THE APPROXIMATION OF DIFFERENTIABLE FUNCTIONS BY POLYNOMIALS.

An iterative procedure is developed for the expeditious determination of accurate approximations to the coefficients of the polynomial of preassigned maximum degree that best approximates a given differentiable function over a finite interval of the argument, in the sense that the greatest absolute deviation of the polynomial from the function is less than that of any other polynomial of the same or smaller degree. Detailed examples are given of the application of this procedure to the optimum approximation by polynomials of several elementary transcendental functions. In the final section of the body of this report a sufficient condition for the convergence of the iterative procedure is given. Briefly stated, this condition is that the difference between the given differentiable function and the polynomial of degree or n with which the iterative procedure is begun should have n 2 points in the given finite interval at which it assumes extreme values which alternate in sign, n being the preassigned maximum degree. Appended to the report are tables of extended approximations to certain Besselfunction values especially useful in applications of this method to trigonometric functions. The tables are preceded by an explanation of the procedure followed in their calculation. Author