[Paper Review] Radial positive definite functions and Schoenberg matrices with negative eigenvalues

Leonid Golinskiĭ, M. M. Malamud+1 more|arXiv (Cornell University)|Feb 25, 2015

Spectral Theory in Mathematical Physics13 references3 citations

TL;DR

This paper investigates radial positive definite functions on R^n that cannot be extended to R^{n+1}, characterizing the class Φ_n \ Φ_{n+1} via Schoenberg's integral representation. It establishes necessary and sufficient conditions on the representing measure ν_n for f ∈ Φ_n \ Φ_{n+1}, proves that such functions have infinitely many negative squares (arbitrarily many negative eigenvalues in Schoenberg matrices), and provides explicit examples including Ω_n and Ω_n²(a·).

ABSTRACT

The main object under consideration is a class $Φ_n\backslashΦ_{n+1}$ of radial positive definite functions on $\R^n$ which do not admit \emph{radial positive definite continuation} on $\R^{n+1}$. We find certain necessary and sufficient conditions for the Schoenberg representation measure $ν_n$ of $f\in Φ_n$ in order that the inclusion $f\in Φ_{n+k}$, $k\in\N$, holds. We show that the class $Φ_n\backslashΦ_{n+k}$ is rich enough by giving a number of examples. In particular, we give a direct proof of $Ω_n\inΦ_n\backslashΦ_{n+1}$, which avoids Schoenberg's theorem, $Ω_n$ is the Schoenberg kernel. We show that $Ω_n(a\cdot)Ω_n(b\cdot)\inΦ_n\backslashΦ_{n+1}$, for $a ot=b$. Moreover, for the square of this function we prove surprisingly much stronger result: $Ω_n^2(a\cdot)\inΦ_{2n-1}\backslashΦ_{2n}$. We also show that any $f\inΦ_n\backslashΦ_{n+1}$, $n\ge2$, has infinitely many negative squares. The latter means that for an arbitrary positive integer $N$ there is a finite Schoenberg matrix $\kS_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X := \{x_j\}_{j=1}^m \subset\R^{n+1}$, which has at least $N$ negative eigenvalues.

Motivation & Objective

• To characterize the class Φ_n \ Φ_{n+1} of radial positive definite functions on R^n that do not admit radial positive definite continuation to R^{n+1}.
• To determine necessary and sufficient conditions on the Schoenberg measure ν_n for a function f ∈ Φ_n to lie in Φ_n \ Φ_{n+1}.
• To demonstrate that functions in Φ_n \ Φ_{n+1} have infinitely many negative squares, i.e., their Schoenberg matrices can have arbitrarily many negative eigenvalues.
• To provide explicit examples of functions in Φ_n \ Φ_{n+1}, including the Schoenberg kernel Ω_n and products like Ω_n(a·)Ω_n(b·) for a ≠ b.
• To establish a transition formula linking the Schoenberg measures ν_n and ν_m for m < n, revealing structural constraints on ν_n for non-extendability.

Proposed method

• Utilizes Schoenberg's integral representation f(r) = ∫₀^∞ Ω_n(rt) ν_n(dt), where Ω_n is the Schoenberg kernel involving Bessel functions.
• Applies the transition formula ν_m(dx) = p_m(x) dx with p_m(x) = [2x^{m-1} / B(m/2, (n−m)/2)] ∫_x^∞ (1 − x²/u²)^{(n−m)/2 − 1} ν_n(du)/u^m for m < n.
• Employs asymptotic expansions of Bessel functions J_{2k}(x) ∼ √(2/πx) cos(x − kπ − π/4) to analyze the sign oscillation of Fourier coefficients.
• Constructs explicit finite sets X ⊂ R^{n+1} to show that Schoenberg matrices S_X(Ω_n) have at least one negative eigenvalue, proving Ω_n ∈ Φ_n \ Φ_{n+1}.
• Analyzes the number of negative eigenvalues in Schoenberg matrices via the behavior of ∫ J_{2k}(2rs) ν(ds) as r → ∞, showing unbounded growth under certain measure conditions.
• Uses the theory of Stieltjes moment problems and properties of the Hankel transform to derive conditions on ν_n for non-extendability.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on the Schoenberg measure ν_n for a radial positive definite function f ∈ Φ_n to not belong to Φ_{n+1}?
• RQ2 Can the Schoenberg kernel Ω_n be shown to lie in Φ_n \ Φ_{n+1} without relying on Schoenberg’s theorem?
• RQ3 What is the maximal extension class for f = Ω_n²(a·), and how does its non-extendability relate to dimension?
• RQ4 How many negative eigenvalues can Schoenberg matrices of functions in Φ_n \ Φ_{n+1} have, and can this number be made arbitrarily large?
• RQ5 What structural properties of the measure ν_n (e.g., singular components or support near zero) imply that f ∉ Φ_{n+1}?

Key findings

• The Schoenberg kernel Ω_n belongs to Φ_n \ Φ_{n+1}, and this is proven directly via construction of a finite set X ⊂ R^{n+1} such that the Schoenberg matrix S_X(Ω_n) has at least one negative eigenvalue.
• For a ≠ b, the product Ω_n(a·)Ω_n(b·) lies in Φ_n \ Φ_{n+1}, showing that the class Φ_n \ Φ_{n+1} is rich and not limited to kernels.
• The square of the kernel satisfies Ω_n²(a·) ∈ Φ_{2n−1} \ Φ_{2n}, indicating a stronger non-extendability property than the original kernel.
• Any f ∈ Φ_n \ Φ_{n+1} with n ≥ 2 has infinitely many negative squares: for every N ∈ ℕ, there exists a finite Schoenberg matrix S_X(f) with at least N negative eigenvalues.
• If the Schoenberg measure ν_n(f) has a singular component or satisfies ν_n((0,ε)) = 0 for some ε > 0, then f ∉ Φ_{n+1}, providing a measure-theoretic obstruction to extension.
• Under the condition ∫_a^∞ ν(ds)/s^{3/2} < ∞ and ∫_x^∞ ν(ds)/s^{3/2} = o(x^{-1}) as x → 0, the number of negative eigenvalues in S_X(f) grows unboundedly as r → ∞, implying κ⁻(f) = ∞.