The envelope of the error for trigonometric and Chebyshev interpolation

The error in Chebyshev or Fourier interpolation is the product of a rapidly varying factor with a slowly varying modulation. This modulation is the envelope of the error. Because this slow modulation controls the amplitude of the error, it is crucial to understand this error envelope. In this article, we show that the envelope varies strongly withx, but its variations can be predicted from the convergence-limiting singularities of the interpolated function f(x). In turn, this knowledge can be translated into a simple spectral correction algorithm for wringing more accuracy out of the same pseudospectral calculation of the solution to a differential equation.