Approximation on disks
Authors: Preskenis Kenneth John 
Published: July 2010

Let

D
D

be a closed disk in the complex plane,

f
f

a complex valued continuous function on

D
D

and

R
f

(
D
)
=

{R_f}(D) =

the uniform closure on

D
D

of rational functions in

z
z

and

f
f

which are finite. Among other results we obtain the following. Theorem. If

f
f

is of class

C
1

{C^1}

in a neighborhood of

D
D

and

|

f

z
¯

|

>

|

f
z

|

|{f_{\bar z}}| > |{f_z}|

everywhere (i.e.,

f
f

is an orientation reversing immersion of

D
D

in the plane), then

R
f

(
D
)
=
C
(
D
)

{R_f}(D) = C(D)

. Theorem. Let

f
f

be a polynomial in

z
z

and

z
¯

\bar z

. If for each a in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D comma f minus normal upper Sigma left-parenthesis j factorial right-parenthesis Superscript negative 1 Baseline upper D Superscript j Baseline f left-parenthesis a right-parenthesis left-parenthesis z minus a right-parenthesis Superscript j Baseline equals left-parenthesis z overbar minus a overbar right-parenthesis Superscript k Baseline g">

D
,
f
−
Σ

(
j
!

)

−
1

D
j

f
(
a
)

(
z
−
a

)
j

=

(

z
¯

−

a
¯

)
k

g

D,f - \Sigma {(j!)^{ - 1}}{D^j}f(a){(z - a)^j} = {(\bar z - \bar a)^k}g

with

|

g

z
¯

|

>

|

g
z

|

|{g_{\bar z}}| > |{g_z}|

at he zeros of

g
g

in

D
D

where

D
f
=

f
z

Df = {f_z}

, then

R
f

(
D
)
=
C
(
D
)

{R_f}(D) = C(D)

. Corollary. Let

f
f

be a polynomial in

z
z

and

z
¯

\bar z

and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f Subscript z z overbar Baseline left-parenthesis 0 right-parenthesis EndAbsoluteValue greater-than StartAbsoluteValue f Subscript z overbar z overbar Baseline left-parenthesis 0 right-parenthesis EndAbsoluteValue slash 2">

|

f

z

z
¯

(
0
)

|

>

|

f

z
¯

z
¯

(
0
)

|

/

2

|{f_{z\bar z}}(0)| > |{f_{\bar z\,\bar z}}(0)|/2

. Then there exists an

r
>
0

r > 0

such that, for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D equals left-parenthesis StartAbsoluteValue z EndAbsoluteValue less-than-or-slanted-equals r right-parenthesis comma upper R Subscript f Baseline left-parenthesis upper D right-parenthesis equals upper C left-parenthesis upper D right-parenthesis">

D
=
(

|

z

|

⩽
r
)
,

R
f

(
D
)
=
C
(
D
)

D = (|z| \leqslant r),{R_f}(D) = C(D)

. The proofs of the theorems use measures and the conditions involved in the theorems are independent of each other. Concerning the corollary, results of E. Bishop and G. Stolzenberg show that

f

z
¯

(
0
)
=
0

{f_{\bar z}}(0) = 0

and

|

f

z
¯

z
¯

(
0
)

|

>

|

f

z

z
¯

(
0
)

|

|{f_{\bar z\,\bar z}}(0)| > |{f_{z\bar z}}(0)|

, then there exists no

r
r

such that

R
f

(
D
)
=
C
(
D
)

{R_f}(D) = C(D)

where

D
=
(

|

z

|

⩽
r
)

D = (|z| \leqslant r)

. Let

F
=
(

f
1

,
⋯
,

f
n

)

F = ({f_1}, \cdots ,{f_n})

be a map on

B
B

= unit polydisk in

C

n

{{\mathbf {C}}^n}

with values in

C

n

,

P
F

{{\mathbf {C}}^n},{P_F}

= uniform closure on

B
B

of polynomials in

z
1

,
⋯

z
n

,

f
1

,
⋯
,

f
n

{z_1}, \cdots {z_n},{f_1}, \cdots ,{f_n}

. Theorem. If

F
F

is of class

C
1

{C^1}

in a neighborhood of

B
,

F

z
¯

B,{F_{\bar z}}

is invertible and if for each

a
=
(

a
1

,
⋯
,

a
n

)

a = ({a_1}, \cdots ,{a_n})

in

B
B

, there exist complex constants

{

c
j

}
,
{

d

i
j

}
,
i
,
j
=
1
,
⋯
,
n

\{ {c_j}\} ,\{ {d_{ij}}\} ,i,j = 1, \cdots ,n

, such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma c Subscript j Baseline left-parenthesis z Subscript j Baseline minus a Subscript j Baseline right-parenthesis left-parenthesis f Subscript j Baseline left-parenthesis z right-parenthesis minus f Subscript j Baseline left-parenthesis a right-parenthesis right-parenthesis plus normal upper Sigma d Subscript i j Baseline left-parenthesis z Subscript i Baseline minus a Subscript i Baseline right-parenthesis left-parenthesis z Subscript j Baseline minus a Subscript j Baseline right-parenthesis">

Σ

c
j

(

z
j

−

a
j

)
(

f
j

(
z
)
−

f
j

(
a
)
)
+
Σ

d

i
j

(

z
i

−

a
i

)
(

z
j

−

a
j

)

\Sigma {c_j}({z_j} - {a_j})({f_j}(z) - {f_j}(a)) + \Sigma {d_{ij}}({z_i} - {a_i})({z_j} - {a_j})

has positive real part for all

z
≠
a

z \ne a

, then

{
(
ζ
,
F
(
ζ
)
)
:
ζ
∈
B
}

\{ (\zeta ,F(\zeta )):\zeta \in B\}

is a polynomially convex set. Corollary. If

F
=
(
f
,
g
)

F = (f,g)

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z comma w right-parenthesis equals z overbar plus c z z overbar plus d z overbar squared plus q z overbar w comma g left-parenthesis z comma w right-parenthesis equals w overbar plus s w w overbar plus t w overbar squared plus p w overbar z">

f
(
z
,
w
)
=

z
¯

+
c
z

z
¯

+
d

z
¯

2

+
q

z
¯

w
,
g
(
z
,
w
)
=

w
¯

+
s
w

w
¯

+
t

w
¯

2

+
p

w
¯

z

f(z,w) = \bar z + cz\bar z + d{\bar z^2} + q\bar zw,g(z,w) = \bar w + sw\bar w + t{\bar w^2} + p\bar wz

and the coefficients satisfy

|

c
¯

+
d

|

+

|

d

|

+

|

q

|

>
1

|\bar c + d| + |d| + |q| > 1

and

|

s
¯

+
t

|

+

|

t

|

+

|

p

|

>
1

|\bar s + t| + |t| + |p| > 1

, then

P
F

=
C
(
B
)

{P_F} = C(B)

. Corollary. If

F
(
z
)
=

z
¯

+
R
(
z
)

F(z) = \bar z + R(z)

where

R
=
(

R
1

,
⋯
,

R
n

)

R = ({R_1}, \cdots ,{R_n})

is of class

C
2

{C^2}

and satisfies the Lipschitz condition

|

R
(
ζ
)
−
R
(
η
)

|

⩽
k

|

ζ
−
η

|

|R(\zeta ) - R(\eta )| \leqslant k|\zeta - \eta |

with

k
>
1

k > 1

, then

P
F

=
C
(
B
)

{P_F} = C(B)

. This last corollary is a result of Hörmander and Wermer. The proof of the theorem uses methods from several complex variables.