Birkhoff Interpolation

Abstract

This reference book provides the main definitions, theorems and techniques in the theory of Birkhoff interpolation by polynomials. The book begins with an article by G. G. Lorentz that discusses some of the important developments in approximation and interpolation in the last twenty years. It presents all the basic material known at the present time in a unified manner. Topics discussed include; applications of Birkhoff interpolation to approximation theory, quadrature formulas and Chebyshev systems; lacunary interpolation at special knots and an introduction to the theory of Birkhoff interpolation by splines.

Differentiation matrices in polynomial bases

“…This is an interesting polynomial interpolation that deserves a better consideration: the ''Hermite interpolation'' (See e.g., [2,10]). It is basically similar to the Lagrange interpolation, but at each node, we have the value of P(x) as well as its derivatives up to a certain order.…”

Approximation Properties of Quadrature Methods for Volterra Integral Equations of the First Kind

“…This then will complete the proof of Theorem 3.1 and Lemma 3.2. For all r > 1, the matrices E can be extended with columns of zeros to Poly a matrices without odd supported sequences, hence they are order regular, and the % exist and are unique, see Lorentz et al [12,Theorem 1.3], or the original reference Atkinson and Sharma [2]. D A consequence of the lemma is that the functions ^ depend linearly on z, so we may write them as (3.9)…”

“…Assume that the method (1.4) for the case $(x, y, z) = z is (i) zero-stable, i.e., (1.13) sup aT1! W"1!^ oo ; n (ii) asymptotically r-cyclic and composite of order p, i.e., (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) \W"{W«rnf-ft,*7)ll< c\\fM\\n-r-\ where / = l,2,...,r-1,…”

On Birkhoff Interpolations with Fractional-order Derivatives

“…But in general, we say the Birkhoff interpolation which means the other case. It turns out that the Birkhoff interpolation problem is difficult and causes many literatures [2,3,4,5,6,7]. In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution.…”