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This paper develops a theory of harmonic analysis (Fourier series, approximation, convolution, and singular integrals) for a general class of Banach function or distribution spaces. Continuity of singular convolution operators and convergence of trigonometric series is shown with respect to the norms of all the spaces in this class; the maximal class supporting a theory of the type developed in this paper is characterized (for other classes other theories exist). Theorems are formulated in category language throughout. However only elementary category theory is needed, and for most results the notions of functor and natural mapping are sufficient. Introduction. Over the past thirty years the application of Banach space theory to approximation theory, to integral equations, and to differential equations has led to efforts to obtain results on approximation, stability, and existence of solutions in as wide a class of Banach spaces as possible. By now results for most questions of approximation and existence of solutions have been proved for the II spaces and spaces constructed from them; in some cases results have been extended to the classes of Orlicz spaces or Lorentz spaces; and for linear ordinary differential equations a theory of stability has been worked out as far as general classes of Banach function spaces. In these applications there are very many cases of Banach spaces all constructed from other Banach spaces according to very similar schemes: a Banach space suited to a given problem will be constructed from other Banach spaces according to a pattern which depends on the problem. As examples one may give the Sobolev spaces or the Lipschitz spaces in partial differential equations; in ordinary differential equations one may adduce the constructions of full or lean spaces in the work of Massera and Shaeffer. For very many of these results on differential equations or approximation theory, the continuity of integral operators from one space to another plays a crucial part. Very much effort has been devoted to proving such results on continuity for many classes of operators and spaces, such as the If spaces, the Orlicz spaces, Lorentz spaces, and generalizations of these. Received by the editors February 23, 1972 and, in revised form, July 18, 1972. AMS (MOS) subject classifications (1970). Primary 46E30, 42-00, 46M15, 43A95; Secondary 42A18, 42A20,42A40, 46E35,43A75.

Harmonic analysis for functors on categories of Banach spaces of distributions