DSpace at VNU: Function algebra on a disk

V N U JO U R N AL OF SCIENCE, M athem atics - Physics T X V III, N0 - 2002 F U N C T IO N ALGEBRA ON A D IS K K ie u P h u o n g C hi D epartm ent, o f M a th em a tics, V inh U niversity A b s tr a c t In this paper we prove the theorem, on a p p ro xim a tio n o f co n tin o u s fu n c ­ tio n algebra o n a disk T h is result is an e xte n sio n o f the W e rn e r *8 one I I n t r o d u c t io n Let D be sm all closed disk in the complex plane, centered a t the 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 Wermer [3] proved th a t if / of class c [ and p ( 0) 7*^ then [z) / ; D\ = (yẨ C ( D ) In 2001, P J fie Paepe [4] show th a t if / of class df , with /(0 ) = 0, -77-(0 ) = and c f — (0) / &z then [zm , / n ;D] = C ( D ) with D small enough and 771,71 are coprime natural num bers T he proof of de Paepe does not work if (0) i= In this paper, we give df conditions such th a t [zm y f n \D] = C( D) w hen (0) ỷ 0* T h e proofs are m aked by the line of [2], the basis tool is S to u t’s version of Kallin’s lemma II 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, A e o r — (0) = a nd — (0) = b ^ Suppose 771,71 are coprim e na tu l n u m b e rs w ith m , 71 > dz and ỡz , \b\ / >2 1+ V max l < * , r < m ; < /, < n + I|ccos o s ((Z7T[ ( f ma : + ^ ) ) | 1l T — —— - ~ k —r I l—s — C s2 r( m + # ) (*) fo r k Ỷ r o r I ^ s T h en [zm, / n ; D] = C ( D ) i f D is a su ffic ien tly s m a ll d isk a ro u n d L e m m a L et X be a com pact subset 0/ C 2, a n d let 7T : c -> c be d efin ed 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 ct, a n d X k i — { ( p k z , T lw) : {zyw) € x m n } f o r I < k < m , < 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 polynomial Q in two variables w ith / o 7r ~ Q on In particular, this is true on Xk i , so f ( z m , w n ) ~ Q { p k z , r l w) := Qk i ( z , w ) Oïl X mn , * Typeset by Ạ/ựịS-TỊ$( K ie u P huong Chi 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 ( z m, ™n ) - P ( m, u; ) on x mn, th a t is, f ~ p on X So P(A*)= C (X ) L e m m a (S to u t’s version of Eva Kallin’s lemma) [4] Suppose that: (1 ) X ] and X ‘i are com pact su bsets o f c n w ith P ( X i) = C ( X i) a n d P ( X 2) = C ( X ); ^ Y\ a n d V2 are p o ly n o m ia lly convex subsets o f c such th a t is boundary p o i n t o f both Y\ a n d Y , a n d Y \ n Y = {0}; (3 ) p is polyn o m ia l su ch th a t p ( X \ ) c Y\ a n d p ( X 2) c Y Ỉ (4) p ( ) 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 Theorem 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 at m and / n separate points near Indeed, first we see th at p o in t s u a n d V w i t h V ^ for all < k < 771 a r e s e p a r a t e d by 2m Now, It e x p suppose th at ( f ( z ) ) n take the sam e value at uexp ự ~ ~ j and u e x f o r u Ỷ 0- Then, there is < r < n such th a t f ị u e x = exp k Ỷ I and f (uexp It implies th at , / x/ (2mk\ ( 2iril = - |„ |e x p ( i* > ) ^ x p ( ^ ) - * p ( m + where u = |u| exp m r \\ n ) ) It follows th at in M ( l + ) < - ) If m , n are coprime, t h e n h is not integer w ith < k ^ I < m and < r < 71, so rn n / k —I r\ o o s r ( h - - ) V 771 71 / 7É Therefore F u n c t i o n algebra o n a d isk Since h( z ) = o(\z\) for every Ố > 0, there exists Ỗ > such th a t |/i(*)| < e\z\ for all z € B(0,