On general Hermite trigonometric interpolation

Abstract

Summary.A sequence of general Hermite trigonometric interpolation polynomials with equidistant interpolation points is given. Integrating these interpolation formulae a sequence of quadrature formulae for the integration of periodic functions is obtained. Derivative-flee remainders are stated for these interpolation and quadrature formulae.Given an analytic function/: ]R-->• with period 2z~ we consider the sequence T~,,,,(/), p -----0, t, 2 ..... n = t, 2, ..., of general Hermite trigonometric interpolation polynomials of order (p + t)n with prescribed values

Tracing the Sources of Cosmic Rays With Molecular Ions

“…(Black 1998;Montmerle 2009)). Among the 18 molecular cations now identified in the interstellar medium, it is the reactive ions H + 3 , OH + , and H 2 O + that are expected to be the most specific tracers of the CR ionization rate (Black 1998(Black , 2007, and it was only during the last year that the latter two species have been detected (Benz et al 2010;Bruderer et al 2010;Gerin et al 2010;Gupta et al 2010;Kre lowski et al 2010;Neufeld et al 2010a;Ossenkopf et al 2010;Schilke et al 2010;Wyrowski et al 2010). Analysis of OH + and H 2 O + absorption in diffuse interstellar gas of low molecular fraction (H 2 /H < 10%) serves to calibrate the background ionizing frequency of interstellar hydrogen at a level ζ Gal = (0.6 − 2.4) · 10 −16 s −1 per atom (Gerin et al 2010;Neufeld et al 2010a,b).…”

The Lebesgue function for generalized Hermite-Fejér interpolation on the Chebyshev nodes

“…This suggests that significant improvements to the asymptotic result (8) will follow only if a different approach to estimating the A k-2m ,n (T, 1) is employed, whereby properties specific to the Chebyshev nodes are used. We have been able to achieve this by using explicit formulas for Hermite trigonometric interpolation based on equidistant points that were developed by KreB [11]. The following results, which are proved in Section 2 of this paper, have been obtained.…”

“…These properies are illustrated in Figure 1, which shows a graph of the Lebesgue function k Oi g(T, x) for Lagrange interpolation on eight Chebyshev nodes. (The graph also illustrates the 1.47 [11] 1.44 result (4) that the maximum of X On (T, x) on [-1, 1] occurs at ±1. However, it is not true in general that the maximum of A.…”

“…where the a m _ r are defined by (9) (see KreB [11,Theorem 1.1]). From this formula and the representation (14), the required expression (10) for the Lebesgue constant A 2m , n (r) follows immediately.…”

On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

“…[8].) The lemma is then established by substituting (2.2) into this representation of A k, 2m, n (cos %).…”