On the Fourier-Borel transformation and spaces of entire functions in a normed space. (English) Zbl 0568.46036

Functional analysis, holomorphy and approximation theory, Proc. Semin., Rio de Janeiro 1981, North-Holland Math. Stud. 86, 139-169 (1984).
[For the entire collection see Zbl 0529.00027.]
The author continues earlier work by himself, Gupta, and Martineau, motivated by the study of convolution operators in spaces of analytic functions on an infinite dimensional normed space [See, for example, the author’s papers Math. Z. 162, 113-123 (1978; Zbl 0402.46028) and Math. Z. 171, 289-290 (1980; Zbl 0429.46028)]. The primary interest here is in the use of the Fourier-Borel transformation in the identification of the strong duals of spaces of entire functions of finite order and exponential type with other spaces of the same kind. Although these results too complicated to be given here, they are essential in proving Malgrange type theorems for existence and approximation of solutions to convolution equations. This application of the work done here will appear elsewhere.
Reviewer: R.M.Aron


46G20 Infinite-dimensional holomorphy
58D25 Equations in function spaces; evolution equations