On best approximation and estimates of Bohr--Favard type on the sphere for functions with bounded Laplacian

Technical Summary: Best Approximation and Estimates of Bohr–Favard Type on the Sphere for Functions with Bounded Laplacian

Problem Statement
This work addresses the determination of sharp constants in estimates for the uniform ($L^\infty$) norm of functions on the unit sphere $S^d$ ($d \ge 2$) with a bounded Laplacian ($\|\Delta f\|_\infty \le 1$). The study focuses on two distinct but related problems, analogous to the classical Bohr and Favard theorems for periodic functions:

1. The Bohr Problem: Estimating the norm of functions whose first $n-1$ spherical harmonic components vanish ($f \perp \mathcal{H}_{<n}$). The goal is to find the sharp constant $H(n, S^d)$ in the inequality $\|f\|_\infty \le H(n, S^d) \|\Delta f\|_\infty$.
2. The Favard Problem: Estimating the best uniform approximation error $E_{n-1}(f)_{L^\infty}$ of such functions by spherical polynomials of degree at most $n-1$. The goal is to find the sharp constant $K(n, S^d)$ in the inequality $E_{n-1}(f)_{L^\infty} \le K(n, S^d) \|\Delta f\|_\infty$.

The paper seeks to determine the exact values of these constants for fixed $n$ and their asymptotic behavior as $n \to \infty$, specifically investigating whether the constants for the approximation problem ($K$) coincide with or are strictly smaller than those for the norm problem ($H$), a distinction that differs from the classical periodic case where they are equal.

Methodology
The primary methodology relies on the Green representation of the inverse Laplacian. For functions of mean zero, $f$ can be expressed as a convolution of its Laplacian $g = -\Delta f$ with a zonal Green kernel $G_d$:
$$f(x) = \int_{S^d} G_d(x \cdot y) g(y) \, d\sigma_d(y).$$
The problem of bounding $\|f\|_\infty$ or the approximation error reduces to finding the best $L^1$ approximation of the Green kernel $G_d$ by polynomials of degree at most $n-1$ with respect to the induced measure on the sphere.

Key technical steps include:

• Kernel Interpolation: Identifying that the best $L^1$ approximant to the Green kernel is the interpolating polynomial at the zeros of a specific monic extremal polynomial $q_{n,d}$. These zeros are determined by vanishing moment conditions against the weight induced by the spherical measure.
• Extremal Polynomials: Utilizing Lemma 4.1 to establish the existence and uniqueness of the monic polynomial minimizing the weighted $L^1$ norm. The sign of the approximation error is shown to be the sign of this extremal polynomial, which allows for the construction of extremal functions (Bohr extremals) that achieve the bounds.
• Dimensional Analysis:
  • $S^2$: The problem is mapped to a trigonometric $L^1$ approximation problem on an equally spaced grid, leading to explicit series formulas involving Chebyshev nodes.
  • $S^3$: The interpolation nodes are determined by a weighted $L^1$ extremal problem. Rescaling the problem as $n \to \infty$ leads to the Hörmander–Bernhardsson function, a unique entire function of exponential type one that minimizes the $L^1$ norm on $\mathbb{R}$.
  • Higher Dimensions ($S^4, S^6$): The Green kernels in even dimensions are expressed as finite sums of poles and logarithms. The asymptotic constants are derived by analyzing the limiting behavior of the rescaled extremal polynomials, which converge to weighted entire extremals.
• Duality and Strict Inequality: To prove $K(n, S^d) < H(n, S^d)$, the paper employs a dual formulation involving measures annihilating $\mathcal{H}_{<n}$. It demonstrates that the equality of constants would require the signs of the kernel approximation errors at different poles to align perfectly, a condition shown to be impossible for $n \ge 2$ due to the geometric intersection properties of the sign-change sets (latitude spheres).

Key Contributions and Results

1. Exact Constants for Fixed $n$:

   • $S^2$: The constant $H(n, S^2)$ is given by an explicit convergent series (Theorem 7.1). The best coefficient for the asymptotic estimate is $H(S^2) = \frac{7\zeta(3)}{4}$.
   • $S^3$: The constant $H(n, S^3)$ is expressed via an integral involving the extremal polynomial $q_n$. The asymptotic constant $H(S^3)$ is identified as a positive integral involving the Hörmander–Bernhardsson function $\phi$ (Theorem 9.2), numerically evaluated as $\beta \approx 2.850999$.
   • $S^4$ and $S^6$: The asymptotic constants are derived in terms of the minimum norms of weighted entire extremal functions (Theorem 13.1). For $S^4$, the limit is $M_4/4$; for $S^6$, it involves a combination of a weighted norm and the second derivative of the extremal function at zero.

2. Uniform Bounds:
   The paper proves that the asymptotic limits derived are indeed the best coefficients valid simultaneously for all $n$. Specifically, for $S^3$, $S^4$, and $S^6$, the supremum of $n^2 H(n, S^d)$ equals the limit as $n \to \infty$ (Theorems 10.1, 14.1).

3. Strict Inequality between Bohr and Favard Constants:
   A central finding is that for every fixed $n \ge 2$ on $S^2$ and $S^3$, the best approximation constant is strictly smaller than the Bohr constant:
   $$K(n, S^d) < H(n, S^d).$$
   This contrasts with the classical periodic case where the constants coincide. The paper provides a quantitative gap for large $n$ on $S^2$ (Proposition 17.1) and proves the strict inequality via a compactness argument on the space of measures (Theorems 15.2 and 18.1).

4. Constructive Approximation:
   The paper provides an explicit linear operator for approximating functions on $S^2$ that is optimal among all linear methods mapping to $\mathcal{H}_{<n}$. This method involves convolving the Laplacian with the best $L^1$ approximant to the Green kernel (Theorem 15.1, Section 16).

Significance and Claims
The paper claims to resolve the determination of sharp Bohr-type constants on spheres for functions with bounded Laplacians, expressing them through weighted $L^1$ extremal polynomials. The most significant observation, described as "unexpected," is the strict separation between the Bohr constant (norm of functions with missing harmonics) and the Favard constant (best approximation error) for every fixed $n \ge 2$ on $S^2$ and $S^3$.

The work establishes that while the coincidence of these constants holds in the classical periodic setting, it fails on spheres. The paper provides explicit formulas for $S^2$, integral representations involving the Hörmander–Bernhardsson function for $S^3$, and asymptotic formulas for $S^4$ and $S^6$. It leaves open the question of whether the strict inequality persists in the limit (i.e., whether $\limsup n^2 K(n, S^d) < \lim n^2 H(n, S^d)$) and does not determine the exact values of $K(n, S^d)$ for $n \ge 2$, only providing lower bounds and proving the strict inequality exists.