Wayne Lawton

Department of Mathematics

National University of Singapore

matwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwml

Convergence of Infinite Products

Fixed Point Formulation

Semigroup cccM ),,(

Sequence Set Mcccc n :),,,( 321

Map ),,(),,,( 321321 ccccccT

Problem What topologies make

?),,,(),,,( 321 ccccT k

Calculus 101

),,(),,,(),0,( 321321 ccccccTR

Sequences and Series

RR log

),,(),,,(),1,( 321321 ccccccTR

related by isomorphism

Infinite Products

Probability

M))(())(( xhxh

)0()( hh

probability measures on

),,(),,,(),,( 321321 TM

,dR

measure defined by

convolution of measures defined by

2121 )()( hh

acts on)(0 RC by

Theorem 1. ),,,( 321 kT converges

if kkk rrB ),()(support

212

102

11 )2(2)(1 xx kk

k 21

412

102

12

Distributions with Compact Support

Frechet space and

21

||

2)(sup||||

Nn

n

xN xD

Definition For open is the space of

[T] Proposition 21.1 A linear function

)(, CRd

0 real,0integer ,compact NKis in

is a)( C

.

CCL )(:

distributions with compact support in

)( C with order N

NhhLCh |||||)(|),(

where ),...,( 1 ddnn

|||||| 1 ddnn

Fourier-Laplace-Borel TransformDefinition For

[H] Thm7.3.1 Paley-Wiener-Schwartz

)( C

)] im(exp[||1)(0 zHzzF KN

is compact and convex and

CC d :̂dCzzxixhhz ),2exp()(),()(ˆ

let

dRK

d

KxK RyyxyH

,sup)(

IfCCF d : is entire, then ̂F

)( C of orderN andwith

Iff K)(support

where

Convergence to Distributionsare complex measures

distribution with compact support.

1Theorem 2. If

),,,( 321 kT converges to a

andk

k cr such that

total variation

,1)1( k,|| k

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.

k

andthen

Interpolatory Subdivision 23

21

21

23 32

1329

021

329

321

1

k 21

)(2)(1 xx kk

2101 2 )2(2)(1 xx kk

)(

)2(

)1(

nh

h

h

h

Jet Representation of Convolution, d=1 jjj

j rjtmRCh !/)()1(),(

00

000

00

0

2

11

21

n

n

n

m

mm

mmm

),()()()( hRhJAhhJ nnnn

)(hJ n )(nA ),( hRn

1

22

11

||||

||||

||||

||||

hr

hr

hr

hr

nn

nn

Sequence Space Proof of Theorem 2

)()()( 2121 SSS

0k

sks kPsP )(|| polynomial

Definition For denote thelet )(Sspace of complex sequences s that satisfyLemma 1

)(s of sums partial seq)( SSs Lemma 2

)(||||,0 iNik ShNi

kk hhRCh 1),(

Proof 011)(

0 ||)(||..|||||||| kkkiikkiik hhJssphJh

)()|||||)((| 1)1( ShmOs nkk

nk

)()|||||)(||||||)((| 2211

)2( ShmhmOs nkknkknk

1 if order dist)(||||induction 0 nk

nk nhSh

Analytic Functionals

|)(|sup|)(|,0 zhhAhz

[H] Definition 9.1.1 For compactis the space of linear forms

such that for every open dCK space

dRK

A)(KA

of entire analytic functions on dC

on the

[M] Paley-Wiener-Ehrenpreis ̂F 0,0,0 CR

)(Im||exp|)(| zHzCzF K

Convergence to Analytic Functionalsare complex measures

analytic functional

1,0 krTheorem 3. If

),,,( 321 kT converges to an

and

1 krRsuch that

total variation

,1)1( k,|| k

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.

k

andthen

)(),( RBKKA

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