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LCKurtz
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Jan26-12, 08:30 PM
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If you want to say the set of polynomials are to represent a basis, you need to talk about the notion of convergence or what topology you are using on the function space. For example, consider C[0,1], the continuous real functions on [0,1] as your function space where convergence ##f_n\rightarrow f## means the sequence ##f_n## converges uniformly to ##f##. Now, it is true that the set of polynomials form a basis in the sense that given any ##f## and ##\epsilon > 0##, you can find a polynomial p with ##\|f-p\|<\epsilon##. We say that the set of polynomials is dense in C[0,1] and they form a topological basis. But that is not the same thing as saying every function in C[0,1], even if it has derivatives of all orders, can be well approximated by a Taylor polynomial. There are infinitely smooth functions whose Taylor expansion just gives 0. Taylor polynomials are too specialized for that particular job.