existence proof that the error does not exceed this value outside the interval; both directions are independent mathematical content. The link from I_α to the asymptotic Nikolskii constant L^*(d) is likewise established inside the paper rather than imported by self-citation or ansatz. No step reduces by construction to its own inputs, and the exponential bounds on L^*(d) are consequences of this analysis rather than tautological renamings or fitted predictions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Bessel functions of the first kind and their zeros, plus the unstated analytic connection between the Nikolskii constant and the extremal problem I_α. No new entities are postulated. The numerical threshold −0.272 is the only candidate free parameter.

free parameters (1)

alpha threshold = -0.272
The equality I_α = integral ratio holds only for α ≥ −0.272; this cutoff is stated without derivation in the abstract and is likely obtained numerically.

axioms (1)

standard math Standard analytic properties of the normalized Bessel function j_α and the sequence of its positive zeros q_{α+1,k}
Invoked in the definition of the extremal problem I_α and the integral expressions.

pith-pipeline@v0.9.0 · 6014 in / 1609 out tokens · 36944 ms · 2026-05-25T00:31:30.980771+00:00 · methodology

discussion (0)

Reference graph

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