Fourier series: Difference between revisions

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===Diriclet Conditions===
===Diriclet Conditions===
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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.
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Let <math>f</math> be a piecewise regular real-valued function defined on some interval [-L,L], such that <math>f</math> has
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.
''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===
===The Fourier Series===
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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.
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.



Revision as of 21:35, 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.

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.