A polynomial interpolation process at quasi-Chebyshev nodes with the FFT

Interpolation polynomial pn at the Chebyshev nodes cosπj/n (0 ≤ j ≤ n) for smooth functions is known to converge fast as n → ∞. The sequence {pn} is constructed recursively and efficiently in O(n log2 n) flops for each pn by using the FFT, where n is increased geometrically, n = 2i (i = 2, 3, . . . ), until an estimated error is within a given tolerance of ε. This sequence {2j}, however, grows too fast to get pn of proper n, often a much higher accuracy than ε being achieved. To cope with this problem we present quasi-Chebyshev nodes (QCN) at which {pn} can be constructed efficiently in the same order of flops as in the Chebyshev nodes by using the FFT, but with n increasing at a slower rate. We search for the optimum set in the QCN that minimizes the maximum error of {pn}. Numerical examples illustrate the error behavior of {pn} with the optimum nodes set obtained.