Closed-Form Projection Method for Regularizing a Function Defined by a Discrete Set of Noisy Data and for Estimating its Derivative and Fractional Derivative
For researchers dealing with noisy data and derivative estimation, this method offers a closed-form solution, but it is incremental as it builds on existing SVD theory.
The paper presents a closed-form projection method to regularize noisy discrete data and estimate derivatives and fractional derivatives using low-degree polynomials, leveraging known singular value decompositions of integration operators.
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.