Analytic Of Khavinson D . Seattle August , 2019 I - - - - PDF document

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Variations ( Eternal ) the Theme on Continuation Analytic Of Khavinson D . Seattle August , 2019 I - - " Between real two the truths of and domain easiest the , quite often shortest path through the complex

slide-1
SLIDE 1

Variations

  • n

the ( Eternal ) Theme

Of

Analytic

Continuation

D .

Khavinson

Seattle

,

August

2019
slide-2
SLIDE 2
  • I
  • "

Between

two truths
  • f
the real domain , the easiest and shortest

path

quite

  • ften

passes

through

the

complex

domain . "

P

.

PAINLEVE

,

1900

slide-3
SLIDE 3
  • 2
  • I
. A warm
  • up
.

43¥

.

When

does the series

⇐ anzn

represent

a rational

function ? A.=

Iff

FN

:
  • V
no , N
  • utta

.÷÷÷÷÷÷)=o

Ex . Fz = I Z " , N =/ . ( Kronecker , 1881 ) . Cii )

I

Anz "

wlroc

=/ extends to El

{

I }

iff

an
  • g
cry ,

g

is entire ,
  • f
the minimal

type

. ( L . Lean
  • 1899
, S . Wigert
  • 1900
, A . Faber
  • 1903
. ) × Ex . Ecoscrjzn.TL

hmm

slide-4
SLIDE 4
  • 3
  • (
iii ) "

%

s . s '

HTS

. ,

f HIE ginn

÷÷÷÷÷÷

:

> cot 6 . log r

J

( D

K
  • '
97

)

(

Held

s also

for

g

  • f
" minimal type " in

Sy

. )

LET

IIEimdeessg.sk

?

A ? ? ? Most

likely

" Yes " .

hmmm

slide-5
SLIDE 5
  • 4
  • I
. " More serious '!

ODE

rs .

PDE

Li ) w ' " t an . , Cz ) w " '

I

. . t Golz ) WH )=fH c ⇒ wco ) = we , . . . , w " ' " C D= Wn
  • i
( Cauchy's Problem

)

thrum

If

{

a ; }! " are

analytic

in r , O E SL , all solutions
  • f

( * )

extend to r .

( ii ) txamplemMIE-z.EE

, C** ) w CZ ,,o) = f ( Z

D.

entire

we

, , =

f

Gz

, )
  • might

develop

singularity

  • n
T : =

{

2- , 2- a = I } 72

my

r
  • Where

do

they

come from ?

L

hTwhyIsiTesT

Z .

lieonthesameraeiet.us?

slide-6
SLIDE 6
  • 5
  • why
III.

potentialtheory.me

( i )

( G

.

Herglotz

, 1914 ) . A = solid in IR ? T
  • Or
  • algebraic
new::÷k÷f.÷ . potential

)

(a)

How far can un be continued C

harmonically )

into r

?

(b)

What

kind

  • f

singularities

does it encounter

inside

?

Ee . T : = { 1×1 a

13

Arising

use

¥ ,

. ( mean value thru

)

.

Surprise

: La)

1123403

for ANY entire Lb

?

Algebraic

when

pdfssi.atd.pt?fzYJaemms.Iaegdefl

.
slide-7
SLIDE 7
  • 6
  • "

High

Ground

" ( OK
  • H
. S . Shapiro , '8g

On

::¥÷

:

:

T n Ix
  • y
I mum

Cauchy 's Problem

: DM =p near T u =

Lump

  • uts

.LI/n--oMtn--oH.J

is the

desired

M
  • Us
,p

inside

continuation The

H

. G . Question : 2- Are the

singularities

  • f
solutions
  • f

( Cp)

C * ) dictated

by

the initial

Variety

T and the DO

%%%a::t÷÷q;m-m)

slide-8
SLIDE 8
  • 7
  • 2-

T.leraybclocal.IT#ryt5os-6s

Exam

ples:C)T-{w=Z32#mmmg

E

"i÷÷:%±⇒¥

( Z

W
  • U
= : M
  • modified
Schwarz potential Caution :

(

  • ,
  • )
is a " bad "

point

( characteristic

)

,

Cauchy

  • Keralerskaya
Than .

fails )

. U =

ZI

+

few

"3 is

ramified

around

{

w
  • 0 }
, a characteristic ( wrt

zw )

plane TANGENT to T at (
  • ,
  • )
.

Kray

's principle ;

#n=u#EnE#eEt:e

:* :p

: :#

vis tic from T Clocally)
slide-9
SLIDE 9
  • 8
  • (2)

Recall

{ IF

  • _

ziff

we ,

,0)=fCz

,)
  • entire

w=fG÷z )

.
  • z=
sing . set is the "

Leroy

tangent

" to

{22--0}

at .
  • 5={77--1}
  • I
(3)

(a) Ellipses

.IE/lipsoids/Spkeres

µ

*

A

EL

III.

  • x. aeg
, has 4 char . points '

+

I
  • tf
R2 I ;

Leroy 's

1 ! I IF

ET

char .

tangent

, , § All

log

potentials

Is

, with entire densities

#←

extend to IRZ a { facing
slide-10
SLIDE 10
  • g
  • Generally

speaking

Leroy's

Principle

is local . But in 2
  • dim
is globed for D. O .

L

' . =D t lower terms . Thanks to Riemann 's method
  • f

integrating

hyperbolic

eggs

in 2
  • var
.
  • In
dim 73 , it is
  • nly
Known to be

global

for

:
  • spheres
,

spherical

cylinders

( Dk
  • H ?
.

Shaping

? 89 , G .

Johnsson

  • '
90 ) .
  • ALL
Quadratic

Surfaces

( G
  • Toh

nsson.194-%57.mg

E

  • When

should

we

expect

bocinded

, algebraic

singularities

?
  • whetisfhereasonbehindm]
unbounded , " polar "

singularities

,

£

.es#eespaere.--

slide-11
SLIDE 11
  • 10
  • (b)
Oblate spheroid : T : =LIItIItII=l ^ a > by
  • µy
" E : =L xs-qxftxEE.bg
  • the
caustic , each point is the

meeting point

  • f
2 char . Cl
  • lines

tangent

to F Prelate spheroid ti
  • { IIe fEtII=
I } I p

f

{

ix. I

EVER

, xEXjo )
  • each

point

  • n
the caustic E is the

meeting

point of 2

tangent

at characteristics .

9nnjegetemr.ee?Thisbehariounistrnehg

  • Tzuq(sortof),in20.T
slide-12
SLIDE 12
  • It
  • €fingularitiesofSelutionstoBVP7
me , .

Dirichlet

's

Problem

T⇒r

  • algebraic
" Date "

f

  • entice
, poleyn . Du =
  • in
A

{

u=f
  • n
, ⇒U is real
  • and
. ins .

⇐

Li )

where

do

singularities

  • f
u
  • ccur
  • utside
SL ?

might

. Cci )
  • How
do

singularities

relate to the

geometry

  • f
P

?

  • Is
there the " worst " data

responsible

for

LnTALL

possible

singularities

that

might

  • ccur?
slide-13
SLIDE 13
  • I
2- Few facts :
  • Classical
( E . Heine , G. Lames , M . Ferrers , . . )

Priephg

r :={x : 72×56; al , a , > . . > an ,
  • }
Du
  • in
r A ) Sr

{

u= f , pot .
  • f
deg N " Then , U is a harmonic

polynomial

  • f degree
E N .

The

. ( Oke H . S . Shapiro , ' 92) For any entire data

f

, the solution a
  • f
L * ) is entire . ( D .

Armitage

, ' 04
  • refinement

)

.

ConjeetuuLhK-HSs)T

Ellipsoids

are the
  • nly
domains for which the THM holds . H . Render

(

' d 5)

TRUE

. . . provided

P=LPcx)=o

, P
  • polynomial
, D=

Pmt

. . . t Po
  • homogeneous
' Pm 7,0 in

IR

"

(

i. e. ,

elliptic )

.
slide-14
SLIDE 14
  • 13
  • Discussing
n
  • 2
, T
  • { C x. y )

:xIyZltePn¥D=§

  • ,
n

? ?7

a "

quintic

" . Ad hoc true

for

cuties C E. Lundberg I H . Render

)

deg T 75 ,
  • dd

? ? ?

" intuition "
  • 2N

{

Data " sits "
  • n
2 pieces
  • f
T , so DP

becomes

  • verdetermined
. C due to '

aimi:F¥:::%:;c÷nT

⇐

P . Eben felt 's " TV screen " T : = { x' ' + y 'Ll} (

!

92 ) ×
  • many

singularities

(

)

x xx . .
  • for
the data . . xx A f = XI y
  • L
s × ( DK
  • EL
, ' 10) the nearest x x are n

2314

f-

nom ( 0,0 )
slide-15
SLIDE 15
  • 13
  • FindingSingularitiesT
( S . Bell , P . Eben

felt ,

DK , H . 5. Shapiro ,
  • E. Lundberg
105
  • ' to
, "

Lightning

Bolts "

( A. Kolmogorov

, V.

Arnold

' 50 s ,

Hilbert 's

!3th

Pro # m) .

I

:#

X X a closed LB .

The

size

  • f
LB in E ' C !) ~ how

close

to the
  • rigin
the

singularity

appears

. 3D , HIGHER Dim .
  • virtually

nothing

is known

except

for

rather

imprecise

estimates
  • f
Render

for

elliptic Surfaces .
slide-16
SLIDE 16
  • 14
  • Applicationsto.i.ir/-hogonalpelynomialsy

try

1122 , r

①

. B ,
  • 8mm
OP :L

pfita.io

2eeurre.nu " ( Pn
  • OP
Of

deg

n )

my

Zpn= an

, put it an

,nPn

+
  • An
  • Neff
  • Nti
"

LET

⇒

2K : { su
  • '
m r u tf Pol . of

deg

n u =

pot

.
  • f

deg

Ent K

The

( M

. Putin ar
  • N
. Staglianopolo us , ' 07 D K
  • N
.

Styli

and polo us , ' lo )

Either

condition ⇒ N =3 , e =

ellipse

,

Lpn }

are Chebyshev . Existence
  • f
SOME domain with no recurrence
  • L
. hemp ert , ' 78 .
slide-17
SLIDE 17
  • =y

I

.

OTHER

BVP
  • Eigenfunctions

C-

"

II :÷÷

T

Q

.

where

are

singularities

  • f a ?
  • Essentially

nothing

is known .
  • A
=

ellipsoid

, ALL

eigenfunctions

are entire ( of

exponential

type )

  • to K
  • H
SS C

unpublished )

, H . Render ( unpublished ) ,

well

  • known
C with

Iad

' '

proofs)

? Conjecture ?

If

ALL

eigenfunctions

an entire , d =

ellipsoid

, i. e. , "

singularities

sound
  • ut
the shape .
slide-18
SLIDE 18
  • I
6-

References

can

be

found

in D Ke E . Lundberg ,

Linear

Holomorphic

PDE

and

Classical

Potential

Theory ,

Math . Surveys a Monographs , 232 , AMS ,

2018

.

HappyEuerything.my

Don

I

John

! !

Ln

FT

CHEERS

!