Birkhoff interpolation

Abstract: This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.

Abstract: I Homogeneous Interpolation Problems with Standard Data.- 1. Null Structure for Analytic Matrix Functions.- 2. Null Structure and Interpolation Problems for Matrix Polynomials.- 3. Local Data for Meromorphic Matrix Functions.- 4. Rational Matrix Functions.- 5. Rational Matrix Functions with Null and Pole Structure at Infinity.- 6. Rational Matrix Functions with J-Unitary Values on the Imaginary Line.- 7. Rational Matrix Functions with J-Unitary Values on the Unit Circle.- II Homogeneous Interpolation Problems with Other Forms of Local Data.- 8. Interpolation Problems with Null and Pole Pairs.- 9. Interpolation Problems for Rational Matrix Functions Based on Divisibility.- 10. Polynomial Interpolation Problems Based on Divisibility.- 11. Coprime Representations and an Interpolation Problem.- III Subspace Interpolation Problems.- 12. Null-Pole Subspaces: Elementary Properties.- 13. Null-Pole Subspaces for Matrix Functions with J-Unitary Values on the Imaginary Axis or Unit Circle.- 14. Subspace Interpolation Problems.- 15. Subspace Interpolation with Data at Infinity.- IV Nonhomogeneous Interpolation Problems.- 16. Interpolation Problems for Matrix Polynomials and Rational Matrix Functions.- 17. Partial Realization as an Interpolation Problem.- V Nonhomogeneous Interpolation Problems with Metric Constraints.- 18. Matrix Nevanlinna-Pick Interpolation and Generalizations.- 19. Matrix Nevanlinna-Pick-Takagi Interpolation.- 20. Nehari Interpolation Problem.- 21. Boundary Nevanlinna-Pick Interpolation.- 22. Caratheodory-Toeplitz Interpolation.- VI Some Applications to Control and Systems Theory.- 23. Sensitivity Minimization.- 24. Model Reduction.- 25. Robust Stabilizations.- Appendix. Sylvester, Lyapunov and Stein Equations.- A.1 Sylvester equations.- A.2 Stein equations.- A.3 Lyapunov and symmetric Stein equations.- Notes.- References.- Notations and Conventions.

Abstract: We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.

Abstract: In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.