A lacunary interpolation algorithm on arbitrary points (Q1895909)

The problem of (0,2,3) interpolation is known to be regular on any \(n\) real nodes and also when the nodes are roots of unity. But it is not known how to find a closed form for the interpolant except in three cases: 1) zeros of \(\pi_n (x)= (1- x^2) P'_{n-1} (x)\), where \(P_n (x)\) is a Legendre polynomial of degree \(n\); 2) zeros of Chebyshev polynomials; 3) roots of unity. Here the author gives an algorithm for finding a closed form of the interpolant. It requires the invertibility of an \(n\times n\) matrix. He also gives an error analysis for this case and shows that if the function is smooth enough and the nodes are equispaced, then the error is \(O(h^{3n})\) where \(h\) is the step size (the distance between two successive nodes). Numerical studies are also presented.