Hi everybody,(adsbygoogle = window.adsbygoogle || []).push({});

I am looking for some help with a problem that has been nagging me for some time now.

I'm going to give you the gist of it, but I can provide more details if needed.

So, after some calculations I am left with a function of the following form

$$

F_L(y) = f(y) -S_L(y), \quad S_L(y) = \sum_{k=1}^{L/2-1} g_L(y_k,y) \Theta(y-y_k)

$$

where ## \Theta(y) ## is the Heaviside function limiting the sum, ##y\in[-1,1]##, and the ##y_k## are related to the roots of the unity, ##y_k = -\cos(2 \pi/L k)##.

The problem is that ##S_L(y)## is very close to the function ##f(y)##, so that ##F_L(y)## is much smaller than both ##f(y)## and ##S_L(y)##.

I tried to trade ##S_L(y)## for a (definite) integral, but I'm not sure that's the way to go.

I actually find that, at the leading order, ##S_L(y)=f(y)##, but I am sort of stuck in evaluating the sub-leading terms.

From other considerations I am pretty sure that ##F_L(y) \approx C_L (1-y^2)^{L+\ell} ## where ##\ell## is a number of the order of 1 ( which could be ignored for large ##L##), and ##C_L## is a constant exponentially decreasing with ##L##. I'm also able to verify this guess numerically, for not too large ##L##.

What I'd like is to actually prove that ##F_L## has the form I expect.

I hope someone can provide some insight.

Thanks a lot in advance

Franz

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# Difference of a function and a finite sum

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