12-04-2013, 04:41 PM
Problems based Exponential and Trigonometric Fourier Series and its applications
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Abstract
Fourier series are of great importance in both theoretical and ap
plied mathematics. For orthonormal families of complexvalued functions
{f }, Fourier Series are sums of the f that can approximate periodic,
complexvalued functions with arbitrary precision. This paper will focus
on the Fourier Series of the complex exponentials. Of the many possi
ble methods of estimating complexvalued functions, Fourier series are
especially attractive because uniform convergence of the Fourier series (as
more terms are added) is guaranteed for continuous, bounded functions.
Furthermore, the Fourier coe cients are designed to minimize the square
of the error from the actual function. Finally, complex exponentials are
relatively simple to deal with and ubiquitous in physical phenomena. This
paper first defines generalized Fourier series, with an emphasis on the se
ries with complex exponentials. Then, important properties of Fourier
series are described and proved, and their relevance is explained. A com
plete example is then given, and the paper concludes by brie y mentioning
some of the applications of Fourier series and the generalization of Fourier
series.
Trigonometric fourier series
In this section, ƒ(x) denotes a function of the real variable x. This function is usually taken to be periodic,of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2π–periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−π, π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.
Applications of Fourier series
To recapitulate, Fourier series simplify the analysis of periodic, realvalued func
tions. Specifically, it can break up a perio dic function into an infinite series of
sine and cosine waves. This property makes Fourier series very useful in many
applications. We now give a few.