Weighted- L 2 polynomial approximation in C
aa r X i v : . [ m a t h . C V ] F e b WEIGHTED- L POLYNOMIAL APPROXIMATION IN C S´EVERINE BIARD, JOHN ERIK FORNÆSS, AND JUJIE WU
Abstract.
We study the density of polynomials in H (Ω , e − ϕ ), thespace of square integrable holomorphic functions in a bounded domainΩ in C , where ϕ is a subharmonic function. In particular, we provethat the density holds in Carath´eodory domains for any subharmonicfunction ϕ in a neighborhood of Ω. In non-Carath´eodory domains, weprove that the density depends on the weight function, giving examples. Mathematics Subject Classification (2010): 30D15, 30E10, 32A10,32E30, 32W05
Keywords : Carath´eodory domain, weighted L -estimates, polynomialapproximation, moon-shaped domain Contents
1. Introduction 12. Proof of Theorem 1.3 43. Proof of Theorem 1.6 84. Proof of Theorem 1.7 145. Example 1.10 186. Proof of Theorem 1.11 20References 221.
Introduction
Let Ω be a domain in C . We denote by L (Ω , e − ϕ ) the space of measurablefunctions f such that k f k ,ϕ := Z Ω | f | e − ϕ dλ < + ∞ , where ϕ is a measurable function on Ω, and dλ is the Lebesgue measure. Let H (Ω , e − ϕ ) (resp. H (Ω , e − ϕ ) ) be the space of holomorphic functions on adomain Ω (resp. holomorphic functions on a neighborhood of Ω), which arein L (Ω , e − ϕ ), that is H (Ω , e − ϕ ) := O (Ω) ∩ L (Ω , e − ϕ ) . Recall that a Carath´eodory domain Ω is a simply-connected boundedplanar domain whose boundary ∂ Ω is also the boundary of an unboundeddomain. An unbounded domain Ω is said to be Carath´eodory if its image un-der the map z ( z − z ) − ( z being a fixed point in C \ Ω) is Carath´eodory.Every Jordan domain is a Carath´eodory domain. The domains, for exam-ple, of a snake winding infinitely often around the outside of a circle andapproaching this circle (“outer snake”) are Carath´eodory, but not a snakewinding infinitely often inside a circle and approaching it from the inside(“inner snake”). For more relavant references about Carath´eodory domain,please see [10] on page 17.Questions of completeness for polynomials were first studied by T. Car-leman [4] in 1923 who proved that if Ω is a Jordan domain and ϕ ≡ L holomorphic function on Ω can be approximated by poly-nomials in L (Ω , e − ϕ is continuous and satisfies some conditions then polynomials aredense in H (Ω , e − ϕ ). For non-Carath´eodory domains, the weighted approx-imation is usually considered when the weight e − ϕ is essentially boundedand satisfies additional conditions (see [3]). Based on H¨ormander’s L -estimates for the ¯ ∂ − operator, Taylor [16] proved that polynomials are densein H ( C n , e − ϕ ) when ϕ is convex, which allows the weight to have singular-ity and can be seen as a major breakthrough for weighted L approximation(see also [18]). Sibony [15] generalized Taylor’s result and obtained thatif ϕ is plurisubharmonic (psh) on C n and complex homogeneous of order ρ >
0, i.e, ϕ ( uz ) = | u | ρ ϕ ( z ) for u ∈ C , z ∈ C n then polynomials are densein H ( C n , e − ϕ ) (see also [9], section 8). It is well known that each convexfunction is psh, but the converse is not true. Thus it is natural to ask Question 1.1.
Can we generalize Taylor’s result to any psh function or canwe find some non-convex psh function ϕ on Ω ⊂ C n satisfying the propertythat H (Ω , e − ϕ ) contains all the polynomials but polynomials are not densein it?Our first result is Proposition 1.2.
Let
Ω = T Nν =0 G ν be a bounded domain in C where G is a bounded Carath´eodory domain and G ν , ≤ ν ≤ N , are unboundedCarath´eodory domains. If ϕ is a subharmonic function on Ω , i.e. in aneighborhood V of Ω , then H (Ω , e − ϕ ) is dense in H (Ω , e − ϕ ) . Our proof depends heavily on the Donnelly-Fefferman L -estimate for the¯ ∂ -operator. In contrast with known results on weighted L approximation ofholomorphic functions, we allow singularities of the weight function, whichmakes the result useful. An application of Proposition 1.2 is the following EIGHTED- L POLYNOMIAL APPROXIMATION 3
Theorem 1.3.
Let Ω be a bounded Carath´eodory domain and ϕ a subhar-monic function on Ω . then polynomials are dense in H (Ω , e − ϕ ) . Especially we will have the following
Corollary 1.4.
Let Ω be a bounded Jordan domain and let ϕ be as in The-orem 1.3. Then polynomials are dense in H (Ω , e − ϕ ) . Remark 1.5.
Under the assumptions of Corollary 1.4, let f ∈ H (Ω , e − ϕ ).Then f can be approximated by polynomials in H (Ω , e − ϕ ) such that theTaylor series of the polynomials around a given point p ∈ Ω agrees with theone for f to any given order.It’s not the case that polynomials are dense for general psh weight func-tions so that the corresponding Hilbert spaces contain the polynomials. Anexample is provided by the following Theorem 1.6.
Let ϕ ( z ) = |ℑ m ( z ) | + | z | p , with < p < . Then the holo-morphic polynomials are in H ( C , e − ϕ ) , but they are not dense in H ( C , e − ϕ ) . A general moon-shaped domain is a bounded domain whose boundaryconsists of two Jordan curves having exactly one point in common. Thispoint is called the multiple boundary point. The moon-shaped domain is anexample of a non-Carath´eodory Runge domain in C . Keldych [12] was thefirst to study the L polynomials approximation property of moon-shapeddomains without weight. Here we generalize his result with singular weightas in the following. Theorem 1.7.
Let Ω be a moon-shaped domain with the origin inside theinner Jordan curve of the boundary ∂ Ω . Let ϕ be a subharmonic functionon Ω . Then polynomials are dense in H (Ω , e − ϕ ) if and only if the function √ z can be approximated arbitrarily well by polynomials in L (Ω , e − ϕ ) . We give two concrete examples of moon-shaped domains where densityholds. The first example is based on Keldych [12] and is an application ofTheorem 1.7.
Definition 1.8.
Let φ be a subharmonic function on C . Let µ denote theLaplacian of π φ. We say that φ satisfies condition (A) if the mass of µ onthe closed unit disc is strictly less than 2. Example 1.9.
There exists a moon-shaped domain with the unit circlebeing the outer Jordan curve, such that for any subharmonic function ϕ on C satisfying condition (A), polynomials are in H (Ω , e − ϕ ) and dense in it. Example 1.10.
There exist a moon-shaped domain bounded by two circlesand a subharmonic function ϕ on Ω so that polynomials are in H (Ω , e − ϕ )and are dense in it. S´EVERINE BIARD, JOHN ERIK FORNÆSS, AND JUJIE WU
Theorem 1.11.
Let Ω be a moon-shaped domain bounded by two circlesand let ϕ be a subharmonic function on Ω which is uniformly bounded above.Then the set of polynomials which is in H (Ω , e − ϕ ) is never a dense subsetof H (Ω , e − ϕ ) . This paper is set-up as follows. In Section 2, we prove Proposition 1.2and Theorem 1.3, exploiting the property of Carath´eodory domains in orderto be able to exhaust them from outside by Jordan domains that are con-formally equivalent to the unit disc. Then, we apply Donnelly-Fefferman’s L -estimates on each of those to obtain the weighted L approximation. InSection 3, we give a counterexample on C where we exhibit a subharmonicfunction ϕ for which the polynomials are in H ( C , e − ϕ ) but they are notdense in it (Theorem 1.6). In Section 4, we prove Theorem 1.7 and give anexample of a moon-shaped domain where the density is proved by approx-imating √ z (Example 1.9). In Section 5, we present Example 1.10 and wefinally prove, in Section 6, Theorem 1.11.2. Proof of Theorem 1.3
We observe that it suffices to prove Proposition 1.2 and Theorem 1.3when the subharmonic function ϕ is globally defined. To see this, let φ bea subharmonic function defined on a bounded open set V containing Ω andchoose an open set U, Ω ⊂ U ⊂⊂ V. Then µ := ∆( φ ) | U is a positive measurewith bounded mass on C . Hence there is a globally defined subharmonicfunction ϕ such that ∆( ϕ ) = µ. But then φ = ϕ + h on U for some har-monic function h . Since h is uniformly bounded on Ω, it follows that theHilbert spaces H (Ω , e − φ ) and H (Ω , e − ϕ ) are the same and the norms areequivalent.We use, in the next Lemma, the following classical result from one complexvariable: Theorem 2.1 (cf. [17], p. 382 ) . Let { Ω n } ∞ n =1 be a sequence of uniformlybounded simply connected domains in C and Ω a bounded simply connecteddomain, all containing the origin, so that the Hausdorff distance between ∂ Ω n and ∂ Ω tends to zero as n → ∞ . If we map D conformally onto Ω n by w = f n ( z ) , f n (0) = 0 , f ′ n (0) > , then f n converges locally uniformly to f ∈ O ( D ) such that w = f ( z ) maps D conformally onto Ω . For a planar domain Ω, let SH − (Ω) denote the set of negative subhar-monic functions on Ω. Our key observation to prove Proposition 1.2 is thefollowing: Lemma 2.2.
Let Ω be a bounded Carath´eodory domain. Then there existsa sequence of bounded simply-connected domains Ω n ⊃ Ω , a sequence ofpositive numbers ε n → n → ∞ ) , and a sequence of continuous functions ρ n ∈ SH − (Ω n ) such that (1) Ω n, − ε n := { z ∈ Ω n : ρ n ( z ) < − ε n } ⊂ Ω , EIGHTED- L POLYNOMIAL APPROXIMATION 5 (2) the volume of Ω \ Ω n, − ε n tends to 0 as n → ∞ . Before proving the Lemma we recall the following L -estimates for the¯ ∂ -operator which will be used here. Proposition 2.3 (Donnelly-Fefferman, [7]) . Let Ω ⊂ C n be a pseudoconvexdomain and ϕ ∈ psh (Ω) . Suppose that ψ is a C strictly psh function whichsatisfies i∂∂ψ ≥ i∂ψ ∧ ∂ψ. (2.1) Then for each ∂ -closed (0 , -form v , there exists a solution u to ∂u = v satisfying Z Ω | u | e − ϕ dλ ≤ C Z Ω | v | i∂∂ψ e − ϕ dλ, (2.2) where C > is an absolute constant, provided that the right hand side of(2.2) is finite. The norm k α k i∂ ¯ ∂ψ for a (0 , α is the smallest function H thatsatisfies iα ∧ α ≤ H ( i∂ ¯ ∂ψ ) . In particular, on C , we have(2.3) | α | i∂ ¯ ∂ψ = (cid:18) ∂ ψ∂z∂ ¯ z (cid:19) − | α | . For more details, see for example [6, 2].
Proof of Lemma 2.2.
Since Ω is a Carath´eodory domain, there exists a se-quence { Ω n } of bounded simply-connected domains such that Ω ⊂ Ω n andΩ n +1 ⊂ Ω n and the Hausdorff distance between ∂ Ω n and ∂ Ω tends to zeroas n → ∞ (see for example [10], p.17). Without loss of generality, we mayassume that 0 ∈ Ω. By virtue of Riemann’s mapping theorem, there areconformal mappings w = f n ( z ) , f n (0) = 0 , f ′ n (0) > n onto D , and w = f ( z ) , f (0) = 0 , f ′ (0) > , which maps Ω onto D . Set ρ ( z ) = | f ( z ) | − , ρ n ( z ) = | f n ( z ) | − . Clearly, ρ (resp. ρ n ) is a negative continuous subharmonic function on Ω(resp. Ω n ). Let ε n := max (cid:8) − | f n ( z ) | : z ∈ Ω n \ Ω (cid:9) . By Theorem 2.1, the sequence of Riemann mappings f − n : D → Ω n con-verges u.c.c. to the Riemann mapping f − : D → Ω . Suppose z ∈ Ω n and ρ n ( z ) < − ε n . Then | f n ( z ) | − < − ε n and hence 1 − | f n ( z ) | > ε n . This
S´EVERINE BIARD, JOHN ERIK FORNÆSS, AND JUJIE WU implies that z ∈ Ω which proves (1). Next we prove that ε n → . Let0 < a < b < c <
1. By the open mapping theorem, there exists n ∈ N sothat f − ( | z | < a ) ⊂ f − n ( | z | < b ) ⊂ f − ( | z | < c ) , ∀ n ≥ n . Hence f − n ( | z | < b ) ⊂ Ω so f n (Ω) ⊃ ( | z | < b ) . Therefore if z ∈ Ω n \ Ω , then | f n ( z ) | ≥ b, so 1 − | f n ( z ) | ≤ − b. This shows that ε n ≤ − b if n ≥ n . It follows that ε n → b tends to 1.Finally we show (2).Let δ > . Then if 0 < a < | f − ( | z | > a ) | < δ. Choose 0 < a < b < c < − b < (1 − a ) / . Then for all large enough n , 2 ε n ≤ − b ) < − a. Suppose that z ∈ Ω \ Ω n, − ε n . Then ρ n ( z ) ≥ − ε n so | f n ( z ) | ≥ − ε n > a which impliesthat (for large n ) | f ( z ) | > a. Hence | Ω \ Ω n, − ε n | < δ. (cid:3) Now we can prove Proposition 1.2.
Proof of Proposition 1.2.
In view of Lemma 2.2, there exist for each 0 ≤ ν ≤ N a sequence of Jordan domains G νn ⊃ G ν , a sequence of positivenumbers ε νn → n → ∞ ), and a sequence of continuous functions ρ νn ∈ SH − ( G νn ) such that(i) G νn, − ε νn := { z ∈ G νn : ρ νn ( z ) < − ε νn } ⊂ G ν ,(ii) vol ( G ν \ G νn, − ε νn ) → n → ∞ .(In (ii) we can use the spherical metric near ∞ .) Set ρ n ( z ) = max ≤ ν ≤ N { ρ νn ( z ) } , ε n = max ≤ ν ≤ N { ε νn } , Ω n = N \ ν =0 G νn . It is easy to verify that Ω ⊂ Ω n , ρ n ∈ SH − (Ω n ) ∩ C (Ω n ) and(iii) Ω n, − ε n := { z ∈ Ω n : ρ n ( z ) < − ε n } ⊂ Ω,(iv) vol (Ω \ Ω n, − ε n ) → n → ∞ .We continue with the proof in a similar way as in the proof in [5]. Choosea family of negative C ∞ subharmonic functions { ρ n,s } on Ω n such that ρ n,s ↓ ρ n uniformly on Ω n +1 as s ↓
0. Put ψ sn = − log( − ρ n,s ). Clearly, wehave(2.4) i∂∂ψ sn ≥ i∂ψ sn ∧ ¯ ∂ψ sn . Now choose a cut-off function χ : R → [0 ,
1] such that χ | ( −∞ , − log 3 / ≡ χ | [0 , ∞ ) ≡
0. Set η sn = χ ( ψ sn + log ε n ) on Ω n . Then we havesupp η sn ⊂ { z ∈ Ω n | ρ n,s ( z ) < − ε n } ⊂ Ω n, − ε n ⊂ Ωand | ∂η sn | i∂∂ψ sn ≤ sup | χ ′ | L POLYNOMIAL APPROXIMATION 7 in view of (2.4). Here | · | i∂∂ψ sn stands for the point-wise norm with respectto the metric i∂∂ψ sn , like in (2.3). For each f ∈ H (Ω , e − ϕ ), we define v sn := f ∂η sn . Clearly, v sn is a well-defined C ∞ , ¯ ∂ − closed (0,1) form on Ω n satisfying Z Ω n +1 | f | | ¯ ∂η sn | i∂∂ψ sn e − ϕ dλ ≤ sup | χ ′ | Z { z ∈ Ω n +1 : − ε n ≤ ρ n,s ( z ) < − ε n } | f | e − ϕ dλ ≤ sup | χ ′ | Z Ω \ Ω n, − εn | f | e − ϕ dλ (2.5)provided s sufficiently small. By Proposition 2.3, there exists a solution u sn to the equation ¯ ∂u sn = v sn on Ω n +1 verifying, by using (2.5), Z Ω n +1 | u sn | e − ϕ dλ ≤ C sup | χ ′ | Z Ω \ Ω n, − εn | f | e − ϕ dλ. Let f sn = f η sn − u sn . Then, f sn ∈ O (Ω n +1 ) and Z Ω | f sn − f | e − ϕ dλ ≤ C Z Ω \ Ω n, − εn | f | e − ϕ dλ → n → ∞ (cid:3) Proof of Theorem 1.3.
Let ϕ be a subharmonic function on Ω. If for each x ∈ Ω, the Lelong number ν ( ϕ )( x ) of ϕ satisfies ν ( ϕ )( x ) < ∈ H (Ω , e − ϕ ). So there exists a constant M > Z Ω e − ϕ dλ < M .Let f ∈ H (Ω , e − ϕ ). According to the proof of Proposition 1.2, for each ε > F ∈ H (Ω n , e − ϕ ) satisfying Z Ω | F ( z ) − f ( z ) | e − ϕ ( z ) dλ < ε . where Ω n is a simply connected domain containing Ω . We apply Runge’stheorem to F and see that for δ = p ε M there exists a polynomial P suchthat | F ( z ) − P ( z ) | < δ, z ∈ Ω . Consequently we have Z Ω | F ( z ) − P ( z ) | e − ϕ dλ < δ · M = ε . Thus we have Z Ω | f ( z ) − P ( z ) | e − ϕ ( z ) dλ ≤ Z Ω | f ( z ) − F ( z ) | e − ϕ ( z ) dλ +2 Z Ω | F ( z ) − P ( z ) | e − ϕ ( z ) dλ< ε S´EVERINE BIARD, JOHN ERIK FORNÆSS, AND JUJIE WU
If there exist finitely many points x , x , · · · , x N with ν ( ϕ )( x j ) ≥ ≤ j ≤ N , then we may choose some polynomial Q so that Q ( x j ) = 0and ϕ = ψ + log | Q | with ν ( ψ ) < x j . Then for each f ∈ H (Ω , e − ϕ ) we have Z Ω (cid:12)(cid:12)(cid:12)(cid:12) fQ (cid:12)(cid:12)(cid:12)(cid:12) e − ψ dλ < ∞ . Since ψ is bounded above, R Ω (cid:12)(cid:12)(cid:12) fQ (cid:12)(cid:12)(cid:12) dλ < ∞ . Hence fQ is holomorphic on Ω.Based on the above discussion, for each ε > P satisfying Z Ω (cid:12)(cid:12)(cid:12)(cid:12) fQ − P (cid:12)(cid:12)(cid:12)(cid:12) e − ψ dλ < ε. That is Z Ω | f − P Q | e − ϕ dλ < ε. (cid:3) Proof of Theorem 1.6
In this section, we denote x = ℜ e ( z ) and y = ℑ m ( z ).Before proving Theorem 1.6 by contradiction, we need a couple of Lem-mas: Lemma 3.1. cos z ∈ H ( C , e − ϕ ) .Proof. Since cos z = e i z + e − i z , we have Z C (cid:12)(cid:12)(cid:12) cos z (cid:12)(cid:12)(cid:12) e − ϕ dλ = Z C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e i z + e − i z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − ϕ dλ = 14 Z C (cid:18)(cid:12)(cid:12)(cid:12) e i z (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) e − i z (cid:12)(cid:12)(cid:12) + e i z e − i z + e − i z e i z (cid:19) e − ϕ dλ ≤ Z C (cid:0) e − y + e y + e ix + e − ix (cid:1) e − ϕ dλ = 14 Z C (cid:0) e − y + e y (cid:1) e −| y |−| z | p dλ + 12 Z C cos x e −| y |−| z | p dλ. (3.1) EIGHTED- L POLYNOMIAL APPROXIMATION 9
Remark that ( e − y + e y ) e −| y | = 1 + e − | y | ≤ Z C (cid:0) e − y + e y (cid:1) e −| y |−| z | p dλ = Z C (cid:16) e − | y | + 1 (cid:17) e −| z | p dλ ≤ Z C e −| z | p dλ ≤ π Z + ∞ e − r p rdr < ∞ , and for the second term of (3.1), Z C | cos x | e −| y |−| z | p dλ ≤ Z C e −| y |−| z | p dλ ≤ π Z + ∞ re − r p dr < ∞ . (cid:3) We will need the following integral representation too:
Lemma 3.2 (see Chapter 7 of [1]) . Let u ( t ) : R → R be a continuousfunction satisfying Z R u ( t )1 + | t | dt < ∞ . Then U ( x + iy ) = 1 π Z + ∞−∞ u ( t ) y ( t − x ) + y dt is a harmonic extension of u to the upper half plane. In particular, if u ( t ) = | t | p , 0 < p <
1, then(3.2) U ( x + iy ) = 1 π Z + ∞−∞ y | t | p ( x − t ) + y dt, y > , is a harmonic extension of | t | p to the upper half plane. Proposition 3.3.
Let U be as in (3.2) . Then there exists constant C p > so that on the upper half plane y > , | z | p < U ( x + iy ) < C p | z | p . Proof.
The right hand side inequality is direct: U ( x + iy ) = 1 π Z ∞−∞ y | t | p ( t − x ) + y dt = 1 π Z ∞−∞ y | s + x | p s + y ds = 1 π Z ∞−∞ y | yτ + x | p ( yτ ) + y ydτ = 1 π Z ∞−∞ | yτ + x | p τ + 1 dτ (3.3) ≤ π Z ∞−∞ y p | τ | p τ + 1 dτ + 1 π Z ∞−∞ | x | p τ + 1 dτ ≤ π | y | p · π sin( p π )sin( pπ ) + | x | p = 1cos( p π ) | y | p + | x | p ≤ p π ) | z | p := C p | z | p . (3.4)To prove the left-hand side inequality, we prove the following inequalities:(i) U ( x + iy ) ≥ | x | p and (ii) U ( x + iy ) ≥ | y | p . We prove (i) as follows: from (3.3) τ inherits the sign of x , then we onlyhave to study x ≥
0. We get U ( x + iy ) = 1 π Z ∞−∞ | yτ + x | p τ + 1 dτ ≥ π Z ∞ | yτ + x | p τ + 1 dτ ≥ π Z ∞ | x | p τ + 1 dτ ≥ | x | p . For (ii), we also start from (3.3) and by a similar argument, we mayassume that x ≥
0. Then,
EIGHTED- L POLYNOMIAL APPROXIMATION 11 U ( x + iy ) = 1 π Z ∞−∞ | yτ + x | p τ + 1 dτ ≥ π Z ∞ | yτ + x | p τ + 1 dτ ≥ π Z ∞ | y | p τ p τ + 1 dτ = | y | p π Z ∞ τ p τ + 1 dτ = 12 | y | p cos( p π ) ≥ | y | p . Finally, combining (i) and (ii) and by concavity for 0 < p <
1, we get U ( x + iy ) ≥
14 ( | x | p + | y | p ) ≥
14 ( | x | + | y | ) p ≥ | z | p . (cid:3) Proof of Theorem 1.6.
First, remark that the holomorphic polynomials arein L ( C , e − ϕ ) thanks to the exponential rate e −| z | p .We prove Theorem 1.6 by contradiction. Assume now that holomorphicpolynomials are dense in H ( C , e − ϕ ). Since cos z ∈ H ( C , e − ϕ ) by Lemma3.1, for all ε >
0, there exist a sequence of polynomials ( P n ) and N ∈ N such that for all n ≥ N ,(3.5) Z C (cid:12)(cid:12)(cid:12) P n ( z ) − cos z (cid:12)(cid:12)(cid:12) e − ϕ dλ < ε. Note that for n sufficiently large, k P n ( z ) k C ,ϕ = (cid:13)(cid:13)(cid:13) P n ( z ) − cos z z (cid:13)(cid:13)(cid:13) C ,ϕ ≤ (cid:13)(cid:13)(cid:13) P n ( z ) − cos z (cid:13)(cid:13)(cid:13) C ,ϕ + (cid:13)(cid:13)(cid:13) cos z (cid:13)(cid:13)(cid:13) C ,ϕ ≤ (cid:13)(cid:13)(cid:13) cos z (cid:13)(cid:13)(cid:13) C ,ϕ . (3.6)We deduce from (3.6) that there exists M > k P n ( z ) k C ,ϕ ≤ M for all n .Since P n is analytic, we have P n ( z ) = 1 π Z | ζ |≤ P n ( z + ζ ) dλ ζ . So | P n ( z ) | ≤ π Z | ζ |≤ | P n ( z + ζ ) | e ϕ ( z + ζ )2 e − ϕ ( z + ζ )2 dλ ζ ≤ π sup | ζ |≤ e ϕ ( z + ζ )2 Z | ζ |≤ | P n ( z + ζ ) | e − ϕ ( z + ζ )2 dλ ζ . (3.7)By Cauchy-Schwarz’s inequality, we get from (3.7) and for all n , | P n ( z ) | ≤ π sup | ζ |≤ e ϕ ( z + ζ )2 Z | ζ |≤ | P n ( z + ζ ) | e − ϕ ( z + ζ ) dλ ζ ! Z | ζ |≤ dλ ζ ! ≤ √ π k P n ( z ) k C ,ϕ sup | ζ |≤ e ϕ ( z + ζ )2 ≤ M √ π sup | ζ |≤ e ϕ ( z + ζ )2 . (3.8)Note that for 0 < p < | ζ |≤ ϕ ( z + ζ ) = sup | ζ |≤ ( |ℑ m ( z + ζ ) | + | z + ζ | p ) ≤ |ℑ m ( z ) | + 1 + ( | z | + 1) p ≤ |ℑ m ( z ) | + 1 + 1 + | z | p = | y | + 2 + | z | p . Plugging it in (3.8), we get for all n | P n ( x + iy ) | ≤ M √ π e (2+ | y | + | x + iy | p ) . Then it follows that on the real axis ( y = 0), | P n ( x ) | ≤ M √ π e (2+ | x | p ) ≤ M e √ π e | x | p , so(3.9) log | P n ( x ) | ≤ log M + 1 − log √ π + 12 | x | p . By Lemma 3.2 applied to ˜ u ( x ) = log M + 1 − log √ π + | x | p , e U ( x + iy ) = 1 π Z + ∞−∞ y (log M + 1 − log √ π + | t | p )( x − t ) + y dt, y > , is a harmonic extension of ˜ u to the upper half plane. And from Proposition3.3, we obtainlog M + 1 − log √ π + 18 | z | p ≤ e U ( z ) ≤ log M + 1 − log √ π + C p | z | p . From the fact that for any positive constant C , C log | z | is much smaller than | z | p for large enough | z | , we deduce that for large enough | z | , log | P n | ( z ) ≤ e U ( z ). Because e U is harmonic and log | P n | is subharmonic, we obtain EIGHTED- L POLYNOMIAL APPROXIMATION 13 (3.10) log | P n ( x + iy ) | ≤ e U ( x + iy ) ≤ C + C p | z | p on { z ∈ C | y > } , where C = log M + 1 − log √ π. From (3.10), we getlog | P n ( z ) | ≤ C + C p | z | p , so | P n ( z ) | ≤ e C e C p | z | p . But remark that lim | z |→ + ∞ e | z | / − e C e C p | z | p = + ∞ so there exists a positive constant Y > | z | > Y ,(3.11) e | z | / − > e C e C p | z | p . Note also that if x < y then 4 y = y + 3 y > x + y = | z | so if inaddition y >
0, we get y > | z | . Hence, on { z ∈ C | x < y , y > } , wehave (cid:12)(cid:12)(cid:12) cos z (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) e ix/ − y/ + 12 e ix/ y/ (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) e ix/ y/ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) e ix/ − y/ (cid:12)(cid:12)(cid:12)(cid:12) = 12 ( e y/ − e − y/ ) ≥
12 ( e | z | / − . (3.12)Combining (3.12) with (3.11), we obtain on W := { z ∈ C | x < y , y > , | z | > Y } = { ( r, θ ) | r > Y, π < θ < π } ,(3.13) (cid:12)(cid:12)(cid:12) cos z − P n ( z ) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) cos z (cid:12)(cid:12)(cid:12) − | P n ( z ) | ≥ e C e C p | z | p ≥ e C e C p Y p . Hence, from (3.13), we have on W , Z C (cid:12)(cid:12)(cid:12) cos z − P n ( z ) (cid:12)(cid:12)(cid:12) e − φ ( z ) dλ ≥ Z W (cid:12)(cid:12)(cid:12) cos z − P n ( z ) (cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ ≥ e C e C p Y p Z W e − ϕ ( z ) dλ = e C +2 C p Y p Z π/ π/ dθ Z + ∞ Y re −| r sin θ |−| r | p dr ≥ π e C +2 C p Y p Z + ∞ Y re −| r |−| r | p dr ≥ π e C +2 C p Y p · e − Y = π e C +2 C p Y p − Y . This is a contradiction with the formula (3 .
5) when n is large enough. (cid:3) Proof of Theorem 1.7
Recall the following classical fact:
Lemma 4.1.
Let Ω be an open set in C , h a holomorphic on Ω and ϕ asubharmonic function on an open set V ⊃ h (Ω) . Then ϕ ◦ h ( z ) = ϕ ( h ( z )) is also subharmonic on Ω .Proof of Theorem 1.7. Since √ z ∈ H (Ω , e − ϕ ), the condition is obviouslynecessary.We then need to prove the sufficiency. The mapping w = √ z transformsΩ into a Jordan domain Ω ′ and for each f ∈ H (Ω , e − ϕ ), Z Ω | f ( z ) | e − ϕ ( z ) dλ z = 4 Z Ω ′ | f ( w ) w | e − ϕ ( w ) dλ w < ∞ . Put h ( w ) = w , we know that wf ( w ) ∈ H (Ω ′ , e − ϕ ◦ h ( w ) ), where ϕ ◦ h ( w ) = ϕ ( h ( w )) is subharmonic on the closure of the bounded Jordan domain Ω ′ .In particular, ϕ ◦ h ( w ) = ϕ ( h ( w )) is subharmonic in a neighborhood of Ω ′ .According to Corollary 1.4, for each ε > P ( w )such that 4 Z Ω ′ | wf ( w ) − P ( w ) | e − ϕ ◦ h ( w ) dλ w < ε. Therefore (4.1) Z Ω |√ zf ( z ) − P ( √ z ) | e − ϕ ( z ) · | z | dλ z = Z Ω (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − P ( √ z ) √ z (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ z < ε. Separate the polynomial P into even and odd parts: P ( √ z ) = P ( z ) + √ zP ( z ) , then the formula (4 .
1) implies that Z Ω (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − P ( z ) − P ( z ) √ z (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ z < ε. In order to find some polynomial Q ( z ) such that Z Ω | f ( z ) − Q ( z ) | e − ϕ ( z ) dλ z < ε. It is sufficient to know that Z Ω (cid:12)(cid:12)(cid:12)(cid:12) √ z − R ( z ) (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ < ε for some polynomial R ( z ) but this holds by assumption. (cid:3) EIGHTED- L POLYNOMIAL APPROXIMATION 15
We give now an application of Theorem 1.7: Example 1.9. First we needthe following lemmas.
Lemma 4.2 (Riesz Decomposition Theorem, see for example Theorem 3.7.9in [14]) . Let u be a subharmonic function on a domain D in C , with u
6≡ −∞ .Then, given a relatively compact open subset U of D , we can decompose u as u = Z ζ ∈ U log | z − ζ | dµ ( ζ ) + h on U , where µ = π ∆ u | U and h is harmonic on U . Let
D, R be positive numbers so that D = πR . Fix α, < α < . Lemma 4.3.
Let z ∈ C and A be a measurable set in C with bounded area D. Then R A | z − z | α dλ ≤ R − α − α . Moreover this estimate is sharp.Proof. It suffices to consider the case when z = 0 . The largest integraloccurs when the area is a disc centered at 0. The radius then is given by πR = D. We get R | z |≤ R | z | α dλ = R R t − α dt = R − α − α . (cid:3) Using the convexity of the exponential function we apply Lemma 2.1 in[19] to obtain:
Lemma 4.4.
Suppose < α i , P i α i = α < . Then Z A Π i | z − z i | αi dλ ≤ R − α − α as in the above Lemma. Corollary 4.5.
Let µ be any nonnegative measure with total mass α, <α < . A be a measurable set in C with bounded area D. Then if ϕ ( z ) = R log | z − ζ | dµ ( ζ ) , we have that R A e − ϕ dλ ≤ R − α − α . Proof.
Define ψ n ( z, ζ ) = max { log | z − ζ | , − n } and ϕ n ( z ) = Z ψ n ( z, ζ ) dµ ( ζ ) . It suffices to show that Z A e − ϕ n ( z ) dλ ( z ) ≤ R − α − α + 1 n ∀ n ∈ N ∗ . We fix n . Let ε > . We apply Lemma 2.4 [19]: there exists a finite positivemeasure β := P Ni =1 α i δ z i with P Ni =1 α i = α such that Z ψ n ( z, ζ ) dβ ( ζ ) ≤ Z ψ n ( z, ζ ) dµ ( ζ ) + ε. By Lemma 4.4 we know that Z A e − P i α i log | z − z i | dλ ≤ R − α − α . So Z A e − R log | z − ζ | dβ ( ζ ) dλ ( z ) ≤ R − α − α . Hence Z A e − R ψ n ( z,ζ ) dβ ( ζ ) dλ ≤ R − α − α . Finally, by choosing ε small enoughwe get Z A e − ϕ n dλ ≤ R − α − α + 1 n . (cid:3) Define for n ∈ N ∗ S n = (cid:26) ϕ is subharmonic on C : the mass of µ (cid:18) | z | < n (cid:19) < − n ,ϕ ( z ) = ψ n + h n , ψ n = Z | ζ | < n log | z − ζ | dµ and | h n | < n on | z | < ) . Lemma 4.6.
Suppose ϕ is subharmonic on C satisfying condition ( A ) , thenfor all large enough n , ϕ ∈ S n .Proof. Pick m so that the mass of µ = π ∆ ϕ is strictly less than 2 on thedisc ∆ (cid:0) m (cid:1) . Increasing m , we may assume µ (cid:18) | z | < m (cid:19) < − m . This remains true for all large m , let ϕ be subharmonic on C satisfyingcondition ( A ), write ϕ = ψ m + h m and set K = sup | z |≤ | h m ( z ) | . For n > m , we may also write ϕ = ψ n + h n . We have that | ψ n − ψ m | ≤ n on | z | < . Hence | h n | ≤ K + 2 log n on | z | < . We may choose n so large that K + 2 log n ≤ n. (cid:3) EIGHTED- L POLYNOMIAL APPROXIMATION 17
Now we begin to construct Example 1.9. By Lemma 4.6, we may assume ϕ ∈ S . Then, ϕ = ϕ + h where ϕ = R | ζ | < log | z − ζ | dµ and µ ( | z | < < D be the domain composed of points satisfying the inequalities | z | < , (cid:12)(cid:12)(cid:12)(cid:12) z − (cid:12)(cid:12)(cid:12)(cid:12) > , π < arg z < π − π . Then D is a bounded simply connected domain so that C \ D is connected.By Corollary 4.5 we know that R D e − ϕ dλ is uniformly bounded. Since | h | <
1, we also get that R D e − ϕ dλ is uniformly bounded for all ϕ ∈ S .By the Runge theorem there exists a polynomial P n ( z ) with degree n suchthat Z D (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ < for all ϕ ∈ S . Choose 0 < α < is sufficiently small so that for the abovepolynomial P n ( z ) satisfying Z ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ ≤ sup ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) Z ∆ e − ϕ − h dλ ≤ sup ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) R − α − α · e< , where ∆ , of area πR being the points satisfying the inequalities(4.2) | z | < , | z − α | > − α , | arg z | ≤ π . Let f D be the domain composed of the points verifying the inequalities | z | < , | z − α | > − α , π < arg z < π − π . Set D = D ∪ f D , this is a Jordan domain. We now consider any ϕ ∈ S .Then there exists a polynomial P n ( z ) with degree n such that Z D (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ < . Choose 0 < α < α so that for the above polynomial P n ( z ) satisfying Z ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ ≤ sup ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) Z ∆ e − ϕ dλ ≤ sup ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) R − α − α · e < , where ∆ , of area πR being the points satisfying the inequalities | z | < , | z − α | > − α , | arg z | ≤ π . Let f D := {| z | < , | z − α | > − α , π < arg z < π − π } . Set D = D ∪ f D . Proceed as above, we can find a sequence of bounded simplyconnected domains D , D , · · · , D n , · · · . Let D be the limit domain of D n .Then the domain D is bounded by the circle | z | = 1 and a simple Jordancurve Γ tangent to | z | = 1 in z = 1. Thus D is a bounded very thin moon-shaped domain and we know that the limit domain D is contained in thelimit domain D k ∪ ∆ k , from which it follows that for all ϕ ∈ S k , Z D (cid:12)(cid:12)(cid:12)(cid:12) √ z − P n k ( z ) (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ ( z ) dλ < k +1 + 12 k +1 = 12 k . Then polynomials are dense in H ( D, e − ϕ ) for any ϕ satisfying condition( A ). 5. Example 1.10
First, we prove the following Lemma, already known for Bergman spaces( ϕ ≡ Lemma 5.1.
Let Ω ⊂⊂ C and ϕ a subharmonic function on Ω . Assumethat ∈ H (Ω , e − ϕ ) and p ∈ Ω . For each n , let f n ∈ H (Ω , e − ϕ ) be afunction f n = a n ( z − p ) n + O (( z − p )) n +1 , a n > maximal with k f n k Ω ,ϕ = 1 .Then { f n } ∞ n =0 is an orthonormal basis for H (Ω , e − ϕ ) .Proof. We show that f n ⊥ { g ∈ H (Ω , e − ϕ ); g = O (( z − p ) n +1 ) } .By contradiction, suppose that there exists such g ∈ H (Ω , e − ϕ ) that is notorthogonal to f n . Then for complex-valued ε small enough, h f n + εg, f n + εg i = 1 − ℜ e ( ε h f n , g i ) + | ε | h g, g i , (5.1) = 1 − | ε ||h f n , g i| + | εg | , (5.2) = t < . (5.3)Hence, f f n = f n + εg √ t = a n √ t ( z − p ) n + O t (( z − p )) n +1 ) contradicts the maxi-mality of the coefficients.Suppose now that g ⊥ { f n } such that g = b n ( z − p ) n + O (( z − p )) n +1 with b n = a n . We get g − f n = O (( z − p )) n +1 that is orthogonal to f n i.e h g − f n , f n i = h g, f n i − h f n , f n i = 0 . but h g, f n i = 0 = 1 = h f n , f n i . (cid:3) EIGHTED- L POLYNOMIAL APPROXIMATION 19
Let M be a moon-shaped domain with multiple boundary point Q , and p be an interior point of M . For each n ≥
2, denote M n = M \ B ( Q, /n )where B ( Q, /n ) is the disc of center Q and radius 1 /n . Let ϕ be a non-negative subharmonic function on C and f ∈ H ( M n , e − ϕ ). By Corollary1.4, for every ε >
0, there exists a polynomial P such that k f − P k M n ,ϕ < ε . Lemma 5.2.
Under the previous assumptions, there exists a subharmonicfunction e ϕ on C such that e ϕ = ϕ on M n and k P k B ( Q, /n ) ∩ M, e ϕ < ε. Proof.
Because M n is polynomially convex, there exists a subharmonic func-tion γ on C such that γ = 0 on M n and γ > γ , note that forevery point q outside M n there exists a polynomial P q such that | P q ( q ) | > | P q | < M n . We choose a convex function χ ( x ) which vanishes when x ≤ x > . Then χ ◦ | P q | is subharmonic andvanishes on M n while it is strictly positive in a neighborhood of q. Then onecan define γ = P m ε m χ ◦| P q m | for suitable choices. By choosing e ϕ = ϕ + Lγ for L large enough, we get k P k B ( Q, /n ) ∩ M, e ϕ ≤ k P k B ( Q, /n ) ∩ M,Lγ . By taking the limit as L tends to + ∞ , the second term of the previousestimate tends to 0. (cid:3) By Lemma 5.2, we can construct inductively an increasing sequence ofnon-negative subharmonic functions ϕ n on M and by Lemma 5.1, we canfind f n , . . . , f nn ∈ H ( M n , e − ϕ n ) such that f nj = a nj ( z − p ) j + O (( z − p ) j +1 )with a nj > k f nj k M n ,ϕ n = 1. Then, there exist polynomials P nj and ϕ n +1 large enough by Lemma 5.2 such that(5.4) k f nj − P nj k M n ,ϕ n < n and(5.5) k P nj k B ( Q, /n ) ∩ M,ϕ n +1 < n Moreover, by Remark 1.5, we can choose P nj ( z ) = a nj ( z − p ) j + O (( z − p ) j +1 ) . Let ϕ := lim n → + ∞ ϕ n . Hence k f nj k M n ,ϕ = 1 and (5.4) and (5.5) hold withrespect to ϕ . Lemma 5.3.
Under the previous assumptions, the polynomials P nj , j =0 , . . . , n built inductively as previously verify the following property k P nj k M,ϕ ≤ n . Proof.
Combining the properties of f nj seen in Lemma 5.1 and Lemma 5.2 k P nj k M,ϕ ≤ k P nj k M n ,ϕ + k P nj k B ( Q, /n ) ∩ M,ϕ , ≤ k P nj k M n ,ϕ n + k P nj k B ( Q, /n ) ∩ M,ϕ n +1 , ≤ k P nj − f nj k M n ,ϕ n + k f nj k M n ,ϕ n + 1 n , ≤ n . (cid:3) We are now able to give the details of the construction of Example 1.10.By Lemma 5.3, P nj converges weakly to P j in H ( M, e − ϕ ) such that k P j k M,ϕ ≤ P j = ( lim n → + ∞ a nj )( z − p ) j + O (( z − p ) j +1 ) . In particular, the limit lim n → + ∞ a nj gives optimal coefficients for P j . Itfollows that k P j k M,ϕ = 1 and P j is an orthonormal basis for H ( M, e − ϕ ) byLemma 5.1.Let f ∈ H ( M, e − ϕ ) and ε >
0. We can express f as f = ∞ X j =0 A j P j .For N large enough, k N X j =0 A j P j − f k M,ϕ < ε . For n large enough, we get finally k N X j =0 A j P nj − f k M,ϕ < ε. Proof of Theorem 1.11
Proof of Theorem 1.11.
The proof relies on [10], Chapter I, section 3. Wereason by contradiction.Let Γ be a circle as in Figure 1. Then there exists a positive constant C such that d ∂ Ω ( z ) ≥ C | z − | for all z ∈ Γ . Choose a function f ∈ H (Ω , e − ϕ )which does not extend holomorphically to the inside of the small circle. (Onecan first choose any nontrivial holomorphic function in the H (Ω , e − ϕ ). Picka suitable point p inside the inner circle and observe that fz − p is still in the H space.)Assume that there exists a sequence of polynomials P n so that k f − P n k Ω ,ϕ tends to 0 as n tends to + ∞ . We obtain a contradiction by showing that f extends analytically in the interior of Γ so in the hole (see Figure 1), which EIGHTED- L POLYNOMIAL APPROXIMATION 21 xy z = 1 G • Q Γ Figure 1.
A general moon-shaped domain including thecurve Γ passing through the multiple boundary point Q .is impossible.For each w ∈ Γ \ { Q } , by the mean value property for subharmonic func-tions, we get for any n, m ∈ N , | P n ( w ) − P m ( w ) | ≤ πd ∂ Ω ( w ) Z B ( w,d ∂ Ω ( w )) | P n ( z ) − P m ( z ) | dλ, ≤ e Cπd ∂ Ω ( w ) Z B ( w,d ∂ Ω ( w )) | P n ( z ) − P m ( z ) | e − ϕ ( z ) dλ, (6.1) ≤ e Cπd ∂ Ω ( w ) k P n ( z ) − P m ( z ) k ,ϕ , (6.2)where we have used in (6.1) the fact that ϕ is bounded above in Ω with e C a positive constant. Because d ∂ Ω ( z ) ≥ C | z − | by assumption, we obtainfrom (6.2) and for each w ∈ Γ \ { Q } , | ( w − ( P n ( w ) − P m ( w )) | ≤ d ∂ Ω ( w ) C p e C √ πd ∂ Ω ( w ) k P n ( z ) − P m ( z ) k Ω ,ϕ , ≤ C ′ k P n ( z ) − P m ( z ) k Ω ,ϕ (6.3)where C ′ >
0. The inequality (6.3) holds for each point w ∈ Γ so in theinterior of Γ. Hence, the sequence (( z − P n ) converges uniformly in theinterior of Γ to ( z − f . (cid:3) Acknowledgements
The first author was supported by Rannis-grant152572-051. The second author and the third author were supported in partby the Norwegian Research Council grant number 240569, the third author was also supported by NSFC grant 11601120. The authors give thanks toDr. Zhonghua Wang for his valuable comments in the proof of theorem 1.6and the referees for their valuable suggestions.
References [1] S. Axler, P. Bourdon and R. Wade, Harmonic function theory. Springer Science &Business Media, 2013.[2] Z. B locki, Cauchy-Riemann meet Monge-Amp`ere.
Bull. Math. Sci (2014), 433–480.[3] J. Brennan, Approximation in the mean of polynomials on non-Caratheodory domains, Ark. Mat , no.1 (1977), 117–168.[4] T. Carleman, ber die Approximation analytischer Funktionen durch lineare Aggregatevon vorgegebenen Potenzen[M]. Almqvist and Wiksell, 1923.[5] B.Y. Chen and J.H. Zhang, On Bergman completeness and Bergman Stability, Math.Ann., , 2000, 517-526.[6] J.P. Demailly, Estimations L pour l’op´erateur ¯ ∂ d’un fibr´e vectoriel holomorphe semi-positif au dessus d’une vari´et´e k¨ahl´erienne compl`ete, Annales Sc. de l’E.N.S , , no.3(1982), 457–511.[7] H. Donnelly and C. Fefferman, L -cohomology and index theorem for the Bergmanmetric, Ann. of Math. (1983), 593-618.[8] O. Farrell, On approximation to an analytic function by polynomials, Bull. Amer.Math. Soc. , no. 12 (1934), 908–914.[9] J.E. Fornaess, F. Forstneric and E.F. Wold, Holomorphic apporximation: the legacyof Weierstrass, Runge, Oka-Weil, and Mergelyan. preprint .[10] D. Gaier, Lectures on Complex Approximation, Boston. Basel. Stuttgart. 1980.[11] L.I. Hedberg, Weighted mean square approximation in plane regions and generators ofan algebra of analytic functions, Arkiv f¨or Mathematik band, , no. 36, (1965), 541–552.[12] M. Keldych, ”Sur l’approximation en moyenne quadratique des fonctions analy-tiques.” Rec. Math. [Mat. Sb.] , no. 2 (1939): 391-401.[13] A.I. Markushevitch, Conformal mapping of regions with variable boundaries for theapproximation of analytic functions by polynomials (russian), PhD. Thesis, (1934).[14] T. Ransford, Potential theory in the complex plane, Vol. 28, Cambridge Universitypress, 1995.[15] N. Sibony, Approximation polynomiale pond´er´ee dans un domaine d’holomorphie de C n , Annales de l’institut Fourier, tome 26, 1976, 71-79.[16] B.A. Taylor, On weighted polynomial approximation of entire functions,
Pacific. J.Math., , 1971, 523-539.[17] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen Co., LTD. Tokyo,1959.[18] D. Wohlgelernter, Weighted L approximation of entire functions, Transactions OfThe American Mathematical Society., , 1975, 211-219.[19] J. Wu and J.E. Fornaess, Weighted approximation in C , arXiv:1712.01086v3. ∗ Corresponding author, Jujie Wu ,
E-mail address: [email protected], Schoolof Mathematics and statistics, Henan University, Jinming Campus of HenanUniversity, Jinming District, City of Kaifeng, Henan Province. P. R. China,475001, ,Department of Mathematical Sciences, NTNU, Sentralbygg 2, Alfred Getzvei 1, 7034 Trondheim, NorwayS´everine Biard,
E-mail address: [email protected], Science Institute, University ofIceland, Dunhagi 3, IS-107 Reykjavik, Iceland
EIGHTED- L POLYNOMIAL APPROXIMATION 23
John Erik Fornæss,