Larry Schumaker's Spline Functions was first published in 1981, then in a second edition with minor changes in 1993. The 1981 Telegraphic Review described the book as
A detailed and comprehensive introduction to the theory of polynomial, generalized, and multidimensional splines emphasizing their algebraic, analytic, and approximation-theoretic properties.
For much more information about the book, see the review in the Bulletin of the AMS, 6 (2) (1982), 238-247.
This new edition, in the Cambridge Mathematical Library series, is basically a reprint of the second edition enriched by a "Supplement" that reports on progress since then. The supplement is 23 pages long and comes with a 17-page bibliography.
1. Introduction; 2. Preliminaries; 3. Polynomials; 4. Polynomial splines; 5. Computational methods; 6. Approximation power of splines; 7. Approximation power of splines (free knots); 8. Other spaces of polynomial splines; 9. Tchebycheffian splines; 10. L-Splines; 11. Generalized splines; 12. Tensor-product splines; 13. Some multidimensional tools; Supplement; References; New references; Index.