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. Benabou) 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.
Keywords: monoidal category, promonoidal category, convolution, Fourier transform
2000 MSC: 18D10, 18A25
Theory and Applications of Categories, Vol. 25, 2011, No. 5, pp 118-141.