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A fast algorithm for Chebyshev, Fourier, and sinc interpolation onto an irregular grid. (English) Zbl 0768.65001
The problem is to evaluate efficiently the interpolatory sum \(f(x) \cong\sum f(x_ j)C_ j(x)\) at a set of \(N\) points which do not coincide with the regularly spaced interpolation points \(\{x_ j\}\). The fast Fourier transform (FFT) technique does not apply in this case and the cost of operations of direct summation becomes \(O(N^ 2)\) instead of \(O(N\log N)\) for FFT.
The author proposes an alternative solution: to use the FFT of length \(3N\) to interpolate the Chebyshev series to a very fine grid and then to apply the \(M\)th order Euler sum acceleration (or (\(2M+1\))-point Lagrangian interpolation) with \(M << N\) to approximate \(f\) on the irregular grid. The cost of interpolation is significantly reduced with no loss of accuracy.

65B10 Numerical summation of series
65D05 Numerical interpolation
65T50 Numerical methods for discrete and fast Fourier transforms
Full Text: DOI
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