Least squaresLet . For given samples , the index set , of frequencies, we construct a -variate trigonometric polynomial
such that . Turning this into matrix vector notation, we aim to solve the system of linear equations
for the unknown vector of Fourier coefficients . We denote the vector of the given sample values by and the nonequispaced Fourier matrix by
For , the linear system (1) is over-determined, so that in general the given data will be only approximated up to a residual . In order to compensate for clusters in the sampling set , it is useful to incorporate weights and to consider the weighted approximation problem
where . This library of C functions computes approximations of (2) with the CGNR method.
The algorithms are implemented by Stefan Kunis in ./solver. Related paper are
Kunis, S. and Potts, D.
Stability Results for Scattered Data Interpolation by Trigonometric Polynomials.
SIAM J. Sci. Comput. 29, 1403 - 1419, (full paper ps, pdf) 2007