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. |
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Suppose that |
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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 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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(1) <math>f(x)</math> is defined and single-valued except possibly at a finite number of points in <math>(-L, L)</math> |
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(2) <math>f(x)</math> is periodic outside <math>(-L, L)</math> with period <math>P = 2L</math> |
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(3) <math>f(x)</math> and <math>f'(x)</math> are piecewise continuous in <math>(-L, L)</math>. |
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Revision as of 21:08, 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.