$f(n)=1$ where n is integer. Fourier Transform is of finite bandwidth and repeated differentiable at $0$

Question

I am looking for an example of a function $f(x)$ that satisfies the following requirements:

1. $f(n) = 1$ for all $n \in \mathbb Z$
2. If Fourier Transform is defined as $$G(y) = FT[f(x)] = \int_{-\infty}^{\infty} f(x) e^{i 2 \pi y x} dx$$ $G(y) = 0$ for $y \lt -1/4$ and $y \gt 1/4$
3. G(y) doesn't have any discontinuities or diracdelta functions in ($-\frac{1}{4},\frac{1}{4}$)
4. G(y) is repeatedly differentiable at $y=0$ (ie $G'(0)$, $G''(0)$, ... are all defined and finite)

For the trivial examples I can think of, at least one condition is not met. For example, for $f(x) = 1$, it meets first 2 conditions, but not the other two. For $f(x) = \cos(2 \pi x)$, the first and last conditions are met, but not the second and third.

Any help figuring out what $f(x)$ will satisy my requirement?