Trigonometric Polynomials Revisited

@inproceedings{RoyerRM2008,
 abstract = { Trigonometric representations are commonly used in geosciences to represent functions sampled regularly by continuous approximations. It can be used for instance to correlate logging data sets acquired on several wells or, like in gOscope, to extract attributes from seismic traces. These approaches involve a kernel representation of the seismic traces by trigonometric polynomials.
In this paper, analytical identities based on truncated geometric series formulas are suggested to simplify the calculation of the trigonometric kernels K. The advantages of the obtained results are that computer time is considerably reduced compared to the calculation based on the series terms. A simple analytical formula is obtained for the analytical signal associated to the Kernel function K. This improves the evaluation of the seismic attributes related on the Hilbert transform.
The same methodology is applied to the trigonometric polynomials covariance function C f g (T) and an analytical expression is derived to evaluate C  f g (τ)  in function of C f G (0) near the origin τ ≃ 0 or in the neighborhood of lags t +/-  τ . These results are applied to suggest new analytical expressions for evaluating the wavelet transform of centered trigonometric polynomials. These methods are then applied to better correlate seismic traces. },
 author = { Royer, Jean-Jacques },
 booktitle = { 28th gOcad Meeting },
 month = { "june" },
 publisher = { ASGA },
 title = { Trigonometric Polynomials Revisited },
 year = { 2008 }
}