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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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If the function is always non-negative, why do you think its partial sums' sequence can be negative?? If at all, it perhaps can be zero...but then the function will have to be identically zero. BTW, where did you read that thing about the non-positiveness of the those operators? I find it weird that they even bothered to mention that so I'd like to know why they did. – DonAntonio Oct 29 '12 at 14:25
@DonAntonio, I am thinking about such when I came to the idea of using Fejer operator in one of my work but before that I have to convince the audience that another theorem cannot be applied to what I am doing. I am sorry, I can't remember the title of the book and I simply jot these down when I am in the library since I can't think of an example that is claim as easy by the author out of curiosity. Yes you are very true indeed that if the function is non-negative, its partial sums always $\geq 0$. I manage to work this out and am satisfied with this conclusion. Now left is the question above. – Sandra Oct 29 '12 at 14:40
@Sandra: You write "if the function is non-negative, its partial sums always $\ge0$", but your question was "I would like to know if there is any simple function that is $\ge0$ but with its partial sums $S_m\le0$" -- does that mean that the question has changed and you now only want to know whether the partial sums can be $0$? If so, you should update the question so that people don't have to read through the comments to understand it. If not, please explain. – joriki Oct 29 '12 at 15:51
@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. – Sandra Oct 29 '12 at 17:02
up vote 1 down vote accepted

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: It seems we're maximally talking past each other. Since my answer says that there's no such function, I can't given an example. Also I didn't say anything about $x=0$. – joriki Oct 30 '12 at 13:22
Oopsy. I read wrongly your post. Sorry about that. Have been reading other posts together and gets confused between posts. – Sandra Oct 30 '12 at 13:29
Why the downvote? – joriki Oct 30 '12 at 14:37
I didn't click the downvote but I click the accepted answer. Thanks. – Sandra Oct 30 '12 at 15:33
@Sandra: Thanks for accepting the answer. My comment wasn't directed at you, just in general at the downvoter. – joriki Oct 30 '12 at 15:34

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