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{{Template:Math-h}}

{{Math-h}}

'''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]].

:<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|groups]], 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 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.   

[[category:Calculus]]

[[Category:Calculus]]