polynomially convex hull

polynomially convex hull

Definition

.

Let

K

⊂

ℂ

be a

compact subset

of the

complex plane

,
then the

polynomially convex hull

of

K

, denoted

K

^

, is defined as

K

^

:=

{

z

∈

ℂ

:

|

p

⁢

(

z

)

|

≤

max

ζ

∈

K

⁡

|

p

⁢

(

ζ

)

|

⁢

for all polynomials

p

}

.

A compact set

K

is said to be

polynomially convex

if

K

=

K

^

Obviously

K

⊂

K

^

. The intuitive idea behind this

definition

is that the polynomially convex hull of

K

fills in any “holes” that may exist in

K

.
The following

proposition

makes that precise.

Proposition

.

If

K

⊂

C

is a polynomially convex set, then
all the components of the interior of

K

are simply connected.

One of the reasons for this definition is the following result.

Proposition

.

Let

f

be a

function

analytic

in an open neighbourhood

N

of a compact set

K

⊂

C

, and suppose that

f

can be approximated by

polynomials

uniformly
on compact subsets of

N

. Then

f

can be extended analytically to a neighbourhood of

K

^

.

For example if we take

K

=

{

z

∈

ℂ

:

|

z

|

=

1

}

(the

unit circle

)
then

K

^

=

{

z

∈

ℂ

:

|

z

|

≤

1

}

(the
closed unit disc). The fact that the inside of the disc belongs to

K

^

follows from the

maximum modulus principle

as polynomials are

analytic functions

. The fact that

K

^

does not contain anything outside the
closed unit disc follows by looking at the polynomial

p

⁢

(

z

)

=

z

which has
always

greater

modulus

outside of the unit disc then anywhere on the unit circle. So if we have a function defined on a neighbourhood of the unit circle
and which we can approximate uniformly on compact subsets of this
neighbourhood
by polynomials, then we can extend this function analytically
to the whole unit disc. So this for example implies that

f

⁢

(

z

)

:=

1

z

cannot be approximated uniformly on compact subsets by polynomials on a
neighbourhood of the unit circle.

The reason why we call

K

^

a “hull” of some is that the conventional

convex hull

of

K

⊂

ℝ

n

can be defined as the set of points

x

such that for all

linear functions

f

:

ℝ

n

→

ℝ

we have

|

f

⁢

(

x

)

|

≤

sup

y

∈

K

⁡

|

f

⁢

(

y

)

|

. This coincides with conventional definition because if

x

is not in the conventional convex hull, then there is a

linear functional

that separates

x

from the hull (by

Hahn-Banach theorem

in general, or more elementarily in

ℝ

n

by Farkas’s lemma), and conversely if

x

is in the convex hull of

K

then such linear function does not exist for the same reason. So, intuitively the conventional convex hull is set of point that are inseparable from from

K

by linear functions. Polynomially convex hull is the same thing, but with polynomials.
Of course similar definitions can be made with respect to other classes of functions. For example, hulls with respect to

plurisubharmonic functions

are very useful in

multivariate

complex analysis

.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

polynomially convex hull

Canonical

name

PolynomiallyConvexHull

Date of creation

2013-03-22 14:21:15

Last modified on

2013-03-22 14:21:15

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C10

Classification

msc 52A01

Related topic

HolomorphicallyConvex

Defines

polynomially convex

Generated on Fri Feb 9 20:16:24 2018 by

LaTeXML