This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures. There is a close resemblance to convolution products and the Wiener algebra (of transforms) in functional analysis. The analysis term "kernel" (of a distribution) has also been adapted below in connection with certain special types of "distributors" (in the terminology of J. Bénabou) or "modules" (in the terminology of R. Street) in category theory. In using the term "graphic, in a very broad sense, we are clearly distinguishing the categorical methods employed in this article from standard Fourier and wavelet mathematics. The term "graphic" also applies to promultiplicative graphs, and related concepts, which can feature prominently in the theory.