Log-concavity, Log-convexity, and Growth Order in White Noise Analysis

Being motivated by the work of Cochran et al. on the Gel'fand triple E] (L 2) E] , we are led to nd elementary functions to replace the exponential generating functions G and G 1== for the characterization of generalized and test functions. We deene the Legendre transformù for u in C +;log and the L-function Lu when u 2 C +;log is (log, exp)-convex. We show that u is equivalent to Lu. The dual Legendre function u is deened for u in C +;1=2. When u 2 C +;1=2 is (log, x 2)-convex, the L #-function L # u is deened and shown to be equivalent to L u. Application to the growth order of holomorphic functions on Ec is given. 1. Background and motivations Let E be a real nuclear space with topology given by a sequence of inner product norms fj j p g 1 p=0. Let E p be the completion of E with respect to the norm j j p. We will assume the following conditions: (a) There exists a constant 0 < < 1 such that j j j j 1 p j j p. (b) For any p 0, there exists some q p such that the inclusion mapping i q;p : E q ! E p is a Hilbert-Schmidt operator. Let E 0 be the dual space of E. By using the Riesz representation theorem to identify E 0 with its dual space we get the Gel'fand triple