Applied Mathematics and Computation
Sum-accelerated pseudospectral methods: the Euler-accelerated sinc algorithm
Applied Numerical Mathematics
Family of spectral filters for discontinuous problems
Journal of Scientific Computing
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Mathematics of Computation
Analysis and Application of Fourier--Gegenbauer Method to Stiff Differential Equations
SIAM Journal on Numerical Analysis
Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method
SIAM Journal on Numerical Analysis
Exponentially Accurate Approximations to Piece-Wise Smooth Periodic Functions
Journal of Scientific Computing
On the Gibbs Phenomenon and Its Resolution
SIAM Review
On a High Order Numerical Method for Solving Partial Differential Equations in Complex Geometries
Journal of Scientific Computing
Determination of the jumps of a bounded function by its Fourier series
Journal of Approximation Theory
On a high order numerical method for functions with singularities
Mathematics of Computation
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
Journal of Computational and Applied Mathematics
Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images
SIAM Journal on Numerical Analysis
Inverse Polynomial Reconstruction of Two Dimensional Fourier Images
Journal of Scientific Computing
On a Rational Linear Approximation of Fourier Series for Smooth Functions
Journal of Scientific Computing
One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation
Journal of Computational Physics
IEEE Transactions on Image Processing
Lattice Sums for the Helmholtz Equation
SIAM Review
Hi-index | 31.45 |
If the coefficients in a Fourier cosine series, f(x)~f"N=@?"n"="0^~a"ncos(nx), decrease as a small negative power of n, then one may need millions of terms to sum the series to high accuracy. We show that if the a"n are known analytically and have a power series in 1/n, then it is straightforward to approximate f(x) as a series of what we shall the Lanczos-Krylov (LK) functions. (We describe the similar methodology for sine series; general Fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled, too.) For cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n, the LK functions may be expressed in terms of Bernoulli polynomials. The LK functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly; these are also known as ''Clausen functions''. We provide rapidly convergent series to compute these Clausen functions to high accuracy. Our method includes the ''endpoint subtraction'' ideas of Lanczos and Krylov, but is more general. The sum @?"n"="1^~(+/-1)^n^+^1(1/(n+@l))cos(nx), where @l0 is a constant, arises in phase transitions in absorbed monolayers on metal surfaces. It is easily summed by our method, which correctly incorporates the logarithmic singularities at x=+/-@p.