Pseudo-differential operators with  the global symbols of Shubin class act continuously on the space of tempered distributions. The extension of this class on the space of tempered ultradistributions of Beurling and Roumieu type was done by the use of the convolution on these spaces studied by the authors. Moreover, the Anti-Wick quantization of a standard symbol $a$  equals to the Weyl quantization of a symbol $b$ through  the convolution of $a$ and the gaussian kernel $e^{-|\cdot|^2}$ is extended to the new class of symbols. We will present, first, new results related to the convolution, second, the properties of the Anti-Wick and Weyl quantization for a new class of symbols and, third,the construction of the largest subspace of ultradistributions for which the convolution with the gaussian kernel exist.