2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123331We 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.accepted for publication in Integral transforms and special functionsFunctional AnalysisPrimary 46F05; Secondary 46F12, 42A16, 35Stext