PX275 - H8 - multidimensional convolutions

\begin{matrix} f (x, y) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{i k_{x} x} \frac{1}{2 π} \int_{- \infty}^{\infty} e^{i k_{y} y} \tilde{f} (k_{x}, k_{y}) d k_{x} d k_{y} \\ f (k_{x}, k_{y}) = \int_{- \infty}^{\infty} e^{- i k_{x} x} \int_{- \infty}^{\infty} e^{- i k_{y} y} f (k_{x}, k_{y}) d x d y \end{matrix}

\begin{matrix} \vec{r} = (x_{1}, x_{2}, \dots x_{N}) \\ \vec{k} = (k_{1}, k_{2}, \dots k_{N}) \\ \tilde{f} (\vec{k}) = \int_{R^{N}} \exp (- i \vec{k} \cdot \vec{r}) f (\vec{r}) d^{N} \vec{r} \\ f (\vec{r}) = {(\frac{1}{2 π})}^{N} \int_{R^{N}} \exp (- i \vec{k} \cdot \vec{r}) \tilde{f} (\vec{k}) d^{N} \vec{k} \end{matrix} $ $ - $ R^{N} $ i n d i c a t e s t h a t t h e r e i s a f a m i l y o f $ N $ i n t e g r a l s, o n e f o r e a c h $ x_{i} : $ $ $ \int_{R} d^{N} \vec{r} = \int_{x_{1}} \int_{x_{2}} \dots \int_{x_{N}} d x_{1} d x_{2} \dots d x_{N}

f * g (\vec{r}) = \int_{R} f (\vec{r}) g (\vec{r} - \vec{z}) d^{N} \vec{z}

\begin{matrix} f (x, y) = \frac{1}{2 π σ^{2}} \exp (- \frac{x^{2} + y^{2}}{2 σ^{2}}) \\ \tilde{f} (k_{x}, k_{y}) = \iint_{- \infty}^{\infty} \exp (- i (k_{x} x + k_{y} y)) f (x, y) d x d y \\ = \exp (\frac{1}{2} (k_{x}^{2} + k_{y}^{2}) σ^{2}) \end{matrix}

so, the gaussian remains a gaussian after FT
the variance in $k$ -space is the inverse of the variance in $x$ -space ( $1 / σ^{2}$ )