Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ 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 $x\geq 0$, 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 $x\gg1$, $0<x\ll 1$ 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.Comment: 32 pages, 6 figures, 4 table

Are best approximations really better than Chebyshev?

Best and Chebyshev approximations play an important role in approximation theory. From the viewpoint of measuring approximation error in the maximum norm, it is evident that best approximations are better than their Chebyshev counterparts. However, the situation may be reversed if we compare the approximation quality from the viewpoint of either the rate of pointwise convergence or the accuracy of spectral differentiation. We show that when the underlying function has an algebraic singularity, the Chebyshev projection of degree n converges one power of n faster than its best counterpart at each point away from the singularity and both converge at the same rate at the singularity. This gives a complete explanation for the phenomenon that the accuracy of Chebyshev projections is much better than that of best approximations except in a small neighborhood of the singularity. Extensions to superconvergence points and spectral differentiation, Chebyshev interpolants and other orthogonal projections are also discussed.Comment: 23 page

On the optimal rates of convergence of Gegenbauer projections

In this paper we present a comprehensive convergence rate analysis of Gegenbauer projections. We show that, for analytic functions, the convergence rate of the Gegenbauer projection of degree $n$ is the same as that of the best approximation of the same degree when $\lambda\leq0$ and the former is slower than the latter by a factor of $n^{\lambda}$ when $\lambda>0$, where $\lambda$ is the parameter in Gegenbauer polynomials. For piecewise analytic functions, we demonstrate that the convergence rate of the Gegenbauer projection of degree $n$ is the same as that of the best approximation of the same degree when $\lambda\leq1$ and the former is slower than the latter by a factor of $n^{\lambda-1}$ when $\lambda>1$. The extension to functions of fractional smoothness is also discussed. Our theoretical findings are illustrated by numerical experiments.Comment: 30 pages; 8 figure