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Daniele Tampieri
Daniele Tampieri
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In dual form, you're asking for conditions that characterize (or at least guarantee) that a given $L^1$ function is the Fourier transform of another function. To quote the introduction of Stein and Weiss' book on Fourier Analysis:

"Theorem 1.2 gives a necessary condition for a function to be a Fourier transform. Belonging to the class $C_0$, however, is far from being sufficient. There seems to be no simply satisfactory condition characterizing Fourier transforms of functions on $L^1(R^n)$$L^1(\Bbb R^n)$."

In dual form, you're asking for conditions that characterize (or at least guarantee) that a given $L^1$ function is the Fourier transform of another function. To quote the introduction of Stein and Weiss' book on Fourier Analysis:

"Theorem 1.2 gives a necessary condition for a function to be a Fourier transform. Belonging to the class $C_0$, however, is far from being sufficient. There seems to be no simply satisfactory condition characterizing Fourier transforms of functions on $L^1(R^n)$."

In dual form, you're asking for conditions that characterize (or at least guarantee) that a given $L^1$ function is the Fourier transform of another function. To quote the introduction of Stein and Weiss' book on Fourier Analysis:

"Theorem 1.2 gives a necessary condition for a function to be a Fourier transform. Belonging to the class $C_0$, however, is far from being sufficient. There seems to be no simply satisfactory condition characterizing Fourier transforms of functions on $L^1(\Bbb R^n)$."

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Mark Lewko
Mark Lewko
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In dual form, you're asking for conditions that characterize (or at least guarantee) that a given $L^1$ function is the Fourier transform of another function. To quote the introduction of Stein and Weiss' book on Fourier Analysis:

"Theorem 1.2 gives a necessary condition for a function to be a Fourier transform. Belonging to the class $C_0$, however, is far from being sufficient. There seems to be no simply satisfactory condition characterizing Fourier transforms of functions on $L^1(R^n)$."