Spectral fidelity of gappy data

A one-dimensional periodic domain is chosen over which four physical variables are presumed to be known exactly at 72 equally spaced points. Nineteen experiments are then designed with differing numbers of observing points thus yielding variable gaps in the data. Utilizing trigonometric functions as a basis, Gram-Schmidt functions are generated with orthogonal properties over the non-equally spaced point domain. The quality of the Fourier coefficients for each allowed scale is assessed for the different experiments in terms of the true Fourier coefficients. Results indicate that accuracy in scale coefficients deteriorates rapidly for scales less than twice the maximum gap in the domain, regardless of point density elsewhere in the domain. Furthermore, for the addition of observations in a gappy region, the most spectral fidelity will be achieved by minimizing the largest resulting gaps.