Basic Examples (2) 

Create an approximate formula for a sequence of values based on the discrete Fourier transform of the data:

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A simpler approximation can be created by using only the three most dominant values from the discrete Fourier transform:

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Scope (2) 

For unevenly-spaced data or a path of points in 2D space, TrigApproximateList generates a pair of expressions using a common variable:

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The output is appropriate for use within ParametricPlot to generate this approximation to a square:

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More terms produce more accurate approximations:

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When the number of terms is not specified, TrigApproximateList uses Length[data]/2 terms:

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This produces the most accurate approximation possible while still preventing high-frequency over-fitting:

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Options (3) 

The data range is assumed to be from 1 to Length[data]. To change this, use DataRange:

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Instead of specifying the number of terms to return, you can specify a criteria for deciding when a Fourier term is significant enough to use:

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When a fixed number of terms is requested, you can specify which terms are more important with "TermOrderingFunction". Here terms with the largest real component are used first:

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Neat Examples (1) 

Create a smooth parametric outline from an image:

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