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Convergence of Infinite Products. Wayne Lawton Department of Mathematics National University of Singapore [email protected] http://www.math.nus.edu.sg/~matwml. Fixed Point Formulation. Semigroup. Sequence Set. Map. Problem What topologies make. Calculus 101. Sequences and Series. - PowerPoint PPT Presentation
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Wayne Lawton
Department of Mathematics
National University of Singapore
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