Convergence of Infinite Products

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

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

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

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

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

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

8. Convergence to Distributions are complex measures Theorem 2. If such that and total variation and then converges to a distribution with compact support. 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.

9. Interpolatory Subdivision

11. Jet Representation of Convolution, d=1

12. Sequence Space Proof of Theorem 2 Definition For let denote the space of complex sequences that satisfy Lemma 1 Lemma 2 Proof

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

14. Convergence to Analytic Functionals are complex measures Theorem 3. If such that and total variation and then converges to an analytic functional 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.

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