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DLR in the Matsubara frequency domain\(\eqref{dlrexp}\) yields

\[\begin{equation} - G(i nu_n) \approx \sum_{l=1}^r K(i \nu_n,\omega_l) \widehat{g}_l, \label{dlrexp_imfreq} \tag{2} + G(i \nu_n) \approx \sum_{l=1}^r K(i \nu_n,\omega_l) \widehat{g}_l, \label{dlrexp_imfreq} \tag{2} \end{equation}\]

so we see that the DLR expansion can be evaluated directly in imaginary time or imaginary frequency, i.e. the Fourier transform is performed analytically. 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