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How can you show that the Fourier series approximation of a function (so $f(x)=\sum\limits_{n=0}^{\infty} (a_n cos(nx) + b_n sin(nx))$ can be approximated to arbitrary precision by a feedforward neural network with one input and output unit and one layer of hidden units with an activation function $F(s)=sin(s)$?

I'm new on this kind of topics, so any tips/solutions are welcome!

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The output of a feedforward neural network that uses $\sin$ as an activation function and linear output neurons with no bias, can be written as $\hat f(x) = \sum_{n=1}^h \alpha_n \sin(\beta_n x - \gamma_n)$, where $h$ is the number of hidden nodes, $\alpha_n$ and $\beta_n$ are the output weights and the input-to-hidden layer weights, and $\gamma_n$ is the bias. This bias can be used to change some of the sines to a cosine, and the input-to-hidden layer weights can represent the frequency. Then you need to choose the output weights to represent the Fourier coefficients, which gives an approximation of arbitrary accuracy by increasing the number of hidden nodes.

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