equispaced

Equispaced refers to a set of points that are evenly distributed along a one-dimensional axis or in a multidimensional grid, with a constant spacing between consecutive points. In one dimension, a sequence {x_i} is equispaced if x_{i+1}-x_i = h for all i, where h>0 is the spacing, typically on a closed interval [a,b] with x_i = a + i h and h = (b-a)/n for i = 0,...,n.

Equispaced points arise frequently in numerical analysis, sampling, and signal processing. They are easy to construct and index, and support simple algorithms. In interpolation, equispaced nodes are convenient for Newton or Lagrange interpolation and for finite-difference schemes. However, they can suffer from Runge's phenomenon when used for high-degree global polynomial interpolation on large intervals, making nonuniform node distributions such as Chebyshev nodes preferable in some cases.

In numerical integration, equispaced nodes underpin composite rules such as the trapezoidal rule and, with appropriate weights, Simpson's rule. The uniform grid makes weight computation straightforward and enables efficient implementation.

In digital signal processing, equispaced samples in time are standard, enabling direct use of the discrete Fourier transform (DFT) and fast Fourier transform (FFT). Irregular or nonuniform sampling requires resampling or specialized transforms.

In higher dimensions, equispaced grids form regular Cartesian grids with uniform spacing along each axis, used in finite-difference and finite-element methods for solving partial differential equations.

---