# Multivariate Smoothing via the Fourier Integral Theorem and Fourier Kernel

arXiv: https://arxiv.org/abs/2012.14482 · HTML: https://mlsanity.com/paper/2012.14482/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2020.

## Built on by (most cited first, 5 of 5)

- [Transformer with Fourier Integral Attentions](https://mlsanity.com/paper/2206.00206/) (2022)
- [Statistical Analysis from the Fourier Integral Theorem](https://mlsanity.com/paper/2106.06608/) (2021)
- [On Integral Theorems and their Statistical Properties](https://mlsanity.com/paper/2107.10947/) (2021)
- [Bayesian Consistency with the Supremum Metric](https://mlsanity.com/paper/2201.03447/) (2022)
- [Smoothing-Based Conformal Prediction for Balancing Efficiency and Interpretability](https://mlsanity.com/paper/2509.22529/) (2025)

## Abstract (arXiv)

Starting with the Fourier integral theorem, we present natural Monte Carlo estimators of multivariate functions including densities, mixing densities, transition densities, regression functions, and the search for modes of multivariate density functions (modal regression). Rates of convergence are established and, in many cases, provide superior rates to current standard estimators such as those based on kernels, including kernel density estimators and kernel regression functions. Numerical illustrations are presented.