Uniform approximation to finite Hilbert transform of oscillatory functions and its algorithm

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For the finite Hilbert transform of oscillatory functions Q(f;c,ω)＝f^1_-1 f(x) e^iωx / (x－c) dt with a smooth function f and real ω ≠ 0, for c ∈ (-1,1) in the sense of Cauchy principal value or for c=±1 of Hadamard finite-part, we present an approximation method of Clenshaw–Curtis type and its algorithm. Interpolating f by a polynomial pn of degree n and expanding in terms of the Chebyshev polynomials with O(n log n) operations by the FFT, we obtain an approximation Q(pn;c,ω) ≅ Q(f;c,ω). We write Q(pn;c,ω) as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree n－1. We efficiently evaluate the oscillatory integral with a combination of authors’ previous method and Keller’s method. For f(z) analytic on the interval ［-1,1］in the complex plane z, the error of Q(pn;c,ω) is bounded uniformly with respect to c and ω. Numerical examples illustrate the performance of our method.

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