SCHWEITZER 1958 Problem 10

Prove that the function $f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta$ where $k$ is a positive integer, satisfies the following conditions:(i) $f(x)=0$ if $\mid x \mid \geq k$ and $f(x) \geq 0$ elsewhere;(ii) in interval $(l,l+1)$ $(l= -k, -k+1, \dots , k-1)$ the function $f(x)$ is a polynomial of degree $2k-1$ at most. (R. 7)