Splines as Linear Combinations of B-splines. a Survey

The layout of the survey is as follows. After a short discussion of cardinal B-splines, i.e., of B-splines on a uniform knot sequence, in Section 2, B-splines for an arbitrary knot sequence are introduced in Section 3 and shown to be a basis for certain spaces of piecewise polynomial functions. Various simple properties of B-splines are listed in Section 4, and the relationship between a spline and its coordinates with respect to a B-spline basis is explored in Section 5. This leads naturally into the discussion of local spline approximation schemes, in Section 6. Results concerning existence and uniqueness of interpolating splines and the related total positivity and variation diminishing properties of B-splines are presented in Section 7. Section 8 describes the connection between splines and certain ”best” interpolation schemes. Finally, Section 9 is devoted to generalized B-splines and ends with a new definition of polynomial B-splines in many variables due to I. J. Schoenberg. No claim of completeness is made, and the author would be grateful to hear of any omissions. The following notation is used throughout the paper, usually without further explanation: ZZ denotes the set of integers, IR the set of real numbers, and A the set of functions on B into A. Thus, IR is the set of real bi-infinite sequences. m(B) is the linear space of bounded real functions on B, normed by ‖f‖∞,B := supx∈B |f(x)|. For 1 ≤ p ≤ ∞, ILp(I) denotes the space of (equivalence classes of) functions f on the interval I for which ‖f‖p := ‖f‖p,I := ( ∫