Fourier series: Difference between revisions
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==Diriclet Conditions== |
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''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. |
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==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 (see *[[Orthogonal Functions]])relationships of the sine and cosine functions. |
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Principle author of this page: [[User:Goeari|Aric Goe]] |
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Revision as of 20:45, 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 (see *Orthogonal Functions)relationships of the sine and cosine functions.
Principle author of this page: Aric Goe