An analogous definition of Fourier transform $\hat f(u) = \int_{-\infty}^{+\infty} f(t) \exp(- i u t) dt$ for sinc-function. We know the definition of Fourier transform $$\hat f(u) = \int_{-\infty}^{+\infty} f(t) \exp(- i u t) dt \ \ \ (*)$$
It is widely used in the analysis in the frequency of dynamical systems, in the resolution of differential equations and in signal theory. For example, in systems theory, the Fourier transform of the impulse response characterizes the frequency response of the system in question. 
The question that I do with this post is rather a curiosity. Is there exist, in literature, a transform similar to (*), where exp-function is replaced by sinc-function? For example
$$\hat f(u) = \int_{-\infty}^{+\infty} f(t) \operatorname{sinc}(i u t) dt$$
or
$$\hat f(u) = \int_{-\infty}^{+\infty} f(t) \operatorname{sinc}\left[i (u - t)\right] dt$$
or similar.
I hope I have guessed the tags. In any case, suggestions are welcome.
Thank you very much.
 A: Read this as a (too large) comment, not a full answer: The sinc-transform appears naturally when we take the Fourier transforms of spherical symmetric functions.
Take for example a function $f:\mathbb{R}^3\to \mathbb{R}$ for which $f(x,y,z) \equiv f(r)$ where $r = \sqrt{x^2+y^2+z^2}$, then the 3D Fourier transform becomes a 1D sinc-transform (of the slightly modified function $f\to 4\pi r^2 f$): 
$$F(\vec{k}) = \int f(\vec{x}) e^{-i\vec{k}\cdot \vec{x}}d^3 x = \int_0^\infty 4\pi f(r) r^2 \cdot\text{sinc}(|k|r) dr$$
see for example this page.
Another comment: the way you have written it, sinc($ix)$ - with the $i$, is not a very interesting transform kernel for real functions since sinc(i$x) \sim e^{|x|}/x$ for large $|x|$ so the integral would not converge for most functions we would be interested in taking the transform of.
A: Key reference is the

Nyquist–Shannon sampling theorem

And especially the
Whittaker–Shannon interpolation formula :
$$
x(t) = \sum_{n=-\infty}^{n=+\infty}x(nT)\operatorname{sinc}\left(\frac{t-nT}{T}\right)
$$
Note that it's not an integral but rather a sum that is meaningful
with $\,\operatorname{sinc}\,$ functions.
