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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:358

Article

Hasegawa, Takemitsu¹  Sugiura, Hiroshi² 

[1] Univ Fukui, Dept Informat Sci, Fukui 9108507, Japan

[2] Nanzan Univ, Dept Mechatron, Showa Ku, Nagoya, Aichi 4668673, Japan

关键词: Quadrature rule;    Principal value integral;    Oscillatory function;    Chebyshev interpolation;    Error analysis;    Uniform approximation;   

DOI  :  10.1016/j.cam.2019.02.012

来源: Elsevier

【 摘 要 】

For the finite Hilbert transform of oscillatory functions Q(f; c, omega) = f(-1)(1) f(x)e(i omega x)/(x-c)dt with a smooth function f and real omega not equal 0, for c is an element of (-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 p(n) 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(p(n); c, omega) congruent to Q(f; c, omega). We write Q(p(n); c, omega) 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(p(n); c, omega) is bounded uniformly with respect to c and omega. Numerical examples illustrate the performance of our method. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free