Series Approximation of Functions: Taylor/McLaurin, Fourier & More

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Series approximations of functions can extend beyond Taylor/McLaurin and Fourier series, with the possibility of using Gaussian functions like Aexp(-bx^2) for expansions. The key requirement is that the functions in the series must form a basis for the function space. This principle is rooted in functional analysis, where the basis functions, such as sine and cosine in Fourier expansions, allow for accurate function representation. The discussion highlights the flexibility in choosing series for function approximation, emphasizing the importance of basis functions. Various series can be explored for effective function representation.
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I wonder if there's any other series that can be used to approximate a function, other that Taylor/McLaurin and Fourier. For instance, can we expand a function in terms of Gaussians (Aexp(-bx^2))? Maybe something else?
 
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From what I remember of functional analysis (it was a while ago), as long as the individual functions in the series form a basis for the function space, then yes, you can expand it in terms of that series. For example, the sine/cosine functions form a basis and are used in Fourier expansions.
 
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