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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.
Table of contents:
Editor's statement; Preface; Introduction; 1. Basic definitions and properties; 2. Further elementary theorems; 3. Coalescence of rows; 4. Applications of coalescence; 5. Rolle extensions and independent sets of knots; 6. Singluar matrice; 7. Zeros of Birkhoff splines; 8. Almost0hermitian matrices; special three-row matrices; 9. Applications; 10. Birkhoff quadrature formulas; 11. Interpolation at the roots of unity; 12. Turán's problem of (0,2) interpolation; 13. Birkhoff interpolation by splines; 14. Regularity theorems and self-dual problems; Bibliography and references; Symbol index; Subject index.
Table of contents:
Editor's statement; Preface; Introduction; 1. Basic definitions and properties; 2. Further elementary theorems; 3. Coalescence of rows; 4. Applications of coalescence; 5. Rolle extensions and independent sets of knots; 6. Singluar matrice; 7. Zeros of Birkhoff splines; 8. Almost0hermitian matrices; special three-row matrices; 9. Applications; 10. Birkhoff quadrature formulas; 11. Interpolation at the roots of unity; 12. Turán's problem of (0,2) interpolation; 13. Birkhoff interpolation by splines; 14. Regularity theorems and self-dual problems; Bibliography and references; Symbol index; Subject index.