Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

Abstract

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.

Generalized Discrete Fourier Transform on the Base of Lagrange and Hermite Interpolation Formulas

“…Our study is based on the Barycentric Lagrange Interpolation Formula, which has the appearance in [2,3,16]…”

“…A number of authors note the connection between DFT and the theory of trigonometric interpolation [2,3,18,19,20]. However, no proposals have been made to put the theory of interpolation as the basis for DFT.…”

“…The sources [1,5,7,14,17,[22][23][24][25][26][27] set out the theoretical foundations of DFT, FFT and FFT-algorithms and applications to different problems. The Lagrange interpolation formula is written in a baricentric form in the papers [2,16] and it is mostly convenient for our purpose. The Hermite interpolation formula is explicitly recorded for the case of setting the values of the signal and its derivatives in [15,18,19], which seems to be convenient for practical applications.…”