Fourier series: Difference between revisions
Jump to navigation
Jump to search
mNo edit summary |
mNo edit summary |
||
| Line 1: | Line 1: | ||
==Diriclet Conditions== |
|||
---- |
---- |
||
| Line 9: | Line 9: | ||
''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== |
|||
---- |
---- |
||
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. |
||
Principle author of this page: [[User:Goeari|Aric Goe]] |
|||
Revision as of 21:45, 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