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Recovery of differential operators from a noisy Fourier transform

Konstantin Yu. Osipenko

TL;DR

The paper addresses recovering linear differential operators from noisy Fourier data in the $L_2$ setting, formulating a general optimal recovery framework for operators with imperfect information. It develops explicit optimal recovery formulas for powers of generalized Laplace operators $\Lambda_\theta^{\eta/2}$ and Weil derivatives, and extends to $D^\alpha$, providing exact error values $E_p$ and concrete Fourier-domain estimators with multipliers. The results include sharp norm inequalities that relate the recovered operator to the noisy Fourier input and the reference operator, under natural integrability conditions, illustrating the stability of optimal recovery in high-dimensional Fourier settings. These contributions advance the theory of stable, data-driven reconstruction of differential operators from spectral measurements, with precise, computable error and estimator formulas.

Abstract

The paper concerns problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of generalized Laplace operators from a noisy Fourier transform in the $L_2$-metric.

Recovery of differential operators from a noisy Fourier transform

TL;DR

The paper addresses recovering linear differential operators from noisy Fourier data in the setting, formulating a general optimal recovery framework for operators with imperfect information. It develops explicit optimal recovery formulas for powers of generalized Laplace operators and Weil derivatives, and extends to , providing exact error values and concrete Fourier-domain estimators with multipliers. The results include sharp norm inequalities that relate the recovered operator to the noisy Fourier input and the reference operator, under natural integrability conditions, illustrating the stability of optimal recovery in high-dimensional Fourier settings. These contributions advance the theory of stable, data-driven reconstruction of differential operators from spectral measurements, with precise, computable error and estimator formulas.

Abstract

The paper concerns problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of generalized Laplace operators from a noisy Fourier transform in the -metric.
Paper Structure (4 sections, 7 theorems, 107 equations)

This paper contains 4 sections, 7 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Optimal recovery methods from a noisy Fourier transform
  3. Optimal recovery methods for $\Lambda_\theta^{\eta/2}$
  4. Optimal recovery methods for $D^\alpha$

Key Result

Theorem 1

Let $2<p\le\infty$, $\gamma\in(0,1)$. Assume that Then The method where is optimal. Moreover, the sharp inequality holds.

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • Corollary 2
  • Corollary 3
  • Theorem 3
  • proof
  • Corollary 4