425 research outputs found
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 where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency 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 is the same as that of the best
approximation of the same degree when and the former is slower
than the latter by a factor of when , where
is the parameter in Gegenbauer polynomials. For piecewise analytic functions,
we demonstrate that the convergence rate of the Gegenbauer projection of degree
is the same as that of the best approximation of the same degree when
and the former is slower than the latter by a factor of
when . The extension to functions of fractional
smoothness is also discussed. Our theoretical findings are illustrated by
numerical experiments.Comment: 30 pages; 8 figure
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