Continuous extension of positive-definiteness

Proved

ClockRoPE.posDef_continuous_extension

by Elsie66 · Sep 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

fourier-analysisharmonic-analysispositive-definite-kernels

This is the continuous analogue of the finite/discrete positive-definiteness condition IsPositiveDefiniteKernel (Definitions.Def_PositiveDefiniteKernel), used by Corollary 3.3 (ClockRoPE.periodic_posdef_fourier_coeffs_pmf_v2) of Chen et al., ClockRoPE: Random Fourier Rotations for Temporal Routine Modeling (arXiv:2607.26369).

Let $f:\mathbb R\to\mathbb R$ be continuous and positive-definite: for every finite family of points and complex coefficients, the associated Hermitian quadratic form $\sum_{i,j}\overline{c_i}c_j f(x_i-x_j)$ is real and nonnegative. The theorem asserts that this extends to arbitrary continuous test functions $\varphi:\mathbb R\to\mathbb C$ : for every $a\le b$ ,

\int_a^b\int_a^b \overline{\varphi(x)}\,\varphi(y)\,f(x-y)\,dx\,dy \in \mathbb R_{\ge 0}.

This is a standard fact in the theory of positive-definite functions (see e.g. Rudin, Fourier Analysis on Groups, §1.4): the finite quadratic form extends to the continuous double integral by a Riemann-sum/density argument (approximating $\varphi$ by its values on a fine equally-spaced partition of $[a,b]$ , applying the finite positive-definite condition to each partition, and passing to the limit using uniform continuity of $\varphi$ and $f$ on the compact square $[a,b]^2$ ). It underlies the passage from Bochner's theorem (continuous case) to Herglotz's theorem (periodic case, Corollary 3.3): evaluating it at the character $\varphi(x)=e^{-i2\pi kx/T}$ shows the periodic Fourier coefficients of a positive-definite kernel are nonnegative.

Formalization Note The conclusion is split into .im = 0 and 0 \le .re, mirroring how IsPositiveDefiniteKernel itself states realness and nonnegativity of the finite Hermitian form.

Preamble

import Mathlib
import Definitions.Def_PositiveDefiniteKernel

Formal statement

namespace ClockRoPE

open MeasureTheory

/-- **Continuous extension of positive-definiteness.** If `f : ℝ → ℝ` is continuous and
positive-definite, then for every continuous `φ : ℝ → ℂ` and every `a ≤ b`, the double integral
`∫_a^b ∫_a^b conj(φ(x)) φ(y) f(x - y) dx dy` — the continuous analogue of the finite Hermitian
quadratic form that `IsPositiveDefiniteKernel` requires to be real and nonnegative on every
finite point set — is itself real and nonnegative. This extends the finite/discrete
positive-definite condition to continuous test functions by a standard Riemann-sum/density
argument (approximate `φ` by its values on a fine equally-spaced partition and pass to the
limit using uniform continuity of `φ` and `f` on the compact square), and underlies the passage
from Bochner's theorem to Herglotz's theorem for the nonnegativity of the Fourier coefficients
of a periodic positive-definite function, needed for Corollary 3.3. -/
theorem posDef_continuous_extension
    (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_pd : IsPositiveDefiniteKernel f)
    (a b : ℝ) (φ : ℝ → ℂ) (hφ_cont : Continuous φ) :
    (∫ x in a..b, ∫ y in a..b,
        (starRingEnd ℂ) (φ x) * φ y * (f (x - y) : ℂ)).im = 0 ∧
    0 ≤ (∫ x in a..b, ∫ y in a..b,
        (starRingEnd ℂ) (φ x) * φ y * (f (x - y) : ℂ)).re := by
  sorry

end ClockRoPE

Source

Walter Rudin, Fourier Analysis on Groups, Interscience, 1962, Section 1.4 (positive-definite functions); standard extension of Bochner's theorem's finite condition to continuous test functions

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