15 Facts About Fourier series

Fourier series is a sum that represents a periodic function as a sum of sine and cosine waves.

Convergence of Fourier series means that as more and more harmonics from the series are summed, each successive partial Fourier series sum will better approximate the function, and will equal the function with a potentially infinite number of harmonics.

However, non-periodic functions can be handled using an extension of the Fourier Series called the Fourier transform which treats non-periodic functions as periodic with infinite period.

Many other Fourier series-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier series analysis.

Therefore an alternative form of the Fourier series, using the Cartesian coordinates, is the sine-cosine form:.

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In engineering applications, the Fourier series is generally presumed to converge almost everywhere since the functions encountered in engineering are better-behaved than the functions that mathematicians can provide as counter-examples to this presumption.

Fourier series is named in honor of Jean-Baptiste Joseph Fourier, who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

Fourier series's idea was to model a complicated heat source as a superposition of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions.

From a modern point of view, Fourier series's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century.

The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, shell theory, etc.

When Fourier series submitted a later competition essay in 1811, the committee concluded:.

Since Fourier series arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis.

Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at if is differentiable at, to Lennart Carleson's much more sophisticated result that the Fourier series of an function actually converges almost everywhere.

Since Fourier series have such good convergence properties, many are often surprised by some of the negative results.

Fourier series later constructed an example of an integrable function whose Fourier series diverges everywhere .