AMALGAMS OF L ' AND l

(i.i) ll/IU=( t \ n » — oo makes (L9 l ) into a Banach space. The idea of considering the amalgam (L, l\ as opposed to the Lebesgue space L = (L,l), is a natural one because it allows us to separate the global behavior from the local behavior of a function. This idea goes back to 1926 and Norbert Wiener who considered the special cases (L,1) and (L, l°°) in [Wl] and (L°°, I) and (L\ l°°) in [W2]. Other special cases have appeared sporadically since then, but the first systematic study of these spaces was undertaken in 1975 by F. Holland [HI]. After giving an account of the basic theory of amalgams on groups in §2, we show in the following sections how amalgams have arisen in various areas of analysis: almost periodic functions [Wl], Tauberian theorems [W2], extending the domain of the Fourier transform [Szl], Fourier multipliers [EHR], integral operators [BiS], product-convolution operators [BuS], positive definite functions [Coop], Fourier transforms of unbounded measures [H2], lacunarity [Fou2], the lower majorant property for H(R) [BaS], approximation theory [JR], algebras and modules [LVW], and the range of the Fourier transform [Ke]. The common theme is that, in many situations, an amalgam space (L9 l ) turns out to be exactly the right space that is needed to solve a problem or formulate a theory. In view of these occurrences, "the amalgam spaces . . . appear to be an idea whose time has come" [GdL]. We hope that the present article will help to make these spaces more widely known in the mathematical community.