# Fourier's Series

@article{GibbsFouriersS, title={Fourier's Series}, author={Josiah Willard Gibbs}, journal={Nature}, volume={59}, pages={606-606} }

I SHOULD like to correct a careless error which I made (NATURE, December 29, 1898) in describing the limiting form of the family of curves represented by the equation as a zigzag line consisting of alternate inclined and vertical portions. The inclined portions were correctly given, but the vertical portions, which are bisected by the axis of X, extend beyond the points where they meet the inclined portions, their total lengths being expressed by four times the definite integral .

## 43 Citations

Bi-orthogonal wavelets for investigating Gibbs effects via oblique extension principle

- MathematicsJournal of Physics: Conference Series
- 2020

Gibbs effect is generally known for Fourier and Wavelets expansions of a function in the neighborhood of its discontinuities points which deals with the nonuniform convergence of its truncated sums…

On the Gibbs–Wilbraham Phenomenon for Sampling and Interpolatory Series

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2019

Abstract We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot…

Particle-without-Particle: A Practical Pseudospectral Collocation Method for Linear Partial Differential Equations with Distributional Sources

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2019

It is proved here that this method yields solutions to any linear PDE the source of which is any linear combination of delta distributions and derivatives thereof supported on a one-dimensional subspace of the problem domain, and generically obtain improved convergence rates relative to typical past implementations relying on delta function approximations.

Error Estimates for the Nearly Singular Momentum-Space Bound-State Equations

- Mathematics
- 2020

We present errors of quadrature rules for the nearly singular integrals in the momentum-space bound-state equations and give the critical value of the nearly singular parameter. We give error…

A Fourier Extension Based Numerical Integration Scheme for Fast and High-Order Approximation of Convolutions with Weakly Singular Kernels

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2019

This paper presents and analyzes an O(n\log n) scheme, based on a Fourier extension approach for removing unwanted oscillations, that not only converges with high-order but is also relatively simple to implement.

A reconstruction-based Chebyshev-collocation method for the Poisson equation: An accurate treatment of the Gibbs-Wilbraham phenomenon on irregular interfaces

- Computer Science, MathematicsJ. Comput. Phys.
- 2020

A reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely-differentiable smooth function and a modified Heaviside function, expressed in a weak form using the jump conditions across the interface, to resolve the discontinuity at the interface.

Gibbs phenomenon for p-ary subdivision schemes

- MathematicsJournal of Inequalities and Applications
- 2019

When a Fourier series is used to approximate a function with a jump discontinuity, the Gibbs phenomenon always exists. This similar phenomenon exists for wavelets expansions. Based on the Gibbs…

Boundary bound diffraction: a combined spectral and Bohmian analysis

- Physics, MathematicsPhysica Scripta
- 2019

The diffraction-like process displayed by a spatially localized matter wave is here analyzed in a case where the free evolution is frustrated by the presence of hard-wall-type boundaries (beyond the…

An exponentially accurate spectral reconstruction technique for the single-phase one-dimensional Stefan problem with constant coefficients

- Mathematics
- 2020

Abstract The Stefan problem represents a large class of physical phenomena ranging from heat diffusion during phase change to the shoreline movement problem. The numerical solution of the Stefan…

Bell-shaped proportional viscous damping models with adjustable frequency bandwidth

- Mathematics
- 2021

Abstract A proportional viscous damping model based on a bell-shaped basis function parameterized by the frequency and damping ratio at its peak has recently been proposed. The basis function in the…