# On the Asymptotic Behavior ofp-Faber Polynomials for Domains with Piecewise  Analytic Boundary

Authors: T. Tunc; M. Kucukaslan
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

LetG⊂ℂ be a domain bounded by a piecewise analytic Jordan's curve L, and let Fn,p denote the p-Faber polynomials associated with G. We
derive estimates of the form Fn,p(z)=O(1/nη), (n→∞) for z∈G, where
η depends on geometric properties of L and the parameter p. Also, we show that O cannot be replaced by o in the relation given above.


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## Body

## 1. Introduction and Main Results

LetG⊂ℂ be a domain bounded by a Jordan curve L, and let Φ be the conformal mapping of the domain Ω:=ℂ∞∖G¯, where ℂ∞:=ℂ∪{∞}, onto Δ:={w:|w|>1} with the usual normalization at infinity:
(1.1)w=Φ(z)=αz+a0+a1z+⋯,α>0,z∈Ω.
Let Ψ:=Φ-1:Δ→Ω denote the inverse conformal map. Then,
(1.2)z=Ψ(w)=βw+b0+b1w+⋯,|w|>1,
where β=1/α gives the capacity cap(L) of L.Let0<p<∞. The p-Faber polynomials Fn,p associated the domain G are defined as the polynomial part of the expansion of
(1.3)Φn(z)(Φ′(z))1/p,n=0,1,2,…,
in a neighborhood of the infinity. Therefore, from (1.1) we have
(1.4)Φn(z)(Φ′(z))1/p:=Fn,p(z)+En,p(z),z∈Ω,
where
(1.5)Fn,p(z):=αn+1/pzn+⋯
is the p-Faber polynomial of degree n and
(1.6)En,p(z)=c0z+c1z2+c2z3+⋯
is the singular part of Φn(z)(Φ′(z))1/p. Indeed, the number c0 vanishes, since z((Φ′(z))1/p-α1/p) tends to zero when z tends to infinity. Also, the p-Faber polynomials associated with G can be defined by the generating function
(1.7)g(w):=(Ψ′(w))1-1/pΨ(w)-z=∑n=0∞Fn,p(z)wn+1,z∈G,|w|>1,
see [1, 2].Ifp tends to infinity, the p-Faber polynomials coincide with the usual Faber polynomials Fn. If p=1, then it follows immediately from (1.4) that
(1.8)Fn,1(z)=F′n+1(z)n+1,
where Fn is the Faber polynomials of G of degree n.The aim of this paper is to find asymptotic behaviors ofp-Faber polynomials Fn,p in G and En,p in Ω.IfL is rectifiable, then Φ′ belongs to the Smirnov class E1(Ω{∞}). In addition, both Ψ′ and Φ′ have nontangential limits almost everywhere on 𝕋 boundary of the unit disc 𝔻:={w∈ℂ:|w|<1} and L; respectively, and they are integrable with respect to the arc length measure; see [3] or [4].Assume now that the boundaryL is rectifiable. For such a boundary, Cauchy’s integral formula yields the following representations for the p-Faber polynomials and their associated singular parts:
(1.9)Fn,p(z)=12πi∫LΦn(ζ)(Φ′(ζ))1/pζ-zdζ,z∈G,En,p(z)=12πi∫LΦn(ζ)(Φ′(ζ))1/pζ-zdζ,z∈G.
Our assumption for this work is that L is a piecewise analytic Jordan’s curve. That is, L=⋃j=1mLj, m∈ℕ, where Lj are a finite number of analytic Jordan’s arcs meeting at corners zj where L has an exterior angle λjπ with 0<λj<2. Cusps are, therefore, excluded, but λj=1 may occur—a “smooth corner”.Letλ:=min⁡j=1,…,m{λj}. Gaier [5] proved that
(1.10)Fn(z)=O(1nλ)(n→∞),
for z∈G, uniformly on compact subsets of G. Stylianopoulos [6] indicated that one can obtain
(1.11)Fn,1(z)≤c(L,F)nλ+1,z∈F
by using (1.8) and (1.13), where F is an arbitrary compact subset of G, and c(L,F) is a constant depending on L and F. Estimates of the form (1.13) have been given by Suetin [2, 7], for boundary L has high degree of smoothness.For simplicity of notation, we write(1.12)η:=η(λ,p):=(1-λ)(1-1p).
It is clear that, if 0<λ<2 and 1/2≤p≤∞, then |η|<1.The main results of the paper are the following.Theorem 1.1.
Let1/2≤p≤∞, and let G⊂ℂ be bounded by a piecewise analytic Jordan curve L. If L has exterior angles λjπ with 0<λj<2 and if λ:=min⁡jλj, then
(1.13)Fn,p(z)=O(1n1-η)(n→∞),
for z∈G, uniformly on compact subsets of G.Remark 1.2.
From the proof of Theorem1.1, it follows that an estimate of the form (1.13) holds also all derivatives of the p-Faber polynomials.Remark 1.3.
IfL has no smooth corners; that is, all λj≠1, O cannot be replaced by o in (1.13).Remark 1.4.
Ifp tends to infinity, we get the result (1.13) of Gaier.
Similar arguments in the proof of Theorem1.1 provide the following theorem.Theorem 1.5.
Under the assumptions of Theorem1.1, we have, for anyn∈ℕ,
(1.14)|En,p(z)|≤c(L)dist⁡(z,L)1n1-η,z∈Ω,
where c(L) depends on L only.Remark 1.6.
The equalities (1.13) and (1.14) are also valid for 0<p<1/2 under condition max⁡j=1,…,mλj<1/(1-p).Remark 1.7.
Ifp=1, we get the result of Stylianopoulos [6]:
(1.15)|En,1(z)|≤c(L)dist⁡(z,L)1n,z∈Ω.
We will prove Theorems 1.1, 1.5 and Remark 1.3 in section 3.


## 2. Auxiliary Results

We are going to follow the analog used by Gaier in [5]. Since the components Lj of L are assumed to be analytic Jordan’s arcs, the mapping Φ from Ω to Δ can be continued analytically beyond each point of L that is not corner; this follows from Schwarz's reflection principle. Since L has no cusps, we can refine this statement. For each j=1,2,…,m, there are (short) line segments sj and sj′ in the interior of L meeting Lj at zj and zj+1, respectively, at a (small) positive angle and a rectifiable arc γj lying entirely in the interior of L and connecting the other two endpoints of sj and sj′ such that Φ has analytic continuation beyond Lj into the strip-like domain gj bounded by Lj,sj,sj′, and γj (see [5] or [6] for figure). Therefore, the arc Φ(γj) is in 𝔻 while Φ(sj) and Φ(sj′) are arcs in 𝔻 except for their endpoints Φ(zj) and Φ(zj+1). More can be said: since reflection preserves angles and since the exterior angle at zj of opening λjπ is mapped onto a angle of opening π at Φ(zj), we see that Φ(sj) lies in a Stolz angle in 𝔻 with corner at Φ(zj):
(2.1)|Φ(z)-Φ(zj)|1-|Φ(z)|≤c1forz∈sj,j=1,2,…,m,
and similarly
(2.2)|Φ(z)-Φ(zj+1)|1-|Φ(z)|≤c1′forz∈sj′,j=1,2,…,m.
Furthermore, we have the following inequalities for z on sj(j=1,2,…,m):
(2.3)|Φ(z)-Φ(zj)|≥const⋅|z-zj|1/λj,(2.4)|Φ′(z)|≤const⋅|z-zj|1/λj-1,(2.5)|Φ(z)|≤1-const⋅|z-zj|1/λj,(2.6)dist⁡(z,L)≥const⋅|z-zj|.
The similar inequalities of (2.3)–(2.6) can be written by replacing sj, zj, and λj with sj′, zj+1, and λj+1, respectively. The inequalities (2.3) and (2.4) emerge from Lehman's asymptotic expansions of conformal mappings near an analytic corner [8]. The inequality (2.5) follows from (2.3), because reflection preserves angles. Finally, (2.6) is a simple fact of conformal mapping geometry.The last auxiliary facts are on the behavior ofg near 𝕋 obtained by using Lehman's asymptotic expansions of conformal mappings near an analytic corner [8]. Assume that L has a corner at ζ with exterior angle λπ, 0<λ<2 and that ζ=Ψ(1). Let further z∈G be fixed. Similar arguments in [5] apply to the g(w)=(Ψ′(w))1-1/p/(Ψ(w)-z) one can obtain that if 0<λ<1, then
(2.7)g(w)≅C(w-1)η,asw→1+,withC≠0.
If 1<λ<2, then
(2.8)g′(w)≅C(w-1)η+1,asw→1+,withC≠0.


## 3. Proofs of Theorems

From now on,c denote positive constants, not necessarily the same at the different places and not depending on n degree of the p-Faber polynomial and the parameter p. We recite that
(3.1)η:=(1-λ)(1-1p).Proof of Theorem1.1.
Here we use the integral representation (1.9) of the p- Faber polynomials Fn,p in G. Given a compact subset F of G, we deform the path of integration by replacing each component Lj of L by some Lj′:=sj∪Lj∪s′j (j=1,2,…,m), as mentioned before, with proper orientation and such that F⊂L′:=⋃j=1mLj′. This gives
(3.2)|Fn,p(z)|≤12π∑j=1m∫Lj′|Φ(ζ)|n|Φ′(ζ)|1/p|ζ-z||dζ|≤C(L,F)∑j=1m∫Lj′|Φ(ζ)|n|Φ′(ζ)|1/p|dζ|=C(L,F)∑j=1m(∫sj+∫sj′+∫Lj)|Φ(ζ)|n|Φ′(ζ)|1/p|dζ|.
The contributions of the arcs Lj to these integrals are obviously O(ρn) for some ρ<1, so that typical integrals
(3.3)Ij=∫sj|Φ(ζ)|n|Φ′(ζ)|1/p|dζ|
extended over the line segments sj (and sj′) remain to be estimated. If s is arc length of sj measured from zj, by using (2.4) and (2.5),
(3.4)|Φ(ζ)|≤1-cs1/λj<exp⁡(-cs1/λj),|Φ′(ζ)|≤cs1/λj-1,
and, therefore,
(3.5)Ij<c∫0∞exp⁡(-cns1/λj)s(1/λj-1)/pds=cn(1-λj)/p+λj∫0∞e-tt(1/p-1)(1-λj)ds.
If 1/2≤p≤∞, then the integral in the last line is convergent; that is,
(3.6)Ij<cn(1-λj)/p+λj.
From this we obtain the Theorem 1.1.
In the case0<p<1/2, similar arguments show that, under the condition λj<1/(1-p), the inequality (3.6) is valid. This proves Remark 1.4.Proof of Remark1.3.
We now show thatO in Theorem 1.1 cannot be replaced by o. Let L be any piecewise analytic Jordan’s curve that has no smooth corners; that is, all λj≠1. We assume that one of them is at Ψ(1) with exterior angle λ. Assume that Fn,p(z)=o(1/n1-η)  (n→∞) for some fixed z∈G.
(a) If0<λ<1 and p>1, we consider for|w|>1(3.7)g(w)=(Ψ′(w))1-1/pΨ(w)-z=∑n=0∞Fn,p(z)wn+1.
Since
(3.8)∑n=1∞xnnα~Γ(1-α)(1-x)1-α,whenx→1-
for α<1, where Γ is Gamma Function (see [9, page 225]), then
(3.9)|g(w)|=o(∑n=1∞1n1-ηwn)=o(1)1(w-1)η,asw→1+.
This contradicts (2.7).
(b) If1<λ<2, then, by using (3.8) and
(3.10)g′(w)=-∑n=0∞(n+1)Fn,p(z)wn+2,
we get
(3.11)|g′(w)|=o(∑n=1∞nn1-ηwn)=o(1)1(w-1)η+1,asw→1+
which is contradiction to (2.8).Proof of Theorem1.5.
The proof goes along similar lines as those taken in the proof of Theorem1.1 with one significant difference: here, z lies in Ω, instead of G, and, thus, can tend to L without need altering the curve L′. As a consequence, the set F defined above does not depend on z, and; thus, dist⁡(z,F)>dist⁡(L,F)=c(L).



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*Source: 263657-2011-10-20.xml*