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. | ||
===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. | ||
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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.