Remarks on Piecewise Polynomial Approximation

The theoretical and experimental background for the approximation of "real-world" functions is reviewed to motivate the use of piecewise polynomial approximations. It is possible in practice ~o achieve the results suggested by the theory for piecewise polynomials. A convergent adaptive algorithm is outlined which effectively and efficiently computes smooth piecewise polynomial approxImations to any member of a broad class of functions. This class includes all functions which are piecewise smooth and have a fini te number of "algebraic ll singularities. Theoretica.lly and experimentally determined properties of this algorithm are indicated. REMARKS ON PIECEWISE POLYNOMIAL APPROXIMATION * John R. Rice I. THE EXPERIMENTAL BACKGROUND. The advent of high speed computers made it both possible and necessary to compute approximations to a large variety of functions. The functions involved are somewhat arbitrarily divided into two classes: the mathematical functions (ex, rex), J1(x), etc.) and the "real world" functions (shape of a turbine blade or airplane wing, specific heat versus temperature for titanium, air pressure versus altitude, etc.) In this discussion we consider only the real-world functions and the remarks do not apply, in general, to the approximations of the elementary and special functions of mathematics. The experience of the 1950's was clear and convincing: classical linear methods of approximation are inadequate for real applications. We take ordinary polynomial approximations (say with L2 or L~ norms) as the prime example of these classical methods. Other methods involving trigonometric functions (Fourier Series), Bessel functions, etc. are no better. To make this position quantitative, we note that Rice made an experiment in the early 1960's as follows. Several dozen tabulated functions were selected at random from the rrHandbook of Chemistry and Physics" which represent a variety of relationships in the real world. All of the *This work was supported in part by grant GP 32940X from the National Science Foundation.