A positive-definite Fourier transform, with applications to fractional Brownian motions and Bayesian probabilistic numerics
Abstract
This article is motivated by the use of Bayesian probabilistic numerics for the construction of numerical schemes for PDEs. To avoid the "flat limit problem" encountered for this problem, we consider Gaussian process priors with homogeneous covariance kernels. To construct such kernels, we introduce the positive definite Fourier transform, which defines an invertible map on the set of tempered positive semi-definite distributions onto itself. This enables us to construct an extension of the fractional Brownian motion kernel with arbitrary Hurst parameter H > 0, and arbitrary space dimension R^d , d > 0. Using such kernels, we show that Gaussian process regression with such kernels and with unknown polynomial mean function is mathematically equivalent to interpolation with fractional polyharmonic splines. This finally allows us to construct numerical schemes for the scalar linear transport equation, which exhibit fractional convergence rates for non-smooth initial data.
Related
- Distributionally Faithful Imputation via Positive Semi-Definite Kernel Density Estimation
- Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold
- Geodesics and intrinsic statistics of isolated connections in Hermitian positive-definite matrices
- Structured Positive-Definite Optimal Control Synthesis of Closed-Loop Recommendation Systems over Social Networks
- Sampling and Density Theorems for the Fractional Opdam-Cherednik Transform
BibTeX
@misc{henderson2026positivedefinite,
title = {{A positive-definite Fourier transform, with applications to fractional Brownian motions and Bayesian probabilistic numerics}},
author = {Iain Henderson},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749753v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A positive-definite Fourier transform, with applications to fractional Brownian motions and Bayesian probabilistic numerics
AU - Iain Henderson
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749753v1
AB - This article is motivated by the use of Bayesian probabilistic numerics for the construction of numerical schemes for PDEs. To avoid the "flat limit problem" encountered for this problem, we consider Gaussian process priors with homogeneous covariance kernels. To construct such kernels, we introduce the positive definite Fourier transform, which defines an invertible map on the set of tempered positive semi-definite distributions onto itself. This enables us to construct an extension of the fractional Brownian motion kernel with arbitrary Hurst parameter H > 0, and arbitrary space dimension R^d , d > 0. Using such kernels, we show that Gaussian process regression with such kernels and with unknown polynomial mean function is mathematically equivalent to interpolation with fractional polyharmonic splines. This finally allows us to construct numerical schemes for the scalar linear transport equation, which exhibit fractional convergence rates for non-smooth initial data.
ER -