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. | |||
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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.