Closed-Form Projection Method for Regularizing a Function Defined by a Discrete Set of Noisy Data and for Estimating its Derivative and Fractional Derivative

TL;DR

This paper introduces a closed-form projection technique for regularizing functions from noisy discrete data and accurately estimating their derivatives and fractional derivatives using low-degree polynomials.

Contribution

The method leverages known infinite-dimensional singular value decompositions to improve regularization and derivative estimation from noisy measurements.

Findings

Effective in regularizing noisy data

Accurate estimation of derivatives and fractional derivatives

Utilizes singular value decompositions for improved results

Abstract

We present a closed-form finite-dimensional projection method for regularizing a function defined by a discrete set of measurement data, which have been contaminated by random, zero mean errors, and for estimating the derivative and fractional derivative of this function by linear combinations of a few low degree trigonometric or Jacobi polynomials. Our method takes advantage of the fact that there are known infinite-dimensional singular value decompositions of the operators of integration and fractional integration.