# Fourier series

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Related to Cosine series: Fourier sine series

## Fourier series

n.
An infinite series whose terms are constants multiplied by sine and cosine functions and that can, if uniformly convergent, approximate a wide variety of functions.

[After Baron Jean Baptiste Joseph Fourier.]

## Fourier series

n
(Mathematics) an infinite trigonometric series of the form a0 + a1cos x + b1sin x + a2cos 2x + b2sin 2x + …, where a0, a1, b1, a2, b2 … are the Fourier coefficients. It is used, esp in mathematics and physics, to represent or approximate any periodic function by assigning suitable values to the coefficients

## Fou′rier se`ries

n.
an infinite series that approximates a given function on a specified domain by using linear combinations of sines and cosines.
[1875–80; see Fourier analysis]
ThesaurusAntonymsRelated WordsSynonymsLegend:
 Noun 1 Fourier series - the sum of a series of trigonometric expressions; used in the analysis of periodic functionsseries - (mathematics) the sum of a finite or infinite sequence of expressions
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Here, we think of the DFT or DCT as approximations for the Fourier series or cosine series of a function, respectively, in order to talk about its "smoothness") However, the implicit periodicity of the DFT means that discontinuities usually occur at the boundaries: any random segment of a signal is unlikely to have the same value at both the left and right boundaries.
Incidentally, it is actually not obvious, but it turns out that the cosine series can be obtained from the sine series by termwise differentiation, as one might hope.
Besides, let us say that the choice of the Mercator projection has been made on purpose, as its map factor expression (hyperbolic cosine) is one of the best suited for developing a Fourier cosine series (Benard, 2004).
The following results about the integrability of cosine series
The gear motion error is a real signal, described by an infinite cosine series with fundamental period f r.

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