Adv. Pure Appl. Math. 3 (2012), 113 – 122 DOI 10.1515/APAM.2011.015 © de Gruyter 2012 A characterisation of the Weyl transform R. Lakshmi Lavanya and S. Thangavelu Abstract. A theorem of Alesker et al. says that the Fourier transform on R n is essentially the only transform on the space of tempered distributions which interchanges convolutions and products. In this note we obtain a similar characterisation for the Weyl transform. Keywords. Schwartz class, tempered distributions, Weyl transform, Fourier–Weyl transform, noncommutative derivations. 2010 Mathematics Subject Classification. 42A85, 43A32. 1 Introduction The Fourier transform on R n has been characterised in different ways by its prop- erties with respect to translations, dilations, products and convolutions. For ex- ample, see the work of Jaming [3] and the references therein. It is well known that the Fourier transform F is a bijection on both the Schwartz space  .R n / and its dual  0 .R n / and it interchanges convolutions and products. In an interesting work Alesker et al. [1] have proved that these properties essentially characterise the Euclidean Fourier transform. Theorem 1.1. Assume that T W  .R n / !  .R n / is a bijection which admits a bijective extension T 0 W  0 .R n / !  0 .R n / such that for all f 2  .R n / and ' 2  0 .R n /, we have T.f  '/ D T.f/T.'/ and T.f'/ D T.f/  T.'/: Then, T is essentially the Fourier transform F , that is, for some B 2 GL n .R/ with j det B jD 1, we have either Tf D F .f ı B/ or Tf D F .f ı B/ for all f in  .R n /. It is interesting to note that the operator T is neither assumed to be continuous nor linear. It is shown that any such T satisfying the hypothesis of the theorem Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/2/15 5:17 AM