Solution of The Problem
Under the above conditions, there exists a solution to the problem for any given set of data points {xk, p(xk)} as long as N, the number of data points, is not larger than the number of coefficients in the polynomial, i.e., N ≤ 2n+1 (a solution may or may not exist if N>2n+1 depending upon the particular set of data points). Moreover, the interpolating polynomial is unique if and only if the number of adjustable coefficients is equal to the number of data points, i.e., N = 2n + 1. In the remainder of this article, we will assume this condition to hold true.
The solution can be written in a form similar to the Lagrange formula for polynomial interpolation:
This can be shown to be a trigonometric polynomial by employing the multiple-angle formula and other identities for sin ½(x − xm).
Read more about this topic: Trigonometric Interpolation
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