Still not a full answer, but maybe pointing towards the solution.
PiecewiseExpand[
FourierTransform[
Sinc[\[Omega]1Sinc[ω1 - \[Omega]2]ω2],
{\[Omega]1ω1, \[Omega]2ω2}, {t1, t2},
FourierParameters -> {1, -1}
],
t1 \[Element]∈ Reals
]
This does not work with Sinc[b (\[Omega]1ω1 - \[Omega]2ω2)], though and I don't know why. It should only be a matter of the transformation formula, but apparently, Mathematica is not clever enough to apply it automatically.
Still not a full answer, but maybe pointing towards the solution.
PiecewiseExpand[
FourierTransform[
Sinc[\[Omega]1 - \[Omega]2],
{\[Omega]1, \[Omega]2}, {t1, t2},
FourierParameters -> {1, -1}
],
t1 \[Element] Reals
]
This does not work with Sinc[b (\[Omega]1 - \[Omega]2)], though and I don't know why. It should only be a matter of the transformation formula, but apparently, Mathematica is not clever enough to apply it automatically.
Still not a full answer, but maybe pointing towards the solution.
PiecewiseExpand[
FourierTransform[
Sinc[ω1 - ω2],
{ω1, ω2}, {t1, t2},
FourierParameters -> {1, -1}
],
t1 ∈ Reals
]
This does not work with Sinc[b (ω1 - ω2)], though and I don't know why. It should only be a matter of the transformation formula, but apparently, Mathematica is not clever enough to apply it automatically.
Still not a full answer, but maybe pointing towards the solution.
PiecewiseExpand[
FourierTransform[
b Sinc[(\[Omega]1Sinc[\[Omega]1 - \[Omega]2)]\[Omega]2],
{\[Omega]1, \[Omega]2}, {t1, t2},
\!\(TraditionalForm\`FourierParametersFourierParameters -> {1, \(-1\)1}\),
Assumptions -> b > 0
],
t1 \[Element] Reals
]
This does not work with Sinc[b (\[Omega]1 - \[Omega]2)], though and I don't know why. It should only be a matter of the transformation formula, but apparently, Mathematica is not clever enough to apply it automatically.
PiecewiseExpand[
FourierTransform[
b Sinc[(\[Omega]1 - \[Omega]2)], {\[Omega]1, \[Omega]2}, {t1, t2},
\!\(TraditionalForm\`FourierParameters -> {1, \(-1\)}\),
Assumptions -> b > 0
],
t1 \[Element] Reals
]
Still not a full answer, but maybe pointing towards the solution.
PiecewiseExpand[
FourierTransform[
Sinc[\[Omega]1 - \[Omega]2],
{\[Omega]1, \[Omega]2}, {t1, t2},
FourierParameters -> {1, -1}
],
t1 \[Element] Reals
]
This does not work with Sinc[b (\[Omega]1 - \[Omega]2)], though and I don't know why. It should only be a matter of the transformation formula, but apparently, Mathematica is not clever enough to apply it automatically.
PiecewiseExpand[
FourierTransform[
b Sinc[(\[Omega]1 - \[Omega]2)], {\[Omega]1, \[Omega]2}, {t1, t2},
\!\(TraditionalForm\`FourierParameters -> {1, \(-1\)}\),
Assumptions -> b > 0
],
t1 \[Element] Reals
]