Fourier series: Difference between revisions

Revision as of 21:08, 27 October 2004

Diriclet Conditions

The conditions for a periodic function $f$ with period 2L to have a convergent Fourier series.

Theorem:

Let $f$ be a piecewise regular real-valued function defined on some interval [-L,L], such that $f$ has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to $f$ when $f$ is continuous and to the arithmetic mean of the left-handed and right-handed limit of $f$ at a point where it is discontinuous.

@@ Line 1: / Line 1: @@
-==Diriclet Conditions==
+===Diriclet Conditions===
-'''Theorem:'''
+The conditions for a periodic function <math>f</math> with period 2L to have a convergent Fourier series.
-Suppose that
+''Theorem:''
-(1)  <math>f(x)</math> is defined and single-valued except possibly at a finite number of points in <math>(-L, L)</math>
+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.
-(2)  <math>f(x)</math> is periodic outside <math>(-L, L)</math> with period <math>P = 2L</math>
-(3)  <math>f(x)</math> and <math>f'(x)</math> are piecewise continuous in <math>(-L, L)</math>.

Fourier series: Difference between revisions

Revision as of 21:08, 27 October 2004

Diriclet Conditions

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