Changes

'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions.  The seeds of the modern theory were developed by [[Joseph Fourier]].

'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions.  The seeds of the modern theory were developed by [[Joseph Fourier]].

The Fourier [[series (mathematics)|series]] of a function ''f(t)'' is of the form:

The Fourier [[series (mathematics)|series]] of a function <math>f(t)</math> is of the form:

:<math> f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] </math>

 

where, ''n'' is an integer and

:<math> f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \bigg[a_n \cos(\omega_n t) + b_n \sin(\omega_n t) \bigg] </math>

 

where, <math>n</math> is an [[integer]] and

 

:<math> \omega_n = n\frac{2\pi}{T}</math>

:<math> \omega_n = n\frac{2\pi}{T}</math>

:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>

:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>

:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>

:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>

Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance the space <math>L^2(\mathbb R^n)</math> of [[square integrable function]]s.  Even more generally, it is possible to carry out Fourier analysis in the setting of [[compact space|locally compact]] [[abelian]] [[topology|topological]] [[group]]s, where the fundamental observation of Pontryagin duality provides the necessary theoretical underpinnings.   

Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance the space <math>L^2(\mathbb R^n)</math> of square integrable functions.  Even more generally, it is possible to carry out Fourier analysis in the setting of [[compact space|locally compact]] [[abelian]] [[topology|topological]] [[group]]s, where the fundamental observation of Pontryagin duality provides the necessary theoretical underpinnings.   

 

 

 

[[Category:Calculus]]

[[Category:Calculus]]