This is the first in a series of books dealing with approximation and interpolation of functions. Many changes have occurred in this theory during the last decades. In what follows, we shall try to describe some of the problems and achievements of this period.

Until about 1955, the leading force in approximation was the Russians, in particular, Bernstein and his school (Ahiezer), Chebyshev, Kolmogorov, and Markov. The development of the subject in Germany, Hungary, and the United States occurred later. The West certainly leads in the number of papers published—see the bulky Journal of Approximation Theory. The twelve sections that follow review the newer developments.

The two classical books dealing with approximation and interpolation are those of Natanson [0–N] and Ahiezer [0–A]. Important recent books include two Russian works devoted to special problems: Korneichuk [0–K2] (see also [0–K3]) deals with best constants in the trigonometric approximation, while Tihomirov [0–T1] treats extremal problems, particularly widths and optimization. The book of Butzer and Berens [0–B2] introduced functional analytic methods into the field; the two books by de Boor [0–B1] and Schumaker [0-S] deal with splines, an American development rich in practical applications. Karlin and Studden [0–K,] treat Chebyshev systems exhaustively. Books on general approximation theory are those of Rice [0–R], Lorentz [0–L], Dzyadyk [0–D], and Timan [0–T2]; the last book contains a wealth of material. Several books will be mentioned in later sections.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive change of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

HISTORICAL REMARKS

The first papers containing regularity results for spline interpolation were [155, 74, 64]. Their characteristic features are similar: The interlacing conditions (13.1.5) are necessary and sufficient for regularity; the proof is based on the total positivity of certain matrices. This approach gives much more than regularity, and this is perhaps the reason why no stronger regularity results have been achieved this way. The strongest theorem (regularity when E, E* are quasi-Hermitian) is stated in Karlin [64, 66].

The next important step was achieved by Melkman [120]; he established the regularity when E is conservative, E* Lagrangian, and announced that he also had a proof when E* is Hermitian. This was soon established by Jetter [57], who sketched a proof relying on Theorem 7.13 of Birkhoff and Lorentz. Pence [134] somewhat improved Melkman's conditions. Like Melkman, Goodman [47] also relied on Budan–Fourier theorems to derive his remarkable theorem.

The use of Rolle's theorem for splines (Lemma 7.10) was inaugurated by Lorentz [93]; de Boor [B, 10] used the ordinary Rolle theorem to derive theorems of [155] and [74]. In §14.2 we give a general theorem [61] that uses diagrams of splines, from which the theorem of Goodman follows easily (by means of a relatively simple Lemma 14.5).

Several methods of proof of regularity theorems are known. In this chapter we illustrate: an application of the zero count theorem in this section, of Rolle's lemmas in §§14.2–14.3, and a completely different, more elementary approach that works for self-dual interpolation problems in §14.4. The reader can consult in [47] the original proof of theorem 14.4, based on the classical Budan–Fourier Theorem 2.2.

INTRODUCTION

Coalescence of two rows of a matrix has been implicitly used by Ferguson [37, p. 14]. It has been formally defined by Karlin and Karon [70], who studied its influence on the regularity of the matrix and gave the Taylor expansion (3.5.2). Lorentz and Zeller [108] proved that the leading coefficient in this formula is different from 0. Lorentz [94] studied the coalescence of several rows. There are many applications of this method, for example, in [70, 91, 94, 103]. We discuss them in Chapter 4.

Let E = [ei, k] be an m × (n + 1) matrix, not necessarily normal, satisfying the Pólya condition (1.4.3). We interpret E as a vertical grid of boxes. If ei, k = 1, then a ball occupies the ith box in the kth column. We place a tray of n + 1 boxes under the column of the grid. Then the balls are permitted to fall from the grid into the boxes of the tray in such a way that if the box immediately below is occupied, the ball rolls to the first available box on the right. The condition (1.4.3) assures us that no ball will roll out of the tray. The distribution of balls in the tray constitutes the one-row matrix obtained by coalescence of the m rows of E. It is to be expected that the final arrangement of the balls in the tray is independent of the manner in which the balls were allowed to fall. Figure 3.1 is an example of coalescence of a two-row matrix.

INTRODUCTION

Several authors have tried to study classes of matrices, of modest generality, that contain the matrices (1.3.7) as special cases. The hope was that some insight into the general problem might be gained by deciding the regularity problem for this class. The failure to find a complete solution for even a relatively simple class of this type illustrates the complexity of the general problem, and to some extent justifies its probabilistic reformulation in §6.4.

A matrix E is almost Hermitian if all of its rows are Hermitian except for one interior row. A subclass consists of the three-row matrices E(p, q; k1, …, ks) where p and q are the lengths of the Hermitian first and third row while 0 < k1 < … < ks are positions of 1's in the second row. We have then p + q + s = n + 1, and we shall always assume 1 ≤ p ≤ q. The regularity problem is not completely solved even for matrices E(p, q; k1, k2).

For almost-Hermitian matrices, many useful results follow from a representation of D(E, X) given by Theorem 8.1. Most notable are a strong singularity criterion due to Lorentz [91] and DeVore, Meir, and Sharma [26] in §8.2 and the reciprocity theorem of Drols [29] in §8.3.

Sections 8.4–8.6 deal with the special matrices E(p, q; k1, k2). In §8.4 we give the “chasing method” of Lorentz, Stangler, and Zeller [104] in a more geometric form. This solves the problem except for the two cases, (8.2.11) and (8.2.12). The second of them admits a complete solution.

Birkhoff, or lacunary, interpolation appears whenever observation gives scattered, irregular information about a function and its derivatives. First discovered by G. D. Birkhoff in 1906, it received little attention until I. J. Schoenberg revived interest in the subject in 1966. Lacunary interpolation differs radically from the more familiar Lagrange and Hermite interpolation in both its problems and its methods. It could even be described as “non-Hermitian” interpolation. The name Birkhoff interpolation is justified also from a historical point of view.

At present, the main definitions and theorems for polynomial Birkhoff interpolation seem to have been found, while the theory for other systems of functions, most notably splines, is in healthy development. Since this material can be found only in research periodicals and proceedings of conferences, it is time for a comprehensive exposition of the material. We have gone to great lengths to unify, simplify, and improve the information already published or in press, and to set the stage for further developments.

The book should be of interest to approximation theorists, numerical analysts, and analysts in general, as well as to computer specialists and engineers who need to analyze functions when their values and those of their derivatives are given in an erratic way. The book could be used as a text for a graduate course, requiring little more than an undergraduate mathematics background.

Many novel ideas and tools have been developed in this theory; interpolation matrices, coalescence of rows in matrices, independent knots, probabilistic methods, diagrams of splines, and the Rolle theorem for splines.