Acta _'vIathematica Academiae Scientiarum Hungaricae Tomus 23 (1--2), (1972), pp. 5--12. SPACES OF GENERALIZED ANALYTIC FUNCTIONS By D. E. MYERS (Tucson) This author in some previous papers [5], [6], [7] has investigated certain pro- perties of complex functions analytic in a strip in the complex plane. The strip was considered because it is the natural domain for functions that are Laplace transforms. MACKEY [3], ELLIOr [21, ARENS and SI~q6ER [1] among others have given definitions of analytic functions defined on some subset of G x ~, the generalized character group of a locally compact Abelian Group. Mackey and Elliot used derivatives with respect to semigroups, Arens and Singer the duality contained in a Poisson represen- tation. None of the above papers contain any elaboration of the properties of the class of analytic functions like the Cauchy-Riemann conditions or preservation of analyticity under uniform convergence. One result of this paper is the existence of necessary and sufficient conditions for analyticity which are analogous to the Cauchy-Riemann equations. In this paper I will also obtain those extensions which allow us to construct a Hilbert space of such functions. NOTATION. Let G be a locally compact group (not necessarily Abelian) and its character group. That is, C consists of those complex-valued functions, con- tinuous on G with modulus identically 1 and with a multiplicative property, d is also a locally compact group and the topology is that of uniform convergence on compact subsets of G. G denotes the set of real, continuous linear functionals on G. A subset 32, of G is said to be large convex if it is convex, contains the zero elem- ent and the closed linear span is G. G x G becomes a complex vector space by defining (u + iv)(x,, x2) = (uxi - vxz, vxl + ux 2) for u + iv complex and (xi, x2) E G x G. Finally for each x g. G, define a one-parameter subgroup of d by x[u] = exp (iux), u real. DEFINII"XON (MACKUV [4]). Let K be a large convex subset of G and x0 an interior point of K (i.e. there exists r > 1 such that rx C K). Let F(x, y) be a complex-valued function defined on Kx G. It is said to be analytic at (x0, Y0) if (i) Lira F(x~ + uxl' yoxz[u]) - F(xo, Yo) = F~: .... (xo, Yo) u~O U exists for every (xl, x2) ~ G. (ii) The above limit is a complex-homogeneous function of (Xl, x2). As will be shown the complex-homogeneity condition is analogous to requiring that the partial derivatives satisfy Cauchy-Riemann like conditions. Acta Mathematica Academiae Scientiarum Hungaricae 23, 1972