  We give a simple proof of the Kernel theorem for the space of tempered
ultradistributions of Beurling - Komatsu type, using the characterization of
Fourier-Hermite coefficients of the elements of the space. We prove in details
that the test space of tempered ultradistributions of Beurling - Komatsu type
can be identified with the space of sequences of ultrapolynomal falloff and its
dual space with the space of sequences of ultrapolynomial growth. As a
consequence of the Kernel theorem we have that the Weyl transform can be
extended on a space of tempered ultradistributions of Beurling - Komatsu type.
