# 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.

• You may be interested in Chapter 7 of Keener's "Principles of Applied mathematics". He takes a general approach to deriving Fourier-type transforms. – Dunham Apr 2 '15 at 16:06
• A $\mathrm{sinc}$ transform is substantially the same as a Fourier $\sin$ transform. @Winther's answer shows that it can be interpreted as a three-dimensional Fourier transform of radially-symmetric functions even if the transform kernel is not $\mathrm{sinc}$ but $x\mapsto x\sin x$. Tables of the Fourier $\sin$-transform are available. For instance the Bateman's legacy books by Erdélyi et al. or the russians Prudnikov et al. – Tom-Tom Apr 7 '15 at 9:02

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 $f\to 4\pi r^2 f$):

$$F(\vec{k}) = \int f(\vec{x}) e^{-i\vec{k}\cdot \vec{x}}d^3 x = \int_0^\infty 4\pi f(r) r^2 \cdot\text{sinc}(|k|r) dr$$

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.
• More generally any Bessel function $J_\nu$ can be the kernel of a Fourier-like transform. In $\mathbf{R}^d$, radially symmetric functions have a Fourier-like transform using $J_{d/2-1}$ as kernel. Note that $J_{1/2}=\mathrm{sinc}$. – Tom-Tom Apr 7 '15 at 9:05
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.
• This nice formula follows from the Fourier transform of the so-called gate function which is defined by $g(x)=1$ if $x\in[0,1]$ and $g(x)=0$ otherwise. It is useful to transform a function defined on integers into a function defined on the reals, but it is not exactly what I would call a Fourier-type transform. – Tom-Tom Apr 7 '15 at 8:26