@article{oai:u-fukui.repo.nii.ac.jp:02000113,
 author = {Hasegawa, Takemitsu and Sugiura, Hiroshi},
 journal = {Journal of Computational and Applied Mathematics},
 month = {Oct},
 note = {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.},
 pages = {327--342},
 title = {Uniform approximation to finite Hilbert transform of oscillatory functions and its algorithm},
 volume = {358},
 year = {2019}
}