Approximation by Generalized Faber Series in Bergman Spaces on Finite Regions with a Quasiconformal Boundary

Journal of Approximation Theory  AT2964 journal of approximation theory 87, 2535 (1996) article no. 0090

Approximation by Generalized Faber Series in Bergman Spaces on Finite Regions with a Quasiconformal Boundary Abdullah Cavus Faculty of Sciences H Arts, Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey Communicated by Hans Wallin Received July 5, 1994; accepted in revised form July 25, 1995

In this work, for the first time, generalized Faber series for functions in the Bergman space A 2 (G) on finite regions with a quasiconformal boundary are defined, and their convergence on compact subsets of G and with respect to the norm on A 2(G) is investigated. Finally, if S n ( f, z) is the n th partial sum of the generalized Faber series of f # A 2 (G), the discrepancy & f &S n ( f, } )& A2 (G) is evaluated by E n ( f, G), the best approximation to f by polynomials of degree n.  1996 Academic Press, Inc.

1. INTRODUCTION AND STATEMENT OF PROBLEM Let G be a finite region with 0 # G bounded by a quasiconformal curve 1. We recall that 1 is called a quasiconformal curve if there exists a quasiconformal homeomorphism of the complex plane onto itself that maps a circle onto 1. The set

{

A 2(G) := f : f is analytic in G and

||

| f (z)| 2 d_ z < +

G

=

is called the Bergman space on G, where d_ z denotes the Lebesque measure in the complex plane. It is well known that A 2(G) is a Hilbert space with the inner product ( f, g) :=

||

f (z) g(z) d_ z ,

G

and the set of polynomials are dense in A 2(G) with respect to the norm & f & A2(G) :=(( f, f ) ) 12, see [8: Ch. I] and [5: p. 420]. Also, for n=1, 2, ..., 25 0021-904596 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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26

ABDULLAH CAVUS

if the degree of the best approximation to f by polynomials of degree n is defined by the formula E n ( f, G) :=Inf[& f &P& A2(G) : P(z) is a polynomial of degree n], there exists a polynomial P *(z), of degree n, such that E n( f, G) := n . P *(z) is called the best approximant polynomial to f # A 2(G). & f &P *& n A2(G) n Let U be the open unit disc and w=.(z) the conformal mapping of CG onto CU with normalization .()= and lim z   (1z) .(z)>0, where CG and CU are the complements of G and U in the extended complex plane respectively. We denote the inverse of .(z) by 9(w). If F m (z) is the m th Faber polynomial for G, then it is known that for every (z, w) # G_CU  9$(w) F m (z) = : 9(w)&z m=0 w m+1

and the series converges uniformly and absolutely on compact subsets of G_CU. From this, we get  9$(w) F$m (z) = : , 2 (9(w)&z) w m+1 m=1

(z, w) # G_CU,

(1)

where the series converges absolutely and uniformly on compact subsets of G_CU. More information for Faber polynomials and Faber expansions can be found in [8] and [11]. In this work, we define a generalized Faber series of a function f # A 2(G) to be of the form   m (z), and we show that it converges m=1 a m ( f ) F$ uniformly on compact subsets of G. Furthermore, if S n ( f, z) :=  n+1 m (z) is the n th partial sum of a generalized Faber series of m=1 a m ( f ) F$ f # A 2(G), then we give an estimation of the discrepancy & f &S n ( f, } )& A2(G) by means of E n ( f, G). Also, if a series   m (z) is convergent to m=1 a m F$ f # A 2(G) with respect to the norm & & A2(G) , we show that the a m are the generalized Faber coefficients a m ( f ) of f. In [9], similar problems were studied by D. M. Israfilov in A(G ), where A(G ) denotes the class of functions which are analytic in G and continuous in G.

2. DEFINITIONS AND SOME AUXILIARY RESULTS In [4], V. I. Belyi gave the following integral representation for the functions f # A(G ) as f (z)= &

1 ?

||

CG

( f b y)(`) y  (`) d_ ` , (`&z) 2 `

z # G,

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(2)

27

GENERALIZED FABER SERIES

and he studied the approximation by polynomials in A(G ). Here y(z) is a K-quasiconformal reflection with respect to 1, i.e., a sense-reversing K-quasiconformal involution of the extended complex plane taking G into CG and keeping every point of 1 fixed, with y(0)= and y()=0. Such a mapping of the plane does exist [10]. It follows from Ahlfors' lemma [1: p. 80] that the reflection y(z) may always be chosen to be differentiable everywhere, except possibly on the points of 1 _ [0]. In this case, I. M. Batchaev generalized the integral representation above to Lebesque integrable and analytic functions, and so to functions f # A 2(G) [3]. The analog of the integral representation (2) for unbounded domains with boundary passing through  was first proved by L. Bers [6]. Quasiconformal mappings and quasiconformal reflections with respect to 1 are examined in [1] and [10] in great detail. From now on, the reflection y(z) will be a differentiable K-quasiconformal reflection with respect to 1. Let f # A 2(G). Substituting `=9(w) in (2), we get f (z)= &

1 ?

||

CU

f ( y(9(w))) 9$(w) y ` (9(w))

9$(w) d_ , (9(w)&z) 2 w

z # G,

(3)

Thus, if we consider (1) and (3), and define coefficients a m ( f ), by a m ( f ) := &

1 ?

||

CU

f ( y(9(w))) 9$(w) y ` (9(w)) d_, w m+1

m=1, 2, ... (4)

then we can associate a formal series   m (z) with the function m=1 a m ( f ) F$ f # A 2(G), i.e., 

f (z)t : a m ( f ) F$m (z).

(5)

m=1

We call this formal series a generalized Faber series of f # A 2(G), and the coefficients a m ( f ) are called generalized Faber coefficients of f. Lemma 2.1.

Let [F m (z)] be the Faber polynomials for G. Then n

: m=1

&F$m & 2A2(G) n?. m

Proof. Let S m (G) be the area of the image of G under F m in the &v , |w| >1, Riemann surface of F m . Since F m (9(w))=w m +  v=1 mb mv w

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28

ABDULLAH CAVUS

[7: p. 43] where the b mv are the Grunsky coefficients, by means of a theorem due to N. A. Lebedev and I. M. Millin in [11: p. 213] we have 

\

+

S m (G)=? m& : vm 2 |b mv | 2 m?. v=1

(6)

On the other hand S m (G)=

||

G

|F$m (z)| 2 d_ z =&F$m& 2A2(G) .

(7)

From (6) and (7), it follows that n

: m=1

&F$m& 2A2(G) n?. m

Let us emphasize that we can, in general, not reduce n? in the inequality above. In fact, if we consider the unit disc U, then F m (z)=z m and  nm=1 &F$m& 2A2(U)m=n?. Remark. Using a completely similar method, it can be proved that this lemma is true for any continuum E whose complement is connected. 2 Lemma 2.2. The series   m (z)| (m+1) is convergent uniformly m=1 |F$ on compact subsets of G.

Proof. Let z be a fixed point in G. Then the power series m+1 defines an analytic function A(z, w) in U, i.e.,  m (z)(m+1)) w m=1 (F$ 

A(z, w) := : m=1

F$m (z) m+1 w , m+1

w # U,

(8)

Thus, by taking derivative of (8) with respect to w and considering (1) we get 

A$(z, w)= : F$m (z) w m = m=1

9$(1w) , (9(1w)&z) 2 w

w # U,

(9)

m Let 0
||

D(0, r)

|A$(z, w)| 2 d_ w =? : m=1

|F$m (z)| 2 2m+2 r . m+1

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(10)

29

GENERALIZED FABER SERIES

(9) and (10) show that 

|F$m (z)| 2 2m+2 r = m+1

? : m=1

||

D(0, r)

}

9$(1w) (9(1w)&z) 2 w

2

} d_ .

(11)

w

Since lim

r  1&

||

D(0, r)

}

9$(1w) (9(1w)&z) 2 w

}

2

d_ w =

||

}

U

9$(1w) (9(1w)&z) 2 w

2

} d_ , w

and

||

U

}

9$(1w) (9(1w)&z) 2 w

2

} d_ < + w

we get 

? : m=1

|F$m (z)| 2 = m+1

||

U

}

9$(1w) (9(1w)&z) 2 w

}

2

d_ w .

On the other hand, it can be easily proved that

||

U

}

9$(1w) (9(1w)&z) 2 w

2

} d_

w

2 is continuous in G. So, by Dini's theorem the series   m (z)| (m+1) m=1 |F$ is convergent uniformly on compact subsets of G.

Lemma 2.3. If f # A 2(G) and y(`) is a differentiable K-quasiconformal reflection with respect to 1, then

||

CG

|( f b y)(`)| 2 | y ` (`)| 2 d_ ` 

& f & 2A2(G) , 1&k 2

where k :=(K&1)(K+1). Proof. Since y(`) is a differentiable K-quasiconformal mapping of the extended complex plane onto itself, we have | y ` |  | y ` | k and | y ` | 2 & | y ` | 2 >0. Also, it is known that | y ` | = | y ` | and | y ` | = | y ` | . Therefore, since | y ` |  | y ` | k and | y ` | 2 & | y ` | 2 >0, we get

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30

ABDULLAH CAVUS

||

|( f b y)(`)| 2 | y ` (`)| 2 d_ ` =

CG

||

|( f b y)(`)| 2 (1&( | y ` |  | y ` | ) 2 ) &1

CG

_(| y ` | 2 & | y ` | 2 ) d_ `

||

|( f b y)(`)| 2 | y ` (`)| 2 d_ ` 

CG

1 1&k 2

||

|( f b y)(`)| 2 (| y ` | 2 & | y ` | 2 ) d_ ` .

CG

Since the Jacobian of y(`) is (| y ` | 2 & | y ` | 2 ), if we substitute z for y(`) on the right side of the inequality above we get

||

|( f b y)(`)| 2 | y ` (`)| 2 d_ ` 

CG

& f & 2A2 (G) . 1&k 2

3. MAIN RESULTS Theorem 3.1. Let f # A 2(G). If   m (z) is a generalized m=1 a m ( f ) F$ Faber series of f, then the series   m (z) converges uniformly to m=1 a m ( f ) F$ f on compact subsets of G. Proof. Let M be a compact subset of G and y(z) a differentiable K-quasiconformal reflection with respect to 1. Since for z # M f (z)= & =&

1 ?

||

( f b y)(`) y (`) d_ ` (`&z) 2 `

1 ?

||

f ( y(9(w))) 9$(w) y ` (9(w))

CG

CU

9$(w) d_ w (9(w)&z) 2

and a m ( f )= &

1 ?

||

CU

f ( y(9(w))) 9$(w) y ` (9(w)) d_ w , w m+1

m=1, 2, ...

we obtain by means of Holder's inequality and Lemma 2.3

}

n

} 9$(w) || \ } (9(w)&z) & :

f (z)& : a m ( f ) F$m (z) m=1



& f & A2(G) ? - 1&k 2

n

CU

2

m=1

F$m(z) w m+1

2

} +

for every z # M.

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d_ w

12

(12)

31

GENERALIZED FABER SERIES

Let 1
}

||

r< |w|
n 9$(w) F$m (z) 2 & : d_ w = 2 (9(w)&z) m=1 w m+1

}

}

||

r< |w|
=?

: m=n+1

m=n+1

F$m (z) 2 d_w w m+1

}

1 1 1 & |F$m (z)| 2 m r 2m R 2m

\



4?



:

: m=n+1

+

|F$m (z)| 2 , m+1

and by letting r  1 + and R  + we get

||

CU

}

n 9$(w) F$m (z) & : 2 (9(w)&z) w m+1 m=1

}

2



d_ w 4?

: m=n+1

|F$m (z)| 2 . m+1

(13)

Therefore, by (12), (13) and Lemma 2.2, we conclude that  m (z) converges uniformly to f on M. This completes the m=1 a m ( f ) F$ proof. Corollary 3.1. If P n (z) is a polynomial of degree n and a m (P n ) are its generalized Faber coefficients, then a m (P n )=0 for all mn+2 and P n (z)= n+1 m (z). m=1 a m (P n ) F$ Proof. Let z # G. By Theorem 3.1, we have P n (z)=  m (z). m=1 a m (P n ) F$ It is obvious that P n (z) can be written in the form P n (z)= n+1 v (z). v=1 A v F$ Let y(z) be a differentiable K-quasiconformal reflection relative to 1. Since y(z) is fixed on 1, we get by Green's formulae

a m (P n )= &

1 ?

CU

n+1

=: & v=1 n+1

=: & v=1 n+1

=: v=1

P n (y(9(w))) 9 (w) y ` (9(w)) d_ w w m+1

||

Av ?

||

F $v ( y(9(w))) 9 $(w) y ` (9(w)) d_ w w m+1

Av ?

||

 F v (y(9(w))) d_ w w w m+1

Av 2?i

CU

|

CU

|w| =1

\

+

F v (9(w)) dw. w m+1

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(14)

32

ABDULLAH CAVUS

Since 1 2?i

|

|w| =1

F v (9(w)) 1, dw= w m+1 0,

if v=m, if v{m,

{

(15)

see [8: p. 43], it follows that a m (P n )=A m , for m=1, ..., n+1, and a m (P n )=0 for all mn+2. So P n (z)= n+1 m (z). m=1 a m (P n ) F$ Theorem 3.2. Let [a m ] be a complex number sequence. If the series 2  m (z) converges to a function f # A (G) in the norm & & A2(G) , then m=1 a m F$ the a m are the generalized Faber coefficients of f. Proof. Let y(z) be a differentiable K-quasiconformal reflection relative be the n th partial sum of to 1, and S n (z) := n+1 m (z) m=1 a m F$  m (z). Using (15), it can be shown that m=1 a m F$ lim n

1 ?

||

CU

S n (y(9(w))) 9 $(w) y ` (9(w)) d_ w =a m , w m+1

m=1, 2, ...,

(16)

So, if m and n are natural numbers, we get by using Holder's inequality and Lemma 2.3 |a m ( f )&a m | 1 ?

[ f ( y(9(w)))&S n ( y(9(w)))] 9 $(w)

y (9(w)) d_ } || } w 1 S (y(9(w))) 9 $(w) + & || y (9(w)) d_ &a } ? } w 1 d_  || + ?\ |w| _ || | f( y(9(w)))&S (y(9(w)))| }|9$(w)| | y (9(w))| d_ \ + 1 S (y(9(w))) 9 $(w) + & || y (9(w)) d_ &a } ? } w 1  |(( f &S ) b y)(`)| | y (`)| d_ + - m? \ || 1 S (y(9(w))) 9 $(w) + & || y (9(w)) d_ &a } ? } w 

`

m+1

CU

w

n

`

m+1

CU

w

m

12

CU

w 2m+2

2

2

2

`

n

w

CU

n

`

m+1

CU

w

m

12

2

2

`

n

CG

`

n

CU

m+1

`

w

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m

12

33

GENERALIZED FABER SERIES



& f &S n& A2(G) - m?(1&k 2 )

}

+&

1 ?

||

CU

S n (y(9(w))) 9 $(w) y ` (9(w)) d_ w &a m . w m+1

}

(17)

Since lim n   & f &S n& A2(G) =0, (16) and (17) show that a m ( f )=a m , and so the proof is completed. Under the assumption of the above theorem, it is seen that the generalized Faber coefficients of the limit function are independent of the choice of the differentiable K-quasiconformal reflection. Theorem 3.3. If f # A 2(G) and S n( f, z)= n+1 m (z) is the n th m=1 a m ( f ) F$ a ( f ) F$ (z), then partial sum of its generalized Faber series   m m m=1 & f &S n ( f, } )& A2(G) - 6n(1&k 2 ) E n ( f, G) for all natural numbers n. Proof. Let y(z) be a differentiable K-quasiconformal reflection with the best approximant polynomial to f # A 2(G) in respect to 1, and P *(z) n the norm & & A2(G) . By means of Holder's inequality, Lemma 2.3 and Corollary 3.1 we obtain | f (z)&S n ( f, z)|  | f (z)&P n*(z)| + |P n*(z)&S n ( f, z)|

}

n+1

 | f (z)&P *(z)| + : (a m (P *)&a n n m ( f )) F$ m (z) m=1

}

 | f (z)&P *(z)| n +



1 ?

1 ?

} ||

n+1

( f b y&P * n b y)(9(w)) 9$(w) y ` (9(w)) :

CU

m=1

\|| _ || \ }:

2 2 2 |( f b y&P * n b y)(9(w))| |9$(w)| | y ` | d_ w

CU

n+1

CU

m=1

F m (z) w m+1

2

} + d_ w

12

+ | f (z)&P n*(z)|

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+

12

F$m (z) d_ w w m+1

}

34

ABDULLAH CAVUS

\

n+1

_ ? : m=1

1 ?

\|| |F$ (z)| m +

 | f (z)&P n*(z)| +

|( f b y&P n* b y)(`)| 2 | y ` (`)| 2 d_ `

CG

2

+

12

12

m

 | f (z)&P n*(z)| +  | f (z)&P n*(z)| +

1 - ?(1&k 2 ) 1 - ?(1&k 2 )

& f &P n*& A2(G) E n ( f, G)

\

\

n+1

: m=1

n+1

: m=1

|F$m (z)| 2 m

|F$m (z)| 2 m

+

+

12

12

for all natural numbers n. This shows that 2 | f (z)&S n ( f, z)| 2 2 | f (z)&P *(z)| + n

n+1 2 |F$m (z)| 2 E 2n ( f, G) : . 2 ?(1&k ) m m=1

Therefore, by integrating both sides over G and considering Lemma 2.1 we get & f &S n ( f, } )& 2A2(G) 2E 2n ( f, G)+

\

 2+ 

n+1 2 &F$m& 2A2(G) 2 2 E n ( f, G) : ?(1&k ) m m=1

2(n+1) E 2n ( f, G) 1&k 2

+

6n E 2 ( f, G), 1&k 2 n

i.e., & f &S n ( f, } )& A2(G) - 6n(1&k 2 ) E n ( f, G) for all natural numbers n. Corollary 3.1. If f # A 2(U), then its generalized Faber series converges to f in the norm & & A2(U) . Proof. This is obvious from the preceding theorem and Theorem 11 and Theorem 1 in [7] and [2], respectively.

ACKNOWLEDGMENTS I express my gratitude to Professor Danyal M. Israfilov for valuable discussions and his encouragement. I am also grateful to the referees for their valuable criticisms and for pointing out L. Bers' work [6].

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GENERALIZED FABER SERIES

35

REFERENCES 1. L. Ahlfors, ``Lectures on Quasiconformal Mapping,'' Wadsworth, Belmont, CA, 1987. 2. S. Ya. Alper, Approximation in the mean of analytic functions of class Ep, Gos. Izdat. FizMat. Lit., Moscow (1960), 273286. 3. I. M. Batchaev, ``Integral Representations in a Region with Quasiconformal Boundary, and Some Applications,'' Doctoral dissertation, Baku, 1980. 4. V. I. Belyi, Conformal mappings and approximation of analytic functions in domains with quasiconformal boundary, USSR-Sb. 31 (1977), 289317. 5. C. A. Berenstein and R. Gay, ``Complex Variables,'' Springer-Verlag, New YorkBerlin, 1991. 6. L. Bers, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113134. 7. E. M. Dyn'kin, ``The Rate of Polynomial Approximation in the Complex Domain,'' Lecture Notes in Mathematics, Vol. 864, pp. 90142, Springer-Verlag, New YorkBerlin, 1981. 8. D. Gaier, ``Lectures on Complex Approximation,'' Birkhauser, BostonBaselStuttgart, 1987. 9. D. M. Israfilov, On approximation by partial sums of generalized Faber series in domains with quasiconformal boundary, Izv. Acad. Nauk. Az. SSR 5 (1981), 1016. [in Russian] 10. O. Lehto and K. I. Virtanen, ``Quasiconformal Mappings in the Plane,'' Springer-Verlag, New YorkBerlin, 1973. 11. P. K. Suetin, ``Series of Faber Polynomials,'' Moscow, Nauka, 1984. [in Russian]

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