Volume 58, pp. 271-288, 2023.

On trigonometric interpolation in an even number of points

Anthony P. Austin


In contrast to odd-length trigonometric interpolants, even-length trigonometric interpolants need not be unique; this is apparent from the representation of the interpolant in the (real or complex) Fourier basis, which possesses an extra degree of freedom in the choice of the highest-order basis function in the even case. One can eliminate this degree of freedom by imposing a constraint, but then the interpolant may cease to exist for certain choices of the interpolation points. On the other hand, the Lagrange representation developed by Gauss always produces an interpolant despite having no free parameters. We discuss the choice Gauss's formula makes for the extra degree of freedom and show that, when the points are equispaced, its choice is optimal in the sense that it minimizes both the standard and 2-norm Lebesgue constants for the interpolation problem. For non-equispaced points, we give numerical evidence that Gauss's formula is no longer optimal and consider interpolants of minimal 2-norm instead. We show how to modify Gauss's formula to produce a minimal-norm interpolant and that, if the points are equispaced, no modification is necessary.

Full Text (PDF) [363 KB], BibTeX

Key words

trigonometric interpolation, Lagrange interpolation, Lebesgue constant

AMS subject classifications

41A05, 42A15, 65D05

< Back