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.
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.
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.