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Question: I would like to know if there is any simple function that is $\geq 0$ but with its partial sums $S_{m} \leq 0$?

Note: After much discussion, it would seem this question is not possible to be true.

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    @joriki: No, there is no change in my question. The question "if the function is non-negative, its partial sums always $\geq 0$" is merely my explanation that I had worked on it and arrive at a satisfactory answer myself. This is shown in my next sentence "I manage to work this out..." The question asked in my original post remain unchanged and is waiting for anyone who can share their thoughts.2012-10-29

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I gather that you're considering functions in $C(0,2\pi)$ and their Fourier series, and you want to know whether there's such a function such that it is non-negative and the partial sums of its Fourier series are non-positive.

The answer is trivially "yes", since the zero function has that property.

Other than that, there is no such function. If the function is non-zero anywhere, it is by continuity non-zero in some neighbourhood. Thus, since it's non-negative, its integral is positive. This is the $0$-th Fourier coefficient, and it is the integral of any partial sum of the Fourier series. Thus the integral of all partial sums is positive, so all partial sums must take positive values somewhere.

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    @Sandra: Thanks for accepting the answer. My comment wasn't directed at you, just in general at the downvoter.2012-10-30