4,698 research outputs found
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Conformal Maps to Multiply-Slit Domains and Applications
By exploiting conformal maps to vertically slit regions in the complex plane, a recently developed rational spectral method [Tee and Trefethen, 2006] is able to solve PDEs with interior layer-like behaviour using significantly fewer collocation points than traditional spectral methods. The conformal maps are chosen to 'enlarge the region of analyticity' in the solution: an idea which can be extended to other numerical methods based upon global polynomial interpolation. Here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries, and apply them to some challenging differential equations
Asymptotic estimates for interpolation and constrained approximation in H2 by diagonalization of Toeplitz operators
Sharp convergence rates are provided for interpolation and approximation schemes in the Hardy space H-2 that use band-limited data. By means of new explicit formulae for the spectral decomposition of certain Toeplitz operators, sharp estimates for Carleman and Krein-Nudel'man approximation schemes are derived. In addition, pointwise convergence results are obtained. An illustrative example based on experimental data from a hyperfrequency filter is provided
Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory
This is a survey of some recent results concerning polynomial inequalities
and polynomial approximation of functions in the complex plane. The results are
achieved by the application of methods and techniques of modern geometric
function theory and potential theory
On the numerical stability of Fourier extensions
An effective means to approximate an analytic, nonperiodic function on a
bounded interval is by using a Fourier series on a larger domain. When
constructed appropriately, this so-called Fourier extension is known to
converge geometrically fast in the truncation parameter. Unfortunately,
computing a Fourier extension requires solving an ill-conditioned linear
system, and hence one might expect such rapid convergence to be destroyed when
carrying out computations in finite precision. The purpose of this paper is to
show that this is not the case. Specifically, we show that Fourier extensions
are actually numerically stable when implemented in finite arithmetic, and
achieve a convergence rate that is at least superalgebraic. Thus, in this
instance, ill-conditioning of the linear system does not prohibit a good
approximation.
In the second part of this paper we consider the issue of computing Fourier
extensions from equispaced data. A result of Platte, Trefethen & Kuijlaars
states that no method for this problem can be both numerically stable and
exponentially convergent. We explain how Fourier extensions relate to this
theoretical barrier, and demonstrate that they are particularly well suited for
this problem: namely, they obtain at least superalgebraic convergence in a
numerically stable manner
Limits of elliptic hypergeometric integrals
In math.QA/0309252, the author proved a number of multivariate elliptic
hypergeometric integrals. The purpose of the present note is to explore more
carefully the various limiting cases (hyperbolic, trigonometric, rational, and
classical) that exist. In particular, we show (using some new estimates of
generalized gamma functions) that the hyperbolic integrals (previously treated
as purely formal limits) are indeed limiting cases. We also obtain a number of
new trigonometric (q-hypergeometric) integral identities as limits from the
elliptic level.Comment: 41 pages LaTeX. Minor stylistic changes, statement of Theorem 4.7
fixe
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