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We construct explicitly the sequence in $L^1 (\\mathbb{R} )$ which is biorthogonal to the system $1$, $\\exp ( i \\pi n x)$, $\\exp ( i \\pi n/ x)$, $n \\in \\mathbb{Z} \\setminus \\{0\\}$, and show that it is complete in $L^1 (\\mathbb{R})$. 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