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Note the derivative of a triangle wave is a square wave. Examine the series coefficients to see this. There are many books and web sites onthe Fourier series that give insight through examples and demos.
Four of the most important theorems in the theory of Fourier analysis are the inversion theorem, the convolution theorem, the differentiationtheorem, and Parseval's theorem [link] .
All of these are based on the orthogonality of the basis function of the Fourier series and integral and all require knowledge of the convergenceof the sums and integrals. The practical and theoretical use of Fourier analysis is greatly expanded if use is made of distributions orgeneralized functions (e.g. Dirac delta functions, $\delta \left(t\right)$ ) [link] , [link] . Because energy is an important measure of a function in signal processing applications, the Hilbert space of ${L}^{2}$ functions is a proper setting for the basic theory and a geometric view can be especiallyuseful [link] , [link] .
The following theorems and results concern the existence and convergence of the Fourier series and the discrete-time Fourier transform [link] . Details, discussions and proofs can be found in the cited references.
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