On the Approximation of Singular Functions by Series of Non-integer Powers
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In this paper, we describe an algorithm for approximating functions of the form f(x)=∫_a^b x^μσ(μ) d μ over [0,1] ⊂ℝ, where 0<a<b<∞ and σ(μ) is some signed Radon measure over [a,b] or some distribution supported on [a,b]. Given the desired accuracy ϵ and the values of a and b, our method determines a priori a collection of non-integer powers {t_j}_j=1^N, so that the functions are approximated by series of the form f(x)≈∑_j=1^N c_j x^t_j, where the expansion coefficients can be found by solving a square, low-dimensional Vandermonde-like linear system using the collocation points {x_j}_j=1^N, also determined a priori by ϵ and the values of a and b. We prove that our method has a small uniform approximation error which is proportional to ϵ multiplied by some small constants. We demonstrate the performance of our algorithm with several numerical experiments, and show that the number of singular powers and collocation points grows as N=O(log1/ϵ).
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