Fourier series: Difference between revisions

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===Diriclet Conditions===
===Diriclet Conditions===



The conditions for a periodic function <math>f</math> with period 2L to have a convergent Fourier series.
The conditions for a periodic function <math>f</math> with period 2L to have a convergent Fourier series.


''Theorem:''
=====''Theorem:''=====

Let <math>f</math> be a piecewise regular real-valued function defined on some interval [-L,L], such that <math>f</math> has
''only a finite number of discontinuities and extrema'' in [-L,L]. Then the Fourier series of this function converges to <math>f</math> when <math>f</math> is continuous and to the arithmetic mean of the left-handed and right-handed limit of <math>f</math> at a point where it is discontinuous.

===Orthogonal Functions===

=====Orthonormal Functions=====
=====Weighing function=====
=====Kronecker delta function=====


===The Fourier Series===

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.


----
Let <math>f</math> be a piecewise regular real-valued function defined on some interval [-L,L], such that <math>f</math> has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to <math>f</math> when <math>f</math> is continuous and to the arithmetic mean of the left-handed and right-handed limit of <math>f</math> at a point where it is discontinuous.

Revision as of 21:31, 27 October 2004

Diriclet Conditions

The conditions for a periodic function with period 2L to have a convergent Fourier series.

Theorem:

Let be a piecewise regular real-valued function defined on some interval [-L,L], such that has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to when is continuous and to the arithmetic mean of the left-handed and right-handed limit of at a point where it is discontinuous.

Orthogonal Functions

Orthonormal Functions
Weighing function
Kronecker delta function

The Fourier Series

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.