A linear continuous right inverse to the representation operator in $(LB)$-spaces

We study the question of the existence of a linear continuous right inverse to the representation operators in $(LB)$-spaces. It is obtained suficient conditions for the existence of such operators in the case of representations in delta-functions in spaces which are dual to weighted Fréchet spaces of entire functions. We state some conditions under which the results can be used for representations in systems of generalized exponential functions. Our study is based on the method developed by S. N. Melikhov for the dual situation and previous works of A. V. Abanin and the author on suffiient sets in weighted Fréchet spaces of entire functions and existence of a linear continuous left inverse for the corresponding restriction operator.