Chui's conjecture in Bergman spaces
aa r X i v : . [ m a t h . C V ] S e p CHUI’S CONJECTURE IN BERGMAN SPACES
EVGENY ABAKUMOV, ALEXANDER BORICHEV, KONSTANTINFEDOROVSKIY
Abstract.
We solve Chui’s conjecture on the simplest fractions(i.e., sums of Cauchy kernels with unit coefficients) in weighted(Hilbert) Bergman spaces. Namely, for a wide class of weights, weprove that for every N , the simplest fractions with N poles on theunit circle have minimal norm if and only if the poles are equis-paced on the circle. We find sharp asymptotics of these norms.Furthermore, we describe the closure of the simplest fractions inweighted Bergman spaces, using an L version of Thompson’s the-orem on dominated approximation by simplest fractions. Introduction
The starting point of our research is the following question: How toput N point charges on the unit circle T of the complex plane C in orderto minimize the average strength of the corresponding electrostaticfield in the unit disk D , assuming forces inversely proportional to thedistance? C. K. Chui [7] conjectured in 1971 that this average strengthis minimal when the charges are equispaced on T , and, surprisingly,this very natural and elementary conjecture is still open.This and related questions call for the study of approximation prop-erties of so-called simplest fractions.We mean by a simplest fraction (the term simple partial fraction isalso used in the literature) a rational function r in the complex variable z having the form r ( z ) = X ≤ k Korevaar’s theorem. Let G be a bounded simply connected domainin C , and let E ⊂ C be such that E ∩ G = ∅ . The following fourstatements are equivalent: The set E is a polynomial approximation set relative to G . HUI’S CONJECTURE IN BERGMAN SPACES 3 For every point w ∈ E , the function z ( z − w ) − can be ap-proximated locally uniformly in G by polynomials having zeros only on E . There exists a system of finite families { a N,k : 0 ≤ k < N } ofpoints in E , such that X ≤ k In connection with this question Chui formulated the following con-jecture. Chui’s Conjecture. For any positive integer N , and for any familyof points { a k } ≤ k 0. Thus, Chui’s conjecture wouldimply that the set SF is not dense in A .The next year after the publication of Chui’s conjecture, D. J. New-man [15] proved that the set SF is not dense in A . More precisely, heestablished that (cid:13)(cid:13)(cid:13)(cid:13) X ≤ k 2. It is proved in [8] that for any Jordan domain D andfor every q > 2, the simplest fractions with poles on the boundary of D are dense in the respective Bers space in D . The results of [8] werelater extended in [9].Despite considerable progress in our knowledge of simplest fractionsproperties, including approximation ones, the original question posed HUI’S CONJECTURE IN BERGMAN SPACES 5 by Chui remains open. In this paper, we resolve a version of Chui’s con-jecture in the context of weighted Bergman spaces of square integrablefunctions, that is in the Hilbert space setting.Throughout the paper we use the following notation: for positive A and B , A . B means that there is a positive numerical constant C such that A ≤ CB , while A & B means that B . A , and A ≍ B means that both A . B and B . A .2. Main results Let us recall that for α > − A α = A α ( D ) consists of all functions f holomorphic in D for which thenorm k f k α is finite, where k f k α = ( α + 1) Z D | f ( z ) | (1 − | z | ) α dm ( z ) . We refer the reader to the book [12] where one can find a thoroughexposition of the theory of standard weighted Bergman spaces.More generally, if g is an integrable positive function on the interval[0 , A g ) = n f ∈ Hol( D ) : k f k g ) = κ g Z D | f ( z ) | g (1 − | z | ) dm ( z ) < ∞ o , where κ g = ( R g ( t ) dt ) − is the normalization constant. It can beverified directly that the fractions ( z − λ ) − , λ ∈ T , belong to A g ) ifand only if(2.1) Z g ( s ) s ds < ∞ . Recall that we denote by SF the set of all simplest fractions with poleson T . For every α > SF ⊂ A α , and for every α ∈ ( − , 0] wehave SF ∩ A α = ∅ . So, in what follows we suppose that α > N point masseson the unit circle, the norm of the corresponding Cauchy transform isthe smallest if and only if these point masses are equispaced on T . E. ABAKUMOV, A. BORICHEV, K. FEDOROVSKIY Theorem 1. Let g be a concave non-decreasing function on [0 , satisfying (2.1) and such that g (0) = 0 . Then for every integer N ≥ and for every family of points { a k } ≤ k For every α ∈ (0 , , for every integer N ≥ , and forevery family of points { a k } ≤ k 1, tends to a positive finitenumber for α = 1, and tends to + ∞ for α > 1. The following resultprovides with the exact asymptotics. As usual, we denote by ζ and Γthe Riemann zeta-function and the Gamma function, respectively. Theorem 3. For every α > we have lim N →∞ N α − k Ψ N k α = Γ( α + 2) ζ ( α + 1) > . In particular, lim N →∞ k Ψ N k = π √ . For general g , we can obtain weaker asymptotical estimates on thenorms k Ψ N k ( g ) . Proposition 4. Let g satisfy (2.1) . Then k Ψ N k g ) ≍ N Z /N g ( t ) dtt + N Z /N (1 − t ) N g ( t ) dt, N → ∞ . HUI’S CONJECTURE IN BERGMAN SPACES 7 Corollary 5. (A) For every c > we have exp( − cN ) . k Ψ N k ( g ) = o ( N / ) , N → ∞ . (B) If g ( t ) = o ( t ) , t → , then k Ψ N k ( g ) = o (1) , N → ∞ . (C) If q > and g ( t ) = log − q (2 /t ) , then k Ψ N k g ) ≍ N log q − N , N → ∞ . (D) If q > and g ( t ) = exp( − t − q ) , then log(1 / k Ψ N k ( g ) ) ≍ N q/ ( q +1) , N → ∞ . We do not know whether the equispaced distribution remains to beoptimal for the spaces A α when α > 1. Nevertheless, we show thatasymptotically this is true up to a constant: Theorem 6. Let α > . For some absolute constant C > and forsome number C ( α ) > , we have αC N − α ≤ min a k ∈ T , ≤ k Let g satisfy (2.1) . Then clos A g ) SF = SF , t = O ( g ( t )) , t → ,A g ) , g ( t ) = o ( t ) , t → . In particular, SF is closed nowhere dense in A α when 0 < α ≤ A α when α > α = 1 we have a more precise result. Set SF N = n X ≤ j Theorem 8. For every f ∈ A , we have lim N →∞ dist A ( f, SF N ) = π √ . This result shows, in particular, that the set SF is a (( π/ √ 3) + ε )-net in the space A , for small ε > 0; considering the functions − Ψ N with large N we see that the set SF is not a (( π/ √ − ε )-net in thespace A , for small ε > f in D , thereexist h n ∈ S N ≥ n SF N , n ≥ 1, converging to f uniformly on compactsubsets of D and such thatsup n ≥ , z ∈ D (1 − | z | ) | h n ( z ) | < ∞ . His proof used the results and the constructions by Mac Lane in [14].Let us formulate a somewhat improved version of Thompson’s theorem.Let H ∞ = H ∞ ( D ) denote the space of bounded analytic functionsin the unit disc. Theorem 9. Let f ∈ H ∞ . For every ε > , for every compact subset K of D , and for every N ≥ N ( f, ε, K ) there exists h ∈ SF N such that k f − h k L ∞ ( K ) ≤ ε, | h ( z ) | ≤ − | z | + C k f k H ∞ log e − | z | , z ∈ D , for some absolute constant C . To prove Theorems 7 and 8 we use an L p version of Thompson’stheorem which we will formulate below. Whereas the simplest fractions h constructed in the proof of Theorem 9 have “almost” equispacedpoles, our Theorem 10 shows that the average growth of h along theconcentric circles r T is not much faster than that of the correspondingsimplest fraction Ψ N .Given β > 0, denote ρ ( β ) = 1 + β ((1 + β ) / ( p − − p − > HUI’S CONJECTURE IN BERGMAN SPACES 9 for 1 < p < ∞ and ρ ( β ) = 1 for p = 1, so that, by a simple calculation,we have ( x + y ) p ≤ (1 + β ) x p + ρ ( β ) y p , x, y ≥ . Theorem 10. Let f ∈ H ∞ , ≤ p < ∞ . For every ε, β > , for everycompact subset K of D , and for every N ≥ N ( f, ε, K ) there exists h ∈ SF N such that k f − h k L ∞ ( K ) ≤ ε, Z | h ( e πis r ) | p ds ≤ (1 + β ) Z | Ψ N ( e πis r ) | p ds (2.2) + ρ ( β ) C p k f k pH ∞ log p e − r , < r < , for C as in Theorem 9. Remark 11. Our estimates on h in Theorem 10 improve on those inTheorem 9. Namely, for < p < ∞ and r ∈ (1 − N − , , Theorem 10gives I r := Z | h ( e πis r ) | p ds . N (1 − r ) − p , which improves on the estimate I r . (1 − r ) − p that we can get fromTheorem 9. If − r = A/N for large fixed A , then Theorem 10 gives I r . N p e − pA while Theorem 9 gives I r . N p A − p , N → ∞ . Our results motivate the following open questions. Question . Does Theorem 1 hold for larger classes of g ? For instance,for g ( t ) = t α , α > Question . It would be of interest to have more information aboutthe mutual location of the sets SF n in the spaces A α . In particular,are pairwise distances between these sets bounded away from 0 in thespace A ? Our conjecture is that the answer is positive, and, moreover,if α > n, k ≥ 1, thendist A α ( SF n , SF n + k ) = k Ψ k k α . We finish this section with a few words about the organization of thepaper and the methods used.Theorems 1, 3, and 6 and Proposition 4 are proved in Section 4.The proof of Theorem 1 uses some classical results on trigonometricseries and a convexity argument, which is discussed in Section 3. Theproofs of Theorem 3 and Proposition 4 are direct calculations. In theproof of Theorem 6 we use moment estimates for systems of unimodularnumbers going back to J. W. S. Cassels.In Section 5 we establish Theorems 9 and 10 generalizing Thompson’stheorem. Using the ideas of [14] and [17], we provide a short argumentwith better pointwise and integral estimates.Theorems 7 and 8 are proved in Section 6. Their proofs use Theo-rem 10. In Remark 15 we indicate an alternative way to get the densityof SF in A α , α > 1. 3. Auxiliary lemmas Let g be a function satisfying the conditions of Theorem 1. Forinteger k ≥ c ( g ) ,k = Z t k g (1 − t ) dt > , and define the function ϕ ( g ) ( t ) = X k ≥ c ( g ) ,k cos(( k + 1) t ) , t ∈ R . Notice that condition (2.1) is equivalent to the fact that ϕ ( g ) (0) < ∞ .Next, for every α > 0, let g α ( t ) = t α , t ≥ c α,k = c ( g α ) ,k , and ϕ α = ϕ g α , so that c α,k = Z t k (1 − t ) α dt, k ≥ ϕ α ( t ) = X k ≥ c α,k cos(( k + 1) t ) , t ∈ R . Notice that for α > c α,k ≍ k − ( α +1) , k → ∞ . HUI’S CONJECTURE IN BERGMAN SPACES 11 Both ϕ ( g ) (for the aforesaid g ) and ϕ α (for α > 0) are 2 π -periodiceven continuous functions.We need the following convexity lemma. Lemma 12. (1) For every function g satisfying the conditions of Theorem 1, thefunction ϕ ( g ) is strictly convex on (0 , π ) . (2) The function ϕ α , α > , is strictly convex on (0 , π ) if and onlyif α ∈ (0 , .Proof. (A) First we prove that the function ϕ = ϕ is strictly convex.We have c ,k = 1( k + 1)( k + 2) , k ≥ , and, hence, ϕ ( t ) = X k ≥ cos( kt ) k ( k + 1) . Therefore, for every t ∈ (0 , π ) we obtain ϕ ′ ( t ) = − X k ≥ sin( kt ) k + 1 = − X k ≥ sin( kt ) k + 1 + X k ≥ sin( kt ) k − π − t X k ≥ sin( kt ) k ( k + 1) − π − t , and, hence, ϕ ′′ ( t ) = 12 + X k ≥ cos( kt ) k + 1 . Now we are going to use the following result from the book by N. Bari[3, Chapter 1, Section 30]. Let { a k } k ≥ be a decreasing convex sequenceof positive numbers, lim k →∞ a k = 0. Then ( a / P k ≥ a k cos( kt ) ≥ t ∈ (0 , π ), because a X k ≥ a k cos( kt ) = 12 X j ≥ ( j + 1)∆ a j F j +1 ( t ) , t ∈ (0 , π ) , where ∆ a j = ∆ a j − ∆ a j +1 , ∆ a j = a j − a j +1 , j ≥ 0, and F j are theFej´er kernels, F j ( t ) = 1 j (cid:16) sin( jt/ t/ (cid:17) ≥ , j ≥ . In our situation, a k = 1 / ( k + 1), ∆ a k > k ≥ 0, and, hence, we have ϕ ′′ ( t ) > , π ).(B) Let g be a function satisfying the conditions of Theorem 1. Thenwe have ϕ ( g ) ( t ) = X k ≥ cos(( k + 1) t ) Z s k g (1 − s ) ds = Re e it Z X k ≥ e itk s k g (1 − s ) ds = Re Z e − it − s g (1 − s ) ds = Z cos t − s s − s cos t g (1 − s ) ds. Hence, ϕ ( g ) ∈ C ∞ ((0 , π )).Furthermore, ϕ ( g ) ( t ) = Z cos t − s s − s cos t g (1 − s ) ds = − Z g (1 − s ) d log(1 + s − s cos t )= − Z log(1 + s − s cos t ) g ′ (1 − s ) ds, for 0 < t < π . Hence, ϕ ′ ( g ) ( t ) = − Z s sin t s − s cos t g ′ (1 − s ) ds, < t < π, and ϕ ′′ ( g ) ( t ) = − Z s cos t (1 + s − s cos t ) − s sin t (1 + s − s cos t ) g ′ (1 − s ) ds = Z s s − (1 + s ) cos t (1 + s − s cos t ) g ′ (1 − s ) ds, < t < π. HUI’S CONJECTURE IN BERGMAN SPACES 13 Thus, ϕ ( g ) is strictly convex on (cid:2) π , π (cid:3) . Next, let us observe that ϕ ′′ ( g ) ( t ) = ϕ ′′ ( g ) (2 π − t ) on (0 , π ). Hence, ϕ ( g ) is strictly convex on (cid:2) π , π (cid:3) .Since the function ϕ is strictly convex on (0 , π ), we obtain that Z h t ( s ) ds > , for 0 < t < π h t ( s ) = s s − (1 + s ) cos t (1 + s − s cos t ) . Take now t ∈ (cid:0) , π (cid:1) and choose (the unique) s t ∈ (0 , 1) such that2 s t = (1 + s t ) cos t . Then h t ( s ) < s ∈ (0 , s t ) and h t ( s ) > s ∈ ( s t , ϕ ′′ ( g ) ( t ) = Z h t ( s ) g ′ (1 − s ) ds ≥ Z s t h t ( s ) g ′ (1 − s t ) ds + Z s t h t ( s ) g ′ (1 − s t ) ds = g ′ (1 − s t ) Z h t ( s ) ds ≥ . Suppose now that ϕ ′′ ( g ) ( t ) = 0. Then g ′ (1 − s t ) = 0, and, hence, g ′ = 0 on the interval [1 − s t , ϕ ′′ ( g ) ( t ) = R s t h t ( s ) g ′ (1 − s ) ds > ϕ ( g ) is strictly convex on(0 , π ). Using once again that ϕ ′′ ( g ) ( t ) = ϕ ′′ ( g ) (2 π − t ), we conclude that ϕ ( g ) is strictly convex on (0 , π ).(C) By the result of (B), the function ϕ α is strictly convex on (0 , π )for α ∈ (0 , α > ϕ α is notconvex. Indeed, for such α we have ϕ α ∈ C ( R ), and, since ϕ α attainsits maximum at the point t = 0, this function cannot be convex on(0 , π ). The lemma is proved. (cid:3) The next lemma pertains to the convex analysis. Lemma 13. Let ϕ be a π -periodic even continuous function strictlyconvex on (0 , π ) . Then for every N ≥ we have (3.1) inf ϑ j ∈ [0 , π ) , ≤ j Proof of Theorem 1. Let N ≥ ϑ k ∈ [0 , π ], 0 ≤ k < N , f ( z ) = X ≤ k Given N ≥ 1, we have k Ψ N k α = ( α + 1) Z D (cid:12)(cid:12)(cid:12)(cid:12) N z N − − z N (cid:12)(cid:12)(cid:12)(cid:12) (1 − | z | ) α dm ( z )= α + 1 π Z (cid:18)Z π dt | − r N e iNt | (cid:19) ( N r N − ) (1 − r ) α r dr. By a direct computation one verifies that Z π dt | − xe it | = 2 π − x , ≤ x < . Therefore, k Ψ N k α = 2( α + 1) Z N r N − (1 − r ) α r − r N dr Using the substitution r = e − s/ (2 N ) , we obtain N α − k Ψ N k α = ( α + 1) Z + ∞ (cid:0) N (1 − e − s/N ) (cid:1) α e s − ds. HUI’S CONJECTURE IN BERGMAN SPACES 17 Since 1 − e − x ≤ x for x ≥ 0, by the Lebesgue dominated convergencetheorem we conclude thatlim N →∞ N α − k Ψ N k α = ( α + 1) Z + ∞ s α e s − ds = ( α + 1) X k ≥ Z + ∞ s α e − ks ds = ( α + 1) X k ≥ k − α − Z + ∞ s α e − s ds = Γ( α + 2) ζ ( α + 1) . (cid:3) Remark 14. The same calculation shows that for every α > , thesequence { N α − k Ψ N k α } N ≥ is monotonically increasing.Proof of Proposition 4. As in the proof of Theorem 3, we have k Ψ N k g ) ≍ Z N r N − g (1 − r ) r − r N dr ≍ N Z r N − g (1 − r )1 − r N dr = N Z − (1 /N )0 r N − g (1 − r )1 − r N dr + N Z − (1 /N ) r N − g (1 − r )1 − r N dr ≍ N Z − (1 /N )0 r N g (1 − r ) dr + N Z − (1 /N ) g (1 − r ) N · (1 − r ) dr = N Z /N (1 − r ) N g ( r ) dr + N Z /N g ( r ) r dr. (cid:3) Proof of Theorem 6. The upper estimate follows from Theorem 3.Fix α > N ≥ 1, and a k ∈ T , 0 ≤ k < N . To establish the lowerestimate, it suffices to verify that for some absolute constant C > I := Z N − < −| z | < N − (cid:12)(cid:12)(cid:12)(cid:12) X ≤ k Since X ≤ k 1, (see [11] for a probabilistic approach and [1] fora deterministic algebraic approach). However, the sum of the squaresof the moduli of S j for j between 1 and (1 + ε ) N (not between 1 and N ) admits a good lower estimate like in (4.2). Our argument here isinspired by that of J. W. S. Cassels in [5].For every M ≥ X ≤ j ≤ M (cid:16) − jM + 1 (cid:17)(cid:12)(cid:12)(cid:12) X ≤ k Choose now M = 2 N . Then X ≤ j ≤ N (cid:12)(cid:12)(cid:12) X ≤ k Proof of Theorem 9. Denote M = k f k H ∞ . Given N ≥ 1, set W N ( t ) = N t − Z t Re( e πiu f ( e πiu )) du, t ≥ . We have W N (0) = 0, W N (1) = N , | W ′ N ( t ) − N | ≤ M, t ≥ . For sufficiently large N , the function W N increases, and we set x N,k = W − N ( k ), 0 ≤ k ≤ N . We have(5.1) | x N,k +1 − x N,k | = N − + O ( N − M ) , ≤ k < N, N → ∞ . Put h N ( z ) = X ≤ k Hence, f ( z ) = Z W ′ N ( t ) z − e πit dt, and | f ( z ) − h N ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12) X ≤ k We use the notation from the proof of Theo-rem 9. Given f ∈ H ∞ , ε > 0, and a compact subset K of D , choose N ≥ N ( f, ε, K ) and h N ∈ SF N constructed in the proof of Theorem 9so that k f − h N k L ∞ ( K ) ≤ ε, It remains to verify the integral estimate (2.2). Fix r ∈ (0 , 1) andset v ( e πit ) = | Ψ N ( e πit r ) | , ≤ t < . Since v ( e πit ) = N r N − | e πitN r N − | , ≤ t < , the function t v ( e πit ) is even and decreases on [0 , / (2 N )]. Fur-thermore, we set v ( e πit ) = v (cid:0) e πit (cid:1) , e πit ∈ U := { e πiu : − πN ≤ u ≤ πN } ,v (cid:0) e πi/N (cid:1) , e πit / ∈ U,w ( t ) = Z t v p ( e πiu ) du, ≤ t ≤ . Then the function t v ( e πit ) is decreasing on [0 , w is concave on [0 , (cid:12)(cid:12) e πis − e πix N,m (cid:12)(cid:12) = min ≤ k 1] into subintervals J k = [ x N,k , ( x N,k + x N,k +1 ) / J k +1 = [( x N,k + x N,k +1 ) / , x N,k +1 ], 0 ≤ k < N . HUI’S CONJECTURE IN BERGMAN SPACES 23 Then Z | h N ( e πis r ) | p ds = X ≤ k Proof of Theorem 7. Denote by S ( g ) the closure of the set SF in A g ) .Since k f k ( g ) ≤ k f k ( g ) when g ≤ g , we need only to consider thecases g ( t ) = t and g ( t ) = o ( t ), t → g ( t ) = t . Then A g ) = A . We are going to verify that(6.1) lim inf N →∞ inf g ∈SF N k f − g k ≥ π , f ∈ A . Since every SF N is compact in A , we can then conclude that S ( g ) = SF .Assume that (6.1) does not hold. Then for some ε ∈ (0 , π / 12) wefind f ∈ A , a sequence { N m } m ≥ such that lim m →∞ N m = ∞ , and asequence { f m } m ≥ , f m ∈ SF N m , m ≥ 1, such that(6.2) k f − f m k ≤ π − ε, m ≥ . Given δ ∈ (0 , g δ ( t ) = min( δ, t ). Since(6.3) | f ( z ) | g δ (1 − | z | ) ≤ | f ( z ) | (1 − | z | ) , z ∈ D , and the function z 7→ | f ( z ) | (1 − | z | ) is integrable on D , by Lebesgue’sdominated convergence theorem we havelim δ → Z D | f ( z ) | g δ (1 − | z | ) dm ( z ) = 0 . Choose δ ∈ (0 , 1) such that Z D | f ( z ) | g δ (1 − | z | ) dm ( z ) ≤ ε . By (6.2) and (6.3) we have Z D | f ( z ) − f m ( z ) | g δ (1 − | z | ) dm ( z ) ≤ π − ε, m ≥ , and, hence, Z D | f m ( z ) | g δ (1 − | z | ) dm ( z ) ≤ (cid:16) ε (cid:17) Z D | f ( z ) − f m ( z ) | g δ (1 − | z | ) dm ( z )+ (cid:16) ε (cid:17) Z D | f ( z ) | g δ (1 − | z | ) dm ( z ) ≤ π − ε, m ≥ . Since the function g δ is concave and non-decreasing, g δ (0) = 0, and Z g δ ( t ) t − dt < ∞ , we can apply Theorem 1 and obtain Z D | Ψ N m ( z ) | g δ (1 − | z | ) dm ( z ) ≤ π − ε, m ≥ . Since the functions Ψ N m tend to 0 uniformly on compact subsets of D as m → ∞ , we conclude that Z D | Ψ N m ( z ) | (1 − | z | ) dm ( z ) ≤ π − ε , m ≥ m ( δ ) , which contradicts to Theorem 3.This contradiction establishes relation (6.1) and, hence, the equality S ( g ) = SF for g ( t ) = t .(B) Let g ( t ) = o ( t ), t → 0, and let f ∈ A g ) . Replacing f by thefunction z f ((1 − δ ) z ) with small positive δ , we can assume that f ∈ H ∞ . By Theorem 10, there exist h N ∈ SF N , such that h N tend HUI’S CONJECTURE IN BERGMAN SPACES 25 to f uniformly on compact subsets of D , N → ∞ , and for r ∈ (0 , 1) wehave Z | h N ( e πis r ) | ds ≤ Z | Ψ N ( e πis r ) | ds + 2 C k f k H ∞ log e − r . By Corollary 5 (B), we conclude that k f − h N k g ) → , N → ∞ . Thus, S ( g ) = A g ) . (cid:3) Remark 15. Given α > , denote by S α the closure of the set SF in A α . An alternative way to prove that S α = A α for α > , follows thescheme proposed by Korevaar in [13] (with a reference to A. Beurling).Namely, in the case α > , we use that lim N →∞ k Ψ N k α = 0 to show thatthe functions z 7→ − ( z − w ) − , w ∈ T , belong to S α . As a consequence,the real linear space R spanned by the family of functions n iw ( z − w ) : w ∈ T o is contained in the set S α . In the case < α ≤ , using anexplicit but more complicated argument we can establish that the reallinear space e R spanned by the family n z − w : w ∈ T o is containedin the set S α . Then an argument based on the Hahn–Banach theorempermits us to show that, for α > , R is dense in A α (for α > , e R isdense in A α ), and to conclude.Proof of Theorem 8. Because of (6.1), we need only to verify that forevery f ∈ A , ε > 0, and for every N ≥ N ( f, ǫ ), there exists h ∈ SF N such that k f − h k ≤ π ε. Replacing f by z f ((1 − δ ) z ) with small positive δ , we can assumethat f ∈ H ∞ . Given 0 < β < 1, choose η ∈ (0 , 1) such that Z D \ (1 − η ) D | f ( z ) | dm ( z ) + Z − η (1 − r ) log e − r dr < β . By Theorem 10, for every N ≥ N ( f, ǫ ), there exists h ∈ SF N suchthat k f − h k L ∞ ((1 − η ) D ) < β and Z | h ( e πis r ) | ds ≤ (1 + β ) Z | Ψ N ( e πis r ) | ds + 1 + ββ C k f k H ∞ log e − r , < r < . Then k f − h k = 2 Z (1 − η ) D | f ( z ) − h ( z ) | (1 − | z | ) dm ( z )+ 2 Z D \ (1 − η ) D | f ( z ) − h ( z ) | (1 − | z | ) dm ( z ) ≤ β + 2(1 + β − ) Z D \ (1 − η ) D | f ( z ) | (1 − | z | ) dm ( z )+ 2(1 + β ) Z D \ (1 − η ) D | h ( z ) | (1 − | z | ) dm ( z ) ≤ β + 2 β (1 + β ) + 2(1 + β ) Z D \ (1 − η ) D | Ψ N ( z ) | (1 − | z | ) dm ( z )+ 2(1 + β ) β Z D \ (1 − η ) D C k f k H ∞ (1 − | z | ) log e − | z | dm ( z ) ≤ (5 + C · C k f k H ∞ ) β + 2(1 + β ) π ≤ π ε, for sufficiently small β . (cid:3) References [1] J. Andersson, On some power sum problems of Montgomery and Tur´an , Int.Math. Res. Not. (2008), Art. ID rnn015.[2] J. M. Anderson, V. Ya. Eiderman, Cauchy transforms of point masses: Thelogarithmic derivative of polynomials , Annals of Math. (2006), 1057–1076.[3] N. K. Bary, A treatise on trigonometric series. I , Pergamon Press, Oxford,NY, 1964.[4] P. A. Borodin, Approximation by simple partial fractions with constraints onthe poles. II , Sb. Math. (2016), no. 3-4, 331–341.[5] J. W. S. Cassels, On the sums of powers of complex numbers , Acta Math.Acad. Sci. Hungar. (1956), 283–289. HUI’S CONJECTURE IN BERGMAN SPACES 27 [6] C. K. Chui, Bounded approximation by polynomials whose zeros lie on a circle ,Trans. Amer. Math. Soc. (1969), 171–182.[7] C. K. Chui, A lower bound of fields due to unit point masses , Amer. Math.Monthly (1971), no. 7, 779–780.[8] C. K. Chui, On approximation in the Bers spaces , Proc. Amer. Math. Soc. (1973), no. 2, 438–442.[9] C. K. Chui, X. C. Shen, Order of approximation by electrostatic fields due toelectrons , Constr. Approx. (1985), no. 2, 121–135.[10] P. V. Chunaev, V. I. Danchenko, M. A. Komarov, Extremal and approximativeproperties of simple partial fractions , Russian Math. (Iz. VUZ) (2018),no. 12, 6–41.[11] P. Erd¨os, A. R´enyi, A probabilistic approach to problems of Diophantine ap-proximation , Illinois J. Math. (1957), 303–315.[12] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces , Graduatetexts in mathematics, vol. 199, Springer–Verlag, New York, 2000.[13] J. Korevaar, Asymptotically neutral distributions of electrons and polynomialapproximation , Annals of Math. (1964), no. 3, 403–410.[14] G. R. Mac Lane, Polynomials with zeros on a rectifiable Jordan curve , DukeMath. J. (1949), 461–477.[15] D. J. Newman, A lower bound for an area integral , Amer. Math. Monthly (1972), no. 9, 1015–1016.[16] Z. Rubinstein, E. B. Saff, Bounded approximation by polynomials whose zeroslie on a circle , Proc. Amer. Math. Soc. (1971), 482–486.[17] M. Thompson, Approximation of bounded analytic functions on the disc ,Nieuw Arch. Wisk. (3) (1967), 49–54. Evgeny Abakumov:Universit´e Gustave Eiffel, Marne-la-Vall´ee, France [email protected] Alexander Borichev:Aix–Marseille University, CNRS, Centrale Marseille, I2M, France [email protected]