Bernsteins analytic continuation of complex p owers

c

Paul Garrett garrettmathumnedu

version January

Analytic continuation of distributions

Statement of the theorems on analytic continuation

Bernsteins pro ofs

Pro of of the Lemma the Bernstein p olynomial

Pro of of the Prop osition estimates on zeros

Garrett Bernsteins analytic continuation of complex p owers

Let f b e a p olynomial in x x with real co ecients For complex s let

n

s

f b e the function dened by

s s

f x f x if f x

s

f x if f x

s

Certainly for s the function f is lo cally integrable For s in this range

s

we can dened a distribution denoted by the same symb ol f by

Z

s s

f x x dx f

n

R

n

R the space of compactlysupp orted smo oth realvalued where is in C

c

n

functions on R

s

The ob ject is to analytically continue the distribution f as a meromorphic

distributionvalued function of s This typ e of question was considered in

several provo cative examples in IM Gelfand and GE Shilovs Generalized

Functions volume I One should also ask ab out analytic continuation as a

temp ered distribution In a lecture at the Amsterdam Congress IM

Gelfand rened this question to require further that one show that the p oles

lie in a nite numb er of arithmetic progressions

Bernstein proved the result in under a certain regularity hyp othesis on

the zeroset of the p olynomial f Published in Journal of Functional Analysis

and Its Applications translated from Russian

The present discussion includes some background material from complex

function theory and from the theory of distributions

Garrett Bernsteins analytic continuation of complex p owers

1 Analytic continuation of distributions

First we recall the nature of the top ologies on test functions and on distributions

Let C U b e the collection of compactlysupp orted smo oth functions with

c

n

supp ort inside a set U R As usual for U compact we have a countable

family of seminorms on C U

c

f sup jD f j

x

where for we write as usual

n

1 n

D

x x

n

It is elementary to show that for U compact C U is a complete lo cally

c

convex top ological space a Frechet space To treat U not necessarily compact

n

eg R itself let

U U

b e compact subsets of U so that their union is all of U Then C U is the

c

union of the spaces C U and we give it with the lo cally convex direct limit

i

c

top ology

n n

The spaces D U and D R of distributions on U and on R resp ec

n n

R For U and D R C tively are the continuous duals of D U C

c c

present purp oses the top ology we put on the continuous dual space V of a

top ological vectorspace V is the weak top ology a subbasis near in V is

given by sets

U f V jv j g

v

In this context a V valued function f on an op en subset of C is holo

morphic on if for every v V the Cvalued function

z f z v

on is holomorphic in the usual sense This notion of holomorphy might b e

more p edantically termed weak holomorphy since reference to the top ology

might b e required

If z for an op en subset of C and f is a holomorphic V valued

o

function on z say that f is weakly meromorphic at z if for every

o o

v V the Cvalued function z f z v has a p ole as opp osed to essential

singularity at z Say that f is strongly meromorphic at z if the orders of

o o

these p oles are b ounded indep endently of v That is f is strongly meromorphic

Garrett Bernsteins analytic continuation of complex p owers

at z if there is an integer n and an op en set containing z so that for all

o o

v V the Cvalued function

n

z z z f z v

o

n

is holomorphic on If n is the least integer f so that z z f is holomorphic

o

at z then f is of order n at z etc

o o

To say that f is strongly meromorphic on an op en set is to require

that there b e a set S of p oints of with no accumulation p oint in so that f

is holomorphic on S and so that f is strongly meromorphic at each p oint

of S

For brevity but risking some confusion we will often say meromorphic

instead of strongly meromorphic

2 Statement of the theorems on analytic continua

tion

n

Let O b e the p olynomial ring Rx x For z R let O b e the lo cal

n z

ring at z ie the ring of ratios P Q of p olynomials where the denominator

do es not vanish at z Let m b e the maximal ideal of O consisting of elements

z z

of O whose numerator vanishes at z Let I dep ending up on f b e the ideal

z z

in O generated by

z

f f

x x

n

n

A p oint z R is simple with resp ect to the p olynomial f if

f z

N

for some N we have I m

z

z

P

f

There are m so that f

i i z

i

x

i

Remarks The second condition is equivalent to the assertion that O I is

z z

nitedimensional The simplest situation in which the second condition holds

is when I O ie some partial derivative of f is nonzero at z The third

z z

condition do es not follow from the rst two For example Bernstein p oints out

that with

f x y x y x y

the rst two conditions hold but the third do es not

Theorem lo cal version If z is a simple p oint with resp ect to f then

there is a neighb orho o d U of x so that the distribution

Z

s s

f f x x dx

U

Garrett Bernsteins analytic continuation of complex p owers

on test functions C U on U has an analytic continuation to a meromor

c

phic element in the continuous dual of C U

c

Theorem global version If all real zeros of f x are simple with resp ect

s

to f then f is a meromorphic distributionvalued function of s C

3 Bernsteins pro of

Let R b e the ring of linear dierential op erators with co ecients in O Note

z z

that R is b oth a left and a right O mo dule for D R for f g O and a

z z

smo oth function near z the denition is

f D g f D g

Lemma There is a dierential op erator D R and a nonzero Bernstein

z

p olynomial H in a single variable so that

n n

D f H nf

for any natural numb er n Pro of b elow

Pro of of Lo cal Theorem from Lemma Let U b e a smallenough neighb or

ho o d of z so that on it all co ecients of D are holomorphic on U For suciently

n

is continuously dierentiable so we large natural numb ers n the function f

have

n

n

D f H nf

For each xed C U consider the function

c

s

s

g s D f H sf

The hyp otheses of the prop osition b elow are satised so the equality for all

largeenough natural numb ers implies equality everywhere

s

s

D f H sf

This gives us

s

D f

s

f

H s

s

Now we claim that for any n Z the distribution f on C U is

c

meromorphic for s n For n this is certain The formula just

derived then gives the induction step Further this argument makes clear that

s

the p oles of f restricted to C U are concentrated on the nite collection

c

of arithmetic progressions

i i i

Garrett Bernsteins analytic continuation of complex p owers

s

where the are the ro ots of H s In particular the order of the p ole of f at

i

a p oint s is equal to the numb er of ro ots so that s lies among

o i o

i i i

s

In particular the distribution f really is strongly meromorphic This proves

the Theorem granting the Lemma and granting the Prop osition

Prop osition attributed by Bernstein to Carlson If g is an analytic

cs

function for s and jg sj be and if g n for all suciently large

natural numb ers n then g

Pro of of Global Theorem Invoking the Lo cal Theorem and its pro of ab ove

n s

for each z R we cho ose a neighb orho o d U of z in which f is meromorphic

z

so that U is Zariskiop en ie is the complement of a nite union of zero sets

z

of p olynomials Indeed writing

X

f

f x

i

x

i

with O as in the pro of of the Lo cal Theorem let g h with

i z i i i

p olynomials g and h and take U to b e the complement of the union of the

i i z

zerosets of the denominators h

i

n

Then Hilb erts Basis Theorem implies that the whole R is covered by

of these Zariskiop ens Then make a partition of U nitelymany U

z z

n 1

unity sub ordinate to this nite cover ie take so that

n i

P

Then and spt U

i i z

i

X

s s

f f

i

i

the righthand side is a nite sum of mero By choice of the neighb orho o ds U

z

i

morphic distributionvalued functions

4 Pro of of the Lemma the Bernstein p olynomial

Now we prove existence of the dierential op erator D and the Bernstein p oly

nomial H This is the most serious part of this pro of The complex function

theory prop osition is not entirely trivial but is approximately standard

Pro of of Lemma Let

X

R P

z i

x

i

where the m are so that

i z

X

f

R f

z i

x i

Garrett Bernsteins analytic continuation of complex p owers

Also put

f

S P f

i

x x

i i

f

P Q

x x

i i

Then we have

X

f

f P f

i

x

i

i

f f

S f f f

i

x x

i i

Thus by Leibniz formula

n n n

P f nf S f

i

Sublemma There is a nonzero p olynomial M in one variable so that M P

can b e written in the form

X

f

M P J

i

x

i

i

for some J R

i z

Pro of of Sublemma Write as usual

j j

n

For a natural numb er m write

X

m

D P

m

j jm

where O That is we move all the co ecients to the right of the

m z

dierential op erators That this is p ossible is easy to see for example

x x

i i

x x

j j

is or as i j or not

Further the co ecients are p olynomials in the Thus

m i

j j

m

m

z

Then taking M P of the form

X X

m

M P b P D b

m m m

m

mq

Garrett Bernsteins analytic continuation of complex p owers

with b R the condition of the sublemma will b e met if

m

X

b I

m m z

m

N

for all indices If j j N where I m then this condition is automatically

fullled Thus there are nitelymany conditions

X

b

m m

m

N

where is the image of in O m Since the latter quotient is by

m m z

z

hyp othesis a nitedimensional vector space the collection of such conditions

gives a nite collection of homogeneous equations in the co ecients b More

m

sp ecically there are

N

dim O m cardf j j N g

z

z

such conditions By taking q large enough we assure the existence of a non

trivial solution fb g This proves the Sublemma

m

Returning to the pro of of the Lemma as an equation in R

z

X X X

P J S J f M P P J

i i i i

x x

i i

Now put

X

D J

i

x

i

H P M P P

Then we have

X

n n

D f J f f

i

x

i

n n

H P f H nf

as desired This proves the Lemma constructing the dierential op erator D

5 Pro of of the Prop osition estimates on zeros

The result we need is a standard one from complex function theory although it

is not so elementary as to b e an immediate corollary of Cauchys Theorem

cs

Prop osition If g is an analytic function for s and jg sj be and

if g n for all suciently large natural numb ers n then g

Pro of of Prop osition Consider

z

c

Gz e g

z

Garrett Bernsteins analytic continuation of complex p owers

Then g is turned into a b ounded function G on the disc with zeros at p oints

n n for suciently large natural numb ers n

We claim that for a b ounded function G on the unit disc with zeros

i

either G or

X

j j

i

i

If we prove this then in the situation at hand the natural numb ers are

mapp ed to

n n

n

n

so here

X X

j j

n

n

n n

Thus we would conclude G as desired

We recall Jensens formula for any holomorphic function G on the unit disc

with G and with zeros for r we have

Z

r

i

jGj exp log jGr e j d

j jr

i

j j

i

Granting this our assumed b oundedness of G on the disc gives us an absolute

constant C so that for all N

r

C jGj

j jr

i

j j

i

We can harmlessly divide by a suitable p ower of z to guarantee that G

Then letting r

j j jGj C

i

For an innite pro duct of p ositive real numb ers j j less than to have a value

i

it is elementary that we must have

X

j j

i

i

as claimed This proves the prop osition

While were here lets recall the pro of of Jensens formula eg as in Rudins

Real and Complex Analysis page Fix r and let

r z

H z Gz

r z z

where the rst pro duct is over ro ots with jj r and the second is over ro ots

with jj r Then H is holomorphic and nonzero in an op en disk of radius

Garrett Bernsteins analytic continuation of complex p owers

r for some Thus log jH j is harmonic in this disk and we have the

mean value prop erty

Z

i

log jH r e j d log jH j

On one hand

r

jH j jGj

jj

On the other hand if jz j r the factors

r z

r z

have absolute value Thus

X

i arg i i

log j e j log jH r e j log jGr e j

jjr

where

i arg

e

As noted in Rudin see b elow

Z

i

log j e j d

Therefore the integral app earing in the assertion of the mean value prop erty

is unchanged up on replacing H by G Putting this all together gives Jensens

formula

And lets do the integral computation following Rudin There is a function

z on the op en unit disc so that

expz z

since the disc is simplyconnected We uniquely sp ecify this by requiring that

We have

z log j z j and jz j

Let b e small Let b e the path which go es counterclo ckwise

i i

around the unit circle from e to e and let b e the path which

i i

go es clo ckwise around a small circle centered at from e to e Then

Z Z

i i

log j e j d lim log j e j d

Garrett Bernsteins analytic continuation of complex p owers

Z

dz

z

i z

Z

dz

z

i z

by Cauchys theorem

Elementary estimates show that the latter integral has a b ound of the form

C log

which go es to as