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- W1974018534 abstract "In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$begin{aligned} H^{+}(f(t)e^{iomega t})(x)=-!!!!!!int nolimits _{!!!0}^{infty }e^{iomega t}frac{f(t)}{t-x},dt,quad omega >0,quad xge 0, end{aligned}$$ where the bar indicates the Cauchy principal value and $$f$$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $$x=0$$ , the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $$omega $$ are derived for each fixed $$xge 0$$ , which clarify the large $$omega $$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $$x$$ , we classify our discussion into three regimes, namely, $$x=mathcal O (1)$$ or $$xgg 1$$ , $$0<xll 1$$ and $$x=0$$ . Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $$omega $$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well." @default.
- W1974018534 created "2016-06-24" @default.
- W1974018534 creator A5022994939 @default.
- W1974018534 creator A5030512618 @default.
- W1974018534 creator A5070318438 @default.
- W1974018534 date "2012-10-16" @default.
- W1974018534 modified "2023-10-09" @default.
- W1974018534 title "Asymptotic expansions and fast computation of oscillatory Hilbert transforms" @default.
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- W1974018534 doi "https://doi.org/10.1007/s00211-012-0501-9" @default.
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