2023/08/21 by Mohan Zhao, Zhao, Mohan, Kirill Serkh +1
Mathematics · #Mathematical functions and polynomials #Mathematical Approximation and Integration #Iterative Methods for Nonlinear Equations
paper · pdf · doi:10.48550/arxiv.2308.10439
In this paper, we describe an algorithm for approximating functions of the form f(x)=∫ab xμ σ(μ) d μ over [0,1], where σ(μ) is some signed Radon measure, or, more generally, of the form f(x) = <σ(μ), xμ>, where σ(μ) is some distribution supported on [a,b], with 0 , where a≤ c ≤ b and m ≥ 0 is an integer. Given the desired accuracy ε and the values of a and b, our method determines a priori a collection of non-integer powers t1, t2, …, tN, so that the functions are approximated by series of the form f(x)≈ ∑j=1N cj xtj, and a set of collocation points x1, x2, …, xN, such that the expansion coefficients can be found by collocating the function at these points. We prove that our method has a small uniform approximation error which is proportional to ε multiplied by some small constants, and that the number of singular powers and collocation points grows as N=O(log\frac1ε). We demonstrate the performance of our algorithm with several numerical experiments.