Â鶹ԼÅÄ

If f(x) is continuous and of bounded variation in the range x is greater than or equal to -1 and less than or equal to +1, it can be expanded in terms of Chebyshev polynomials. The numerical procedure for such expansion is given in detail. The reverse process of evaluating an expression for which the 'Chebyshev coefficients' are known is also fully considered, for cases in which these coefficients can be more easily derived from the data than a simple, explicit formula for f(x). The process of expansion is closely related to that of Fourier analysis. Such expansion has marked advantages, notably that the error committed by terminating the series at any point can be easily estimated, not only for f(x) but for its integral and derivative. The goodness of fit between this terminated series and f(x) is discussed. Such expansion can be used for the approximate solution of ordinary linear differential equations, particularly those having coefficients which are polynomials of low degree in x.

This matter is fully discussed by Clenshaw. What follows is essentially a simplified version of a paper by Clenshaw on the 'Chebyshev series for mathematical functions', designed to help the engineer who wishes to work to an accuracy of at most three or four significant figures, and to take advantage of the ideas and techniques Clenshaw has so thoroughly established.