V N U JO U R N A L OF SCIENCE, M athem atics - Physics T X V III, N0 3 - 2002
F U N C T I O N A L G E B R A O N A D I S K
K ie u P h u o n g C h i
D epartm ent, o f M a th e m a tic s, V inh U n iversity
A b s t r a c t In this paper w e prove the theorem, on a p p ro x im a tio n o f c o n tin o u s fu n c tio n algebra o n a disk T h is result is an e x te n sio n o f the W e r n e r *8 one
I I n t r o d u c t i o n
Let D be sm all closed disk in the complex plane, centered a t th e origin and / €
C ( D ) By (2, / ; D] we denote th e function algebra consisting of uniform lim its on I) of all polynom ials in z an d /
In 1964, J W ermer [3] proved th a t if / of class c [ and 7p ( 0) 7*^ 0 th en [z) / ; D\ =
(y Ẩ
d f
C ( D ) In 2001, P J fie Paepe [4] show th a t if / of class , with /( 0 ) = 0, -77-(0 ) = 0 and
c
f
— (0) / 0 then [zm , / n ;D] = C ( D ) w ith D small enough and 771,71 are coprime n atu ral
&z
num bers T he proof of de Paepe does not work if (0) i= 0 In th is paper, we give
d f
co n d itio n s such t h a t [zm y f n \ D] = C( D) w hen 0 (0) ỷ 0* T h e proofs a re m aked by the
line of [2], the basis tool is S to u t’s version of K allin’s lemma
I I T h e m a in r e s u lt
T h e o r e m L e t f be a fu n c tio n o f class c l defined in a neighbourhood o f Ớ, w ith /( 0 ) = 0,
— (0) = 1 a n d — (0) = b ^ 0 Sup p ose 771,71 are coprim e n a tu ra l n u m b e rs w ith m , 71 > 1
and.
V l < * , r < m ; 1 < /, 3 < n 1 — C 0 s 2 7 r (
l + |c o s ( 2 ( f a : + ^ ) ) |
+ # )
m
k —r I l—s
m
(*)
fo r k Ỷ r o r I ^ s T h e n [zm, / n ; D] = C ( D ) i f D is a su ffic ie n tly s m a ll d isk a ro u n d 0.
L e m m a 1 L e t X be a com pact su b set 0/ C 2, a n d let 7T : c 2 -> c 2 be d e fin e d by n ( z , w ) =
( z m , w n ) Let 7T~l ( X ) = X u u u Xfci u u X m n w ith X m n c o m p a c t, a n d X k i — { ( p k z , T lw) : {zyw) € x m n } f o r I < k < m , 1 < I < n, where p = exp ( — J and
r = e x p ( ^ i ) I f P ( n ~ l ( X ) ) — C ( n ~ l ( X ) ) , th e n P ( X ) = C { X )
Proof. L et / € C ( X ) T hen /077 € C(7T_1(X )), so there is a polynom ial Q in two variables
w ith / o 7r ~ Q on In particular, this is true on Xk i , so
mn *
f ( z m , w n ) ~ Q { p k z , r l w) := Q k i ( z , w ) Oïl X,
1
Trang 22 K i e u P h u o n g C h i
I t fo llo w s t h a t
Now, if Q ( z , w ) = Y ^ a p%qz pw q, the right hand side above equals ^ d p m qnZprnuỉqn (all other term s drop out), so equals P ( z rn) w n )y where p is polynomial in two variables So
f ( zm, ™n ) - P (2m, u; ) on x mn, th a t is, f ~ p o n X So P(A*) = C (X )
L e m m a 2 (S to u t’s version of Eva K allin’s lemma) [4] Suppose that:
(1 ) X ] a n d X ‘i are co m p a ct su b se ts o f c n w ith P ( Xi) = C ( Xi) a n d P ( X2) = C ( X2);
^ Y\ a n d V 2 are p o ly n o m ia lly co n vex subsets o f c su ch th a t 0 is boundary p o i n t o f both Y\ a n d Y 2 , a n d Y \ n Y 2 = {0};
(3 ) p is p o ly n o m ia l s u c h th a t p ( X \ ) c Y \ a n d p ( X2) c Y 2 Ỉ
(4) p ( 0 ) n ( X i U X 2) = X, n x i -
T h e n P ( X l u x 2) = C ( X i u x 2).
P ro o f o f T h eo rem The conditions on / imply th a t f ( z ) = z + bz + h ( z), w ith h( z ) o f class
c l and h ( z ) = o (|2;|)
F irst, we show th a t 2m and / n separate points near 0 Indeed, first we see th a t
p o i n t s u a n d V w i t h V ^ It e x p for all 1 < k < 771 a r e s e p a r a t e d by 2m N ow,
suppose th at (f ( z ) ) n take th e sam e value at u ex p ự ~ ~ j and u e x f o r k Ỷ I and
u Ỷ 0- Then, there is 1 < r < n such th a t f ị u e x = exp f ( u e xp
It implies th at
= - | „ | e x p ( i * > ) ^ x p ( ^ ) - * p ( m + n ) )
where u = |u| exp It follows th a t
If m , n are coprime, t h e n - h is not integer w ith 1 < k ^ I < m and 1 < r < 71, so
/ k — I r \
Trang 3Since h ( z ) = o(\ z\ ) for every Ố > 0, there exists Ỗ > 0 such th a t |/i(*)| < e\z\ for all
z € B(0,<5) := { z € c : \z\ < Ỗ}. So, for D is small enough, we have
|0|2 < 21 V m w/ I + 2Ê L
V l - c o s 2 7 r ( ^ + ^ y 1 - + n)
It follows th at
1 - c o i f f a i - ỉ )
I6f < 2
1 - c o s2^ ( ! ^ i + s ) y ’
because e is arbitrary This contradicts to (*) So 2m and f n sej)arate points near 0 Now, let X = {(<2™, / n ) : z € £)} Furtherm ore, let 77 be as in Lemma 1, and
7r"1(X ) = X n U U X mn w ith X mn = { ( z , f ( z ) ) : z € D}
By W erm er’s theorem P ( X k i ) = C ( X k i ) for 1 < fc < m, 1 < / < n Next we consider polynomial p( Zj t v) = z w and put
Yki ■■= p { Xk i ) = {p kT l ( z 2 + b\z\2 + z h ( z ) ) : z € £>}
W e sh o w t h a t Yki n Yr& = {0} fo r all 1 < k , r < m \ 1 < lyS < n w i t h k ^ T or I ^ s
Indeed, it is easy to see th a t 0 € Yki for all 1 < k < m and 1 < I < 71. Suppose, there is
0 Ỷ y € Yki n Yrs w ith k Ỷ r or l Ỷ s • Then, there are = a e x p (ia ) and Z 2 = cexp(tjS), where a, c are real num bers w ith ac Ỷ 0 such that
p kT l (ba2 4- a 2 e x p ( 2 i a ) 4- a e x p ( i a ) / i ( a e x p ( i a ) ) )
= p r Ts (bc2 + c2 exp(2Í/Ỉ) 4- cexp(z/3)/i(cexp(i/3)))
It implies th a t
b(pr r 9c2 — p k Tl a 2) = pk T la2e x p ( 2 ia ) — pr T 3c 2e \ p ( 2 i p )
+ a ex p (2 ia)h (aex p (2 ia)) — cexp(2i/3)/i(cexp(2i/J))
P u t i4 = c o s2 ^ (q — /3) + 7T ^ - f -— and H = c o s2 7 r^ ~ ~ + By the coprime ness of 771, n we see th a t 4- — is not integer for all 1 < A:, r < m and 1 < /, s < n
w ith Ả: ^ r or l Ỷ s ) so B Ỷ 1- We obtain
2 / a 4 + c4 — 2a2c2i4 \ / q2ị/ì(aexp(2iq))|2 + c2 |/i(cexp(2z/?))|2 \
1 1 - V a 4 + c 4 - 2 a 2c2 5 > 1 I a 4 + c4 - 2 a 2 c2 B y
Since /i(z) = o (|z|), for every e > 0, there exists D is sufficiently small disk such th at
a 2|/i(a e x p (2 ia )|2 4- c2|/i(cexp(2i/?)|2 4(a4 + c4)e
a 4 + c4 — 2a? c* B ~ a 4 4* c4 — 2a 2c2 H ~
where M = 4 if B < 0 and M = if B > 0 So, for D small enough, we obtain
a4 + c4 — 2a2c2i4 \ I,i9 rt/ a + c - za c / i \ '6| î 2( ^ t ^ 5 b)
Trang 4+M£-K i e u P h u o n g C h i
W e have
IM2 < 2 ( t 2 - 2A t + 1 '
t 2 - 2 h t -t- r \ 4- M £ — ‘2^1 + t 2 - 2t i t + 1
2 ( 8 - i4)t
, we o b ta in m ax ợ (í) = —— It
(0,Hhoo) 2(1 - « )
Since £ is arb itra ry , we conclude th a t
T h is inequality co n trad icts to (*)
Furtherm ore, it is easy t o see th a t c \ Yki is connected set if u small enough, so
Yfct is polynoưiially convex [Ca] and p ~ l (0) n Xfci — (0,0) for 1 < k < m and I < I < n
Therefore p ~ x(0) n (X/t/ u X r s ) = X*/ n for all 1 < /c,r < m and 1 < z,s < n with Ả: Ỷ r or r ~f~ s - Now apply S to u t’s version of K allin’s lem m a repeatedly, to obtain
theorem is proved
A c k n o w le d m e n ts T h e a u th o r is in d eb ted to Dr N guyen Q u a n g D ieu for proposing the problem a n d for his generous guidance My th a n k s also go to Dr D inh Iluy H oang for
m any valuable discussions
R e fe re n c e s
Variables, 4 7 (5) (2002) 447 451
2 T w Gamelin, U n ifo rm Agcbras, Prentice-H all (1984)
3 J Wermer A pproxim ation on a Disk, M ath A n n 155(1964) 331-333
4 P J de P æ p e , Eva K allin’s lem m a on polynoraially convexity, Bull L o n d o n M ath Soc., 33(2001), 1-10