Fourier series: Difference between revisions
Jump to navigation
Jump to search
mNo edit summary |
|||
| Line 1: | Line 1: | ||
===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. |
||
| Line 8: | Line 8: | ||
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. |
||
===Orthogonal Functions=== |
|||
=====Orthonormal Functions===== |
|||
=====Weighing function===== |
|||
=====Kronecker delta function===== |
|||
===The Fourier Series=== |
===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 relationships of the sine and cosine functions. |
||
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.