5.2.A. Multidimensional Fourier transform, Fourier integral

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5.2.A. Multidimensional Fourier transform, Fourier integral

Definition 1. Multidimensional Fourier transform is defined as \begin{align} & \hat{f}(\mathbf{k})= \left(\frac{\kappa}{2\pi}\right)^n\iiint_{\mathbb{R}^n} f(\mathbf{x}) e^{-i\mathbf{k} \cdot \mathbf{x}}\,d^nx \tag{FT}\\ & \check{F}(x)= \left(\frac{1}{\kappa}\right)^n \iiint_{\mathbf{R}^n} F(\mathbf{k}) e^{i\mathbf{k} \cdot \mathbf{x}} \,d^n \mathbf{k} \tag{IFT} \end{align} with $\kappa=1$ (but here we will be a bit more flexible).

All the main properties of $1$-dimensional Fourier transform are preserved (with obvious modifications) but some less obvious modifications are mentioned:

Remark 1. Theorem 5.2.3(e) is replaced by \begin{equation} g(\mathbf{x})=f(Q \mathbf{x})\implies \hat{g}(\mathbf{k})= |\det Q|^{-1}\hat{f}(Q^{*\,-1} \mathbf{k}) \label{eq-5.2A.1} \end{equation} where $Q$ is a non-degenerate linear transformation.

Remark 2. Example 2. is replaced by the following: Let $f(x)=e^{-\frac{1}{2}A\mathbf{x}\cdot \mathbf{x}}$ where $A$ is a symmetric (but not necessarily real matrix) $A^T=A$ with positive definite real part: \begin{equation*} \Re (A\mathbf{x}\cdot \mathbf{x}) \ge \epsilon |\mathbf{x}|^2 \qquad\forall \mathbf{x} \end{equation*} with $\varepsilon >0$. One can prove that inverse matrix $A^{-1}$ has the same property and \begin{equation*} \hat{f}(\mathbf{k})= \left(\frac{\kappa}{\sqrt{2\pi}}\right)^n |\det A|^{-\frac{1}{2}} e^{-\frac{1}{2}A^{-1}\mathbf{k}\cdot\mathbf{k}}. \end{equation*}

Remark 3. Poisson summation formula (Theorem 5.) is replaced by \begin{equation} \sum_{\mathbf{m}\in \Gamma} f(\mathbf{m}) = \sum_{\mathbf{k}\in \Gamma^*} (2\pi )^n |\Omega|^{-1} \hat{f}(\mathbf{k}) . \label{eq-5.2A.2} \end{equation} (in notations of Section 4.B.

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