Translation and modulation invariant Hilbert spaces

Abstract Let $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Rd which contains at least one non-zero element the Feichtinger algebra $$S_0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">S0 and is continuously embedded in $$\mathscr {D}'$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">D′ . If translation modulation invariant, also sense its norm, then we prove that H= L^2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">H=L2 , with same norm apart from multiplicative constant.