Asian-European Journal of Mathematics Vol. 4, No. 1 (2011) 127–144 c  World Scientific Publishing Company DOI: 10.1142/S1793557111000113 FOURIER FORMULAE FOR EQUIDISTANT HERMITE TRIGONOMETRIC INTERPOLATION Arnak Poghosyan Institute of Mathematics, National Academy of Sciences, 24b Marshal Baghramian ave. Yerevan, 0019, Republic of Armenia arnak@instmath.sci.am Communicated by C. C. Yang Received February 15, 2010 Revised April 29, 2010 A sequence of Hermite trigonometric interpolation polynomials with equidistant inter- polation nodes and uniform multiplicities is investigated. We derive relatively compact formula that gives the interpolants as functions of the coefficients in the DFTs of the derivative values. The coefficients can be calculated by the FFT algorithm. Correspond- ing quadrature formulae are derived and explored. Convergence acceleration based on the Krylov-Lanczos method for accelerating both the convergence of interpolation and quadrature is considered. Exact constants of the asymptotic errors are obtained and some numerical illustrations are presented. Keywords : Hermite interpolation; trigonometric interpolation; Krylov-Lanczos interpo- lation; Bernoulli polynomials; convergence acceleration; Hermite-Krylov-Lanczos inter- polation; quadrature formula. AMS Subject Classification: 65T40, 42A15 1. Introduction For a given smooth function f ∈ C p-1 [−1, 1] we consider the sequence T p,N (f )(x), p ≥ 1, N ≥ 1, of Hermite trigonometric interpolation polynomials with prescribed values T (s) p,N (f )  2k 2N +1  = f (s)  2k 2N +1  ,s =0, ··· ,p − 1; |k|≤ N. The case p = 2 was first treated by Jackson [15]. The usual trigonometric interpo- lation is included as the particular case p = 1 (see Kress [18]). Salzer [32] considered the general case of full Hermite trigonometric interpo- lation with non-equidistant interpolation points. Trigonometric divided differences were used by Lyche [23] to derive a trigonometric analog of the Newton form of the Hermite polynomial. Interpolation methods of Hermite type in translation invariant 127 Asian-European J. Math. 2011.04:127-144. Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/04/15. For personal use only.