Multiple Fourier Series
William O. Bray
Herein we will explore a few examples of multiple Fourier sine series on the square D=[0,π]×[0,π]. These have the form:
The following command forms the square partial sums of the Fourier sine series with Fourier coefficients given as a table.
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Example 1
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This function is actually a product of a function of x with a function of y, hence the double Fourier sine coefficients are the products of the one dimensional Fourier coefficients. We can implement this as follows.
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Visually, we cannot see much difference between the partial sum plot and that of the actual function. The following is a depiction of the error.
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Notice the growth in error along the boundaries; undoubtably this is due to the fact that the partial derivatives of the function are discontinuous across the boundary (think in terms of the periodic extension of this function).
Example 2
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For a look at the error, consider the following.
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Example 3
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A view of the partial sums relative to the periodic extension of this function is seen as follows.
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