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

Uniform Asymptotic Methods for Integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.Comment: 31 pages, 3 figure

Ping Pong Balayage and Convexity of Equilibrium Measures

In this presentation we prove that the equilibrium measure of a finite union of intervals on the real line or arcs on the unit circle has convex density. This is true for both, the classical logarithmic case, and the Riesz case. The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of many “electrons” will have convex density on every subinterval. Applications to external field problems and constrained energy problems are presented

On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented via nonoscillatory phase functions, even in the high-frequency regime. Our algorithm achieves near machine precision accuracy and the time required to compute one root of a solution is independent of the frequency of oscillations of that solution. Moreover, despite its great generality, our approach is competitive with specialized, state-of-the-art methods for the construction of Gaussian quadrature rules of large orders when it used in such a capacity. The performance of the scheme is illustrated with several numerical experiments and a Fortran implementation of our algorithm is available at the author's website

Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases

When solving differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials $T_{n}(x)$ with coefficients $a_{n}$ to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, $u(\pm 1)=0$, popular choices include the ``Chebyshev difference basis", $\varsigma_{n}(x) \equiv T_{n+2}(x) - T_{n}(x)$ with coefficients here denoted $b_{n}$ and the ``quadratic-factor basis functions" $\varrho_{n}(x) \equiv (1-x^{2}) T_{n}(x)$ with coefficients $c_{n}$. If $u(x)$ is weakly singular at the boundaries, then $a_{n}$ will decrease proportionally to $\mathcal{O}(A(n)/n^{\kappa})$ for some positive constant $\kappa$, where the $A(n)$ is a logarithm or a constant. We prove that the Chebyshev difference coefficients $b_{n}$ decrease more slowly by a factor of $1/n$ while the quadratic-factor coefficients $c_{n}$ decrease more slowly still as $\mathcal{O}(A(n)/n^{\kappa-2})$. The error for the unconstrained Chebyshev series, truncated at degree $n=N$, is $\mathcal{O}(|A(N)|/N^{\kappa})$ in the interior, but is worse by one power of $N$ in narrow boundary layers near each of the endpoints. Despite having nearly identical error \emph{norms}, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic-factor and difference basis sets is nearly uniform oscillations over the entire interval in $x$. Meanwhile, for Chebyshev polynomials and the quadratic-factor basis, the value of the derivatives at the endpoints is $\mathcal{O}(N^{2})$, but only $\mathcal{O}(N)$ for the difference basis

An explicit kernel-split panel-based Nystr\"om scheme for integral equations on axially symmetric surfaces

A high-order accurate, explicit kernel-split, panel-based, Fourier-Nystr\"om discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the evaluation of fields close to the boundary of the computational domain.Comment: 30 pages, 5 figures, errata correcte

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface