Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method
SIAM Journal on Numerical Analysis
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Accumulation of Round-Off Error in Fast Fourier Transforms
Journal of the ACM (JACM)
Numerical integration of functions with boundary singularities
Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions
Journal of Scientific Computing
Applied Numerical Mathematics
Uniform convergence of Fourier---Jacobi series
Journal of Approximation Theory
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Journal of Computational Physics
GNU Scientific Library Reference Manual - Third Edition
GNU Scientific Library Reference Manual - Third Edition
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We present a simple and fast algorithm for the computation of the Gegenbauer transform, which is known to be very useful in the development of spectral methods for the numerical solution of ordinary and partial differential equations of physical interest. We prove that the coefficients of the expansion of a function f(x) in Gegenbauer (also known as ultraspherical) polynomials coincide with the Fourier coefficients of a suitable integral transform of the function f(x). This allows to compute N Gegenbauer coefficients in O(Nlog"2N) operations by means of a single Fast Fourier Transform of the integral transform of f(x). We also show that the inverse Gegenbauer transform is expressible as the Abel-type transform of a suitable Fourier series. This fact produces a novel algorithm for the fast evaluation of Gegenbauer expansions.