A Fourier series is the expansion of any other function in terms of the Fourier functions of sine (nx) and cosine (nx). (These can also be expressed as a combination of positive and negative exponentials.) Since these functions compose a Hilbert space or a basis set, any continuous function can be composed of a combination of them as long as it has a finite number of finite discontinuities and a finite number of extrema in the interval over which the Fourier series is summed, usually 0 to 2 Pi. That is, as long as there's a set number of places where the function is not continuous, and as long as there are not an infinite number of local maxima and minima, the Fourier series can express the function. Each function will have a unique Fourier series expansion.
The Fourier coefficients, that is, the numbers multiplied by each sine or cosine term, may be found by taking the integral of the function times the desired trigonometric term. In many cases, these integrals will be very similar from term to term, providing a generalized way of expressing the coefficients. The theory that allows this formula to work is called Sturm-Liouville Theory. The interval in which the series holds can be changed by simply dividing the original denominator in the sine or cosine expression by the length of the new interval.
At a discontinuity, the Fourier series will converge on the arithmeticmean of the terms on either side of the discontinuity. The discontinuity must be finite for this formulation (or the series itself) to work: a jump from one to zero is fine, but from one to infinity is not.
Fourier series yield a continuous function as long as they are expressions of continuous functions. In this case, they will also be uniformly convergent. On the average of any given function, even discontinuous functions will yield convergent Fourier series results.
Fourier series expansions are used in most applications dealing with wave mechanics, including optics and acoustics. Since any periodic function meeting the above conditions can be described as the superposition of waves in a Fourier series, sometimes one will speak of "Fourier decomposing" a function, or breaking it up into its component Fourier series parts. Fourier analysis is also common for filtering in optics, although this generally uses Fourier transformations rather than Fourier series. They are also used in electronics applications when dealing with a periodic signal. With a wide variety of physical applications, Fourier series are one of the most commonly used mathematical tools of physicists and engineers.
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