= Flat-top kernel for density estimation
{title2=$\widehat K(t)=1\text{ near }0$}
Choose a real even smooth compactly supported <Fourier transform> $\eta$ equal to one near zero and let $K$ be its inverse <Fourier transform>. Repeated integration by parts makes $K$ a <Schwartz function>. It is real, integrable and square-integrable, with unit integral. <Moment differentiation of the Fourier transform> gives $\int x^jK(x)dx=i^j\eta^{(j)}(0)=0$ for every positive integer $j$. Such a <kernel for density estimation> therefore satisfies arbitrary finite-order moment cancellation, and necessarily takes negative values. If the transform plateau contains the frequency support of a <bandlimited function> $f$, the <convolution theorem> proves $K*f=f$.
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