On the Accuracy of Trigonometric Interpolation

Let / ( x ) be a function of x, with the period 2ir , which takes on known values, determined by physical measurement or otherwise, at 2n + 1 points equally spaced throughout a period. It is well known that a finite trigonometric sum, of the «th order at most, which coincides with f (x) at the points in question, is given by formulae analogous to those which characterize the Fourier's development of / ( x ), the integrals of the latter development being replaced by finite sums involving the known quantities. It has been found by Faber that in the case of continuous functions the ordinary sufficient conditions, in the way of restrictions on / ( x ), for the convergence of the Fourier's series are sufficient also to insure the convergence of the interpolating function to the value f (x) at all points, as the number n is indefinitely increased; although in cases not covered by these explicit conditions one development may converge while the other does not.f In the present note it is shown that a method by which the author has studied the rapidity of convergence of Fourier's series is adapted to the treatment of the corresponding problem in interpolation, X and yields similar results. In fact, these results are obtained by a simple combination of materials already at hand. It is further pointed out how a finite number of observed values may be used to define a formula of approximation which converges more rapidly in certain cases than the ordinary interpolation formula.