Difference-differential fields of continuous functions

The set C of complex-valued continuous functions on [0,\infty) is a ring by the addition and the convolution. It has the quotient field Q©, by which J. Mikusinski developed his operational calculus. In this paper, we revisit a derivation and a transforming operator for Q© written in his textbook, and define another transforming operator related to the q-shift operator, which gives structures of a q-difference field and a difference field of Mahler type to Q©. Appropriate derivatives are also considered.