Part 1: The following function (for $N=256$) has the plot shown nexr
$$ G(x) = \frac{1}{N}\text{exp} \bigg( j \frac{\pi}{2} \, x(N-1)\bigg) \frac{\sin (\frac{\pi N}{2} x)}{\sin (\frac{\pi}{2} x )}$$
$$G(0)=1$$
A good approximation of this function would be the piecewise function given next $$G_a(x) = \left\{ \begin{array}{ll} 1 & \mbox{if $ - \frac{2}{N}\leq x \leq \frac{2}{N} $};\\ 0 & \mbox{if $\mbox{OW}$}.\end{array} \right. $$
Part 2: Assume I have a function $$f(y) = \int_{-2}^{2}\frac{f_X(x)}{1+y^{-3}|G(x)|^2}\, dx$$ with $$0<f_X(x) <1, \,\,\,\,\,\,\ \int_{-2}^{2}f_X(x)\, dx=1 $$ $$f_X(x) =\int_{\max\{-1,-1-x\}}^{\min\{1,1-x\}} \left(\frac{1}{\pi^2 \sqrt{1-(x+v)^2}}\frac{1}{\sqrt{1-v^2}}\right) dv $$
If I use the approximate $G_a(x)$ ( which is way easier than the original function) then I can approximate $f(y)$ by $$f_a(y)= \int_{-2}^{N/2} f_X(x)\, dx + \int_{N/2}^{2} f_X(x)\, dx + \int_{-N/2}^{N/2}\frac{f_X(x)}{1+y^{-3}} \, dx $$
Question Which is correct $$f_a(y)>f(y)$$ $$f_a(y)<f(y)$$
Thanks