1. Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml Convergence of Infinite Products

2. Fixed Point Formulation Semigroup Sequence Set Map Problem What topologies make

3. Calculus 101 Sequences and Series related by isomorphism Infinite Products

4. Probability probability measures on measure defined by convolution of measures defined by acts on by Theorem 1.converges if

6. Distributions with Compact Support Frechet space and Definition For open is the space of [T] Proposition 21.1 A linear function is in is a distributions with compact support in with order where

7. Fourier-Laplace-Borel Transform Definition For [H] Thm7.3.1 Paley-Wiener-Schwartz is compact and convex and let If is entire, then of orderand with Iff where

8. Convergence to Distributions are complex measures distribution with compact support. Theorem 2. If converges to a and such that total variation Proof First proved in [DD] using the Paley-Wiener-Schwartz Theorem. [L] gave another proof, based on the Taylor expansion, and used it to generalize the theorem to Lie groups. and then

9. Interpolatory Subdivision

11. Jet Representation of Convolution, d=1

12. Sequence Space Proof of Theorem 2 Definition Fordenote thelet space of complex sequencesthat satisfy Lemma 1 Lemma 2 Proof

13. Analytic Functionals [H] Definition 9.1.1 For compact is the space of linear forms such that for every open space of entire analytic functions on on the [M] Paley-Wiener-Ehrenpreis

14. Convergence to Analytic Functionals are complex measures analytic functional Theorem 3. If converges to an and such that total variation Proof First proved in [U] using the Paley-Wiener-Ehrenpreis Theorem. We gave another proof, based on the Taylor expansion, and used it to generalize the theorem to Lie groups. and then

15. References F.Treves, Topological Vector Spaces, Distributions, and Kernels, 1967 G.Deslauriers and S.Dubuc,Interpolation dyadic, in Fractals, Dimensions Non Entiers et Applications (edited by G. Cherbit), 1987 W.Lawton, Infinite convolution products and refinable distributions on Lie groups, Trans. Amer. Math. Soc., 352, p. 2913-2936, 2000. L.Hormander,The Analysis of Linear Partial Differential OperatorsI,1990 M.Uchida, On an infinite convolution product of measures, Proc. Japan Academy, 77, p. 20-21, 2001 M.Morimoto,Theory of the Sato hyperfunctions,Kyoritsu-Shuppan,1976