Window transform

Let us consider the RHS of Eq. (33) term by term.
1) $\bf {f_0F_0}$

$$\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\text... ...extbf{m})\tilde{a}_{\textbf{p},\textbf{x}_1}d\textbf{p}d\textbf{m}d\textbf{x}_1$$

$$=\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\tex... ...{x}}d\textbf{m}=\omega _{\textbf{k}\textbf{x}}\tilde{a}_{\textbf{k}\textbf{x}},$$ (87)

2) $\bf {f_1F_0}$

$$\left(\frac{1}{\sqrt{\pi}}\right)^d\int\textbf{m}\cdot\nabla_\tex... ...}F(\textbf{k},\ae)\tilde{a}_{\textbf{p},\textbf{x}_1}dpd\textbf{m}d\textbf{x}_1$$

$$=-\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\te... ...xtbf{m})\tilde{a}_{\textbf{p},\textbf{x}_1} d\textbf{p}d\textbf{m}d\textbf{x}_1$$

$$~~-\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\t... ...textbf{p}\tilde{a}_{\textbf{p},\textbf{x}_1}d\textbf{p}d\textbf{x}_1d\textbf{m}$$

$$=-\textbf{x}\cdot\nabla_\textbf{x}\omega _{\textbf{k}\textbf{x}}\... ... _{\textbf{k}\textbf{x}}\cdot\nabla_\textbf{k}\tilde{a}_{\textbf{k}\textbf{x}},$$ (88)

3) $\bf {f_0F_1}$

$$\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\text... ...extbf{m})\tilde{a}_{\textbf{p},\textbf{x}_1}d\textbf{p}d\textbf{m}d\textbf{x}_1$$

$$=-i\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\t... ...i\textbf{m}\cdot\textbf{x}}\nabla_\textbf{k}F(\textbf{k},\textbf{m})d\textbf{m}$$

$$+\left(\frac{1}{\sqrt{\pi}}\right)^d\int \hat{f}((\textbf{k}-\tex... ...}\textbf{x}}\textbf{m}\cdot\nabla_\textbf{k}F(\textbf{k},\textbf{m})d\textbf{m}$$

$$=-i\nabla_\textbf{k}\omega _{\textbf{k}\textbf{x}}\cdot\nabla_\te... ...la_{\textbf{x}})\omega _{\textbf{k}\textbf{x}}\tilde{a}_{\textbf{k}\textbf{x}},$$ (89)

4) $\bf {f_1F_1}$

$$\int\textbf{m}\cdot\nabla_\textbf{p}\hat{f}((\textbf{k}-\textbf{p... ...extbf{m})\tilde{a}_{\textbf{p},\textbf{x}_1}d\textbf{p}d\textbf{m}d\textbf{x}_1$$

$$=\int \hat{f}((\textbf{k}-\textbf{p})/\varepsilon^*)ie^{i(\textbf... ...t\textbf{x}}\textbf{m}\cdot\nabla_\textbf{k}F(\textbf{k},\textbf{m})d\textbf{m}$$

$$+\int \hat{f}((\textbf{k}-\textbf{p})/\varepsilon^*)e^{i(\textbf{... ...t\textbf{x}}\textbf{m}\cdot\nabla_\textbf{k}F(\textbf{k},\textbf{m})d\textbf{m}$$

$$+\int \hat{f}((\textbf{k}-\textbf{p})/\varepsilon^*)e^{i(\textbf{... ...t\textbf{x}}\textbf{m}\cdot\nabla_\textbf{k}F(\textbf{k},\textbf{m})d\textbf{m}$$

$$=i(\nabla_{\textbf{k}}\cdot\nabla_{\textbf{x}})\omega _{kx}\textb... ...}(\nabla_{\textbf{k}}\cdot\nabla_{\textbf{x}})\tilde{a}_{\textbf{k}\textbf{x}}.$$ (90)