Consider the Fourier transform defined as
$$ \mathcal{F}[f](p) = \int_{-\infty}^{+\infty}f(x)e^{-j2\pi xp}dx $$
Assume that
$$ f(x) = \left\{ \begin{array}{ll} 1 & |x| < \frac{1}{2} \\ 0 & \text{otherwise} \end{array} \right. $$
under such hypothesis we now that
$$ \mathcal{F}[f](p) = \hat{f}(p)= \frac{\sin(\pi p)}{\pi p} = \text{sinc(p)} $$
I'm trying to build a sequence whose convergence will be studied later, for now I'm just interested in the construction. Let
$$ p_{k+1} = \left\{ \begin{array}{ll} p_0 & k = -1 \\ p_{k} + \delta_{k} & \text{otherwise} \end{array} \right. $$
where $\delta_k$ is a convergent sequence, this should imply that $p_k$ converges as well. So for all $k$ we have fixed $\delta_k$. Say I'd like to evaluate $\hat{f}$ at $p_{k+1}$ then I'd have
$$ \hat{f}(p_{k+1}) = \int_{-\frac{1}{2}}^{+\frac{1}{2}}e^{-j2\pi xp_{k+1}}dx = \int_{-\frac{1}{2}}^{+\frac{1}{2}}e^{-j2\pi xp_k}e^{-j2\pi x\delta_k}dx $$
Is there any manipulation that maybe I'm missing that would allow to express the last integral in a useful form? It does resemble a bit the fourier series, if $\delta_k$ would be $k$ (though it wouldn't make the sequence converge) that would be a fourier coefficient. But I'm sure there's something there that can be useful.
I'm mostly looking for a recursive expression.