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What is the inverse Fourier transform of $$ \tilde f(k) = \frac{\sin(ck)}{ck} $$ where $k =|\mathbf{k}|$ and $\mathbf{k} \in \mathbb R^n$. The inverse transform should be $$ f(\mathbf x) = \mathcal{F}^{-1}[\tilde f](\mathbf x)=\frac{1}{(2\pi)^\frac{n}{2}}\int_{\mathbb R^n}\frac{\sin(ck)}{ck} e^{i \mathbf x \cdot \mathbf k} dk_1\dots dk_n. $$ My attempt was based on considering $$\mathbf x = |\mathbf x|(\cos \alpha_1,...,\cos \alpha_n)\quad \text{and} \quad \mathbf k = |\mathbf k|(\cos \beta_1,...,\cos \beta_n) $$ so I get $$ \mathbf x \cdot \mathbf k = xk (\cos \alpha_1 \cos \beta_1,...,\cos \alpha_n\cos \beta_n) = xk\ \phi(\alpha,\beta) $$ where $$ \sum_{j=1}^n \cos^2(\alpha_j) = \sum_{j=1}^n \cos^2(\beta_j) = 1 $$ so they are related, and somehow $dk_1\dots dk_n$ becomes proportional to $k^{n-1}dk$, that is $$ f(\mathbf x) = \mathcal{F}^{-1}[\tilde f](\mathbf x) = \frac{1}{(2\pi)^\frac{n}{2}}\int_{[0,2\pi]^{n-1}}\left( \int_{\mathbb R^+}\frac{\sin(ck)}{ck} e^{i xk\ \phi(\alpha,\beta)} k^{n-1}dk \right) \psi(\beta)d\beta_1\dots\beta_{n-1} $$ My problem is the integral wrt $k$, since for 3-dimensional case I can use the spherical coordinates, so $\psi(\beta)$ is known. But I don't know yet how to proceed, this integral doesn't seem to converge.

Physor
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  • For even $n$, the FT of $\sin(r)/r$ is a regularization of $C_n (1 - \rho^2)^{(1 - n)/2} [\rho < 1]$. The exact form of the regularization doesn't seem to be particulary simple. For odd $n \geq 3$, the FT is $C_n \delta^{((n - 3)/2)}(1 - \rho^2)$. – Maxim Sep 26 '20 at 17:02
  • @Maxim I wish I understand what you mean, could you explain more ? – Physor Sep 26 '20 at 17:05
  • In the above, $r = |\boldsymbol x|, , \rho = |\boldsymbol k|$ and $C_n$ are some constants. For all $n \geq 3$, the FT is a singular distribution, not an ordinary function. $(1 - \rho^2)^{(1 - n)/2} [\rho < 1]$ is defined as $$((1 - \rho^2)^{(1 - n)/2} [\rho < 1], \phi) = \ \int_0^1 (1 - \rho^2)^{(1 - n)/2} \left( \phi(\rho) - \phi(1) - \ldots - \frac {\phi^{(m)}(1)} {m!} (\rho - 1)^m \right) d\rho + \ a_0 \phi(1) + \ldots + a_m \phi^{(m)}(1),$$ where $m$ and $a_i$ are also some constants (for a given $n \in 2 \mathbb N$). – Maxim Sep 26 '20 at 17:30
  • @Maxim could you please name a book or the field that treats this point exaclty, because I think there are a lot of gaps in my knowlodge, that I can't understand you answer well, thanks in advance – Physor Sep 26 '20 at 17:34
  • I don't have a reference for these particular formulas. For the theory of distributions, see Gelfand, Shilov, Generalized functions, Vol. 1. – Maxim Sep 26 '20 at 17:46

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