This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.

Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the Finite Element method) or the modeling of smooth sur faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems to be solved in the construction of approximations are 'banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, s, of their domain G ~ IRs, i. e. , the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the num ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.

Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the Finite Element method) or the modeling of smooth sur faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems to be solved in the construction of approximations are 'banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, s, of their domain G ~ IRs, i. e. , the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the num ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.

Der Begriff der Splinefunktionen wurde von I. J. Schoenberg 1946 eingefUhrt. "Spline" ist der Name eines Zeichengerates, welches auf mechanischem Weg Interpolatio- aufgaben lost. Dieses Gerat besteht aus einer flexiblen, oft mehrere Meter langen Latte, die auf dem Zeichenbrett aufliegt und dort an bestimmten Stellen durch Gewichte festgehalten wird. Die Form, die die Latte annimmt, hangt von den Elastizitatseigenschaften der Latte abo -, " , , , , , , \ , \ \ I , , , ," -"', , , , , J::>----" , I I , I I I , , , , , , , , , , , " ) Fig. 1: Latteninterpo1ation Po1ynominterpo1ation _ - - - - - - - --0 Wir konnen natUrlich versuchen, ein mathematisches Modell fUr dieses mechanische Zeichengerat zu machen, d. h. die Gestalt solcher Kurven mathematisch zu erfassen. - 2 - Die Theorie der Balkenbiegung verlangt, dass die mittlere 2 K quadratische KrUmmung, ("strain energy", Spannungs- J energie) minimiert wird. Lasst sich die Kurve als Graph einer Funktion f auf dem Intervall [a,b] schreiben, so erhalt man mit dem bekannten Ausdruck fUr die Krlimmung K [f" (t) P --------------dt ~ min (1) t [1 +f' (t)2J5/2 a Statt dieses schwierige Extremalproblem zu losen, begnUgt man sich damit, (2) zu minimieren. Die Extremalfunktion fUr das Funktional (2) ist stUckweise ein kubisches Polynom; die Polyn- stUcke gehen an den Bruchstellen so glatt ineinander Uber, dass die Funktion zweimal stetig differenzierbar ist.

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Theophanis Chronographia, Volume 2; Theophanis Chronographia; Anastasius (the Librarian) Theophanes (the Confessor), Anastasius (the Librarian), Carl de Boor B.G. Teubnneri, 1885 World history