Algebraic Interpolation in Uniform Norm

Abstract

Chapter 4 treats algebraic interpolation processes in the uniform norm, starting with the so-called optimal system of nodes , which provides Lebesgue constants of order log n and the convergence of the corresponding interpolation processes. Moreover, the error of such an approximation is near to the error of the best uniform approximation. Beside two classical examples of the well-known optimal system of nodes (zeros of the Jacobi polynomials P ^(α,β) _n (x) (−1<α,β≤ − 1/2) and the so-called Clenshaw’s abscissas), we introduce more general results for constructing interpolation processes at nodes with an arc sine distribution having Lebsgue constants of order log n. The so-called additional nodes method with Jacobi zeros is presented in Sect. 4.2.2. Some other optimal interpolation processes and some simultaneous interpolation processes are analyzed in Sects. 4.2.3 and 4.2.4, respectively. The third section of this chapter is devoted to the weighted interpolation in the corresponding weighted spaces (Jacobi, Laguerre and Hermite cases). In addition, the weighted interpolation of functions with internal isolated singularities is considered in the last part of this chapter.