International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 263657 |

T. Tunc, M. Kucukaslan, "On the Asymptotic Behavior of p-Faber Polynomials for Domains with Piecewise Analytic Boundary", International Scholarly Research Notices, vol. 2011, Article ID 263657, 8 pages, 2011.

On the Asymptotic Behavior of p-Faber Polynomials for Domains with Piecewise Analytic Boundary

Academic Editor: X.-L. Gao
Received03 Jul 2011
Accepted08 Aug 2011
Published20 Oct 2011


Let 𝐺⊂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), (ğ‘›â†’âˆž) 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.

1. Introduction and Main Results

Let 𝐺⊂ℂ be a domain bounded by a Jordan curve 𝐿, and let Φ be the conformal mapping of the domain Ω∶=â„‚âˆžâ§µğº, where ℂ∞∶=ℂ∪{∞}, onto Δ∶={𝑤∶|𝑤|>1} with the usual normalization at infinity: 𝑤=Φ(𝑧)=𝛼𝑧+ğ‘Ž0+ğ‘Ž1𝑧+⋯,𝛼>0,𝑧∈Ω.(1.1) Let Ψ∶=Φ−1∶Δ→Ω denote the inverse conformal map. Then, 𝑧=Ψ(𝑤)=𝛽𝑤+𝑏0+𝑏1𝑤+⋯,|𝑤|>1,(1.2) where 𝛽=1/𝛼 gives the capacity ğ‘ğ‘Žğ‘(𝐿) of 𝐿.

Let 0<𝑝<∞. The 𝑝-Faber polynomials 𝐹𝑛,𝑝 associated the domain 𝐺 are defined as the polynomial part of the expansion of Φ𝑛Φ(𝑧)(𝑧)1/𝑝,𝑛=0,1,2,…,(1.3) in a neighborhood of the infinity. Therefore, from (1.1) we have Φ𝑛Φ(𝑧)(𝑧)1/𝑝∶=𝐹𝑛,𝑝(𝑧)+𝐸𝑛,𝑝(𝑧),𝑧∈Ω,(1.4) where 𝐹𝑛,𝑝(𝑧)∶=𝛼𝑛+1/𝑝𝑧𝑛+⋯(1.5) is the 𝑝-Faber polynomial of degree 𝑛 and 𝐸𝑛,𝑝𝑐(𝑧)=0𝑧+𝑐1𝑧2+𝑐2𝑧3+⋯(1.6) 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 Ψ𝑔(𝑤)∶=(𝑤)1−1/𝑝=Ψ(𝑤)âˆ’ğ‘§âˆžî“ğ‘›=0𝐹𝑛,𝑝(𝑧)𝑤𝑛+1,𝑧∈𝐺,|𝑤|>1,(1.7) see [1, 2].

If 𝑝 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 𝐹𝑛,1𝐹(𝑧)=î…žğ‘›+1(𝑧)𝑛+1,(1.8) where 𝐹𝑛 is the Faber polynomials of G of degree 𝑛.

The aim of this paper is to find asymptotic behaviors of 𝑝-Faber polynomials 𝐹𝑛,𝑝 in 𝐺 and 𝐸𝑛,𝑝 in Ω.

If 𝐿 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 boundary 𝐿 is rectifiable. For such a boundary, Cauchy’s integral formula yields the following representations for the 𝑝-Faber polynomials and their associated singular parts: 𝐹𝑛,𝑝1(𝑧)=2𝜋𝑖𝐿Φ𝑛Φ(𝜁)(𝜁)1/𝑝𝐸𝜁−𝑧𝑑𝜁,𝑧∈𝐺,𝑛,𝑝1(𝑧)=2𝜋𝑖𝐿Φ𝑛Φ(𝜁)(𝜁)1/𝑝𝜁−𝑧𝑑𝜁,𝑧∈𝐺.(1.9) Our assumption for this work is that 𝐿 is a piecewise analytic Jordan’s curve. That is, ⋃𝐿=𝑚𝑗=1𝐿𝑗, 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”.

Let 𝜆∶=min𝑗=1,…,𝑚{𝜆𝑗}. Gaier [5] proved that 𝐹𝑛1(𝑧)=𝑂𝑛𝜆(ğ‘›â†’âˆž),(1.10) for 𝑧∈𝐺, uniformly on compact subsets of 𝐺. Stylianopoulos [6] indicated that one can obtain 𝐹𝑛,1(𝑧)≤𝑐(𝐿,𝐹)𝑛𝜆+1,𝑧∈𝐹(1.11) 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 1𝜂∶=𝜂(𝜆,𝑝)∶=(1−𝜆)1−𝑝.(1.12) It is clear that, if 0<𝜆<2 and 1/2â‰¤ğ‘â‰¤âˆž, then |𝜂|<1.

The main results of the paper are the following.

Theorem 1.1. Let 1/2â‰¤ğ‘â‰¤âˆž, 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∶=min𝑗𝜆𝑗, then 𝐹𝑛,𝑝1(𝑧)=𝑂𝑛1−𝜂(ğ‘›â†’âˆž),(1.13) for 𝑧∈𝐺, uniformly on compact subsets of 𝐺.

Remark 1.2. From the proof of Theorem 1.1, it follows that an estimate of the form (1.13) holds also all derivatives of the 𝑝-Faber polynomials.

Remark 1.3. If 𝐿 has no smooth corners; that is, all 𝜆𝑗≠1, 𝑂 cannot be replaced by 𝑜 in (1.13).

Remark 1.4. If 𝑝 tends to infinity, we get the result (1.13) of Gaier.
Similar arguments in the proof of Theorem 1.1 provide the following theorem.

Theorem 1.5. Under the assumptions of Theorem 1.1, we have, for any𝑛∈ℕ, ||𝐸𝑛,𝑝||≤(𝑧)𝑐(𝐿)1dist(𝑧,𝐿)𝑛1−𝜂,𝑧∈Ω,(1.14) 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 max𝑗=1,…,𝑚𝜆𝑗<1/(1−𝑝).

Remark 1.7. If 𝑝=1, we get the result of Stylianopoulos [6]: ||𝐸𝑛,1||≤(𝑧)𝑐(𝐿)1dist(𝑧,𝐿)𝑛,𝑧∈Ω.(1.15) 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 ğ‘ î…žğ‘— 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 ğ‘ î…žğ‘— such that Φ has analytic continuation beyond 𝐿𝑗 into the strip-like domain 𝑔𝑗 bounded by 𝐿𝑗,𝑠𝑗,ğ‘ î…žğ‘—, and 𝛾𝑗 (see [5] or [6] for figure). Therefore, the arc Φ(𝛾𝑗) is in 𝔻 while Φ(𝑠𝑗) and Φ(ğ‘ î…žğ‘—) 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): ||Φ𝑧(𝑧)−Φ𝑗||||||1−Φ(𝑧)≤𝑐1for𝑧∈𝑠𝑗,𝑗=1,2,…,𝑚,(2.1) and similarly ||Φ𝑧(𝑧)−Φ𝑗+1||||||1−Φ(𝑧)â‰¤ğ‘î…ž1forğ‘§âˆˆğ‘ î…žğ‘—,𝑗=1,2,…,𝑚.(2.2) Furthermore, we have the following inequalities for 𝑧 on 𝑠𝑗(𝑗=1,2,…,𝑚): ||𝑧Φ(𝑧)−Φ𝑗||||≥const⋅𝑧−𝑧𝑗||1/𝜆𝑗,||Φ(2.3)||||(𝑧)≤const⋅𝑧−𝑧𝑗||1/𝜆𝑗−1||||||,(2.4)Φ(𝑧)≤1−const⋅𝑧−𝑧𝑗||1/𝜆𝑗||,(2.5)dist(𝑧,𝐿)≥const⋅𝑧−𝑧𝑗||.(2.6) The similar inequalities of (2.3)–(2.6) can be written by replacing 𝑠𝑗, 𝑧𝑗, and 𝜆𝑗 with ğ‘ î…žğ‘—, 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 of 𝑔 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 𝐶𝑔(𝑤)≅(𝑤−1)𝜂,as𝑤→1+,with𝐶≠0.(2.7) If 1<𝜆<2, then ğ‘”î…žğ¶(𝑤)≅(𝑤−1)𝜂+1,as𝑤→1+,with𝐶≠0.(2.8)

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 1𝜂∶=(1−𝜆)1−𝑝.(3.1)

Proof of Theorem 1.1. 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 ğ¿î…žğ‘—âˆ¶=𝑠𝑗∪𝐿𝑗∪𝑠′𝑗 (𝑗=1,2,…,𝑚), as mentioned before, with proper orientation and such that ⋃𝐹⊂𝐿′∶=𝑚𝑗=1ğ¿î…žğ‘—. This gives ||𝐹𝑛,𝑝||≤1(𝑧)2𝜋𝑚𝑗=1𝐿′𝑗||||Φ(𝜁)𝑛||||Φ′(𝜁)1/𝑝||||||||𝜁−𝑧𝑑𝜁≤𝐶(𝐿,𝐹)𝑚𝑗=1𝐿′𝑗||||Φ(𝜁)𝑛||||Φ′(𝜁)1/𝑝||||𝑑𝜁=𝐶(𝐿,𝐹)𝑚𝑗=1𝑠𝑗+𝑠′𝑗+𝐿𝑗||||Φ(𝜁)𝑛||||Φ′(𝜁)1/𝑝||||.𝑑𝜁(3.2) The contributions of the arcs 𝐿𝑗 to these integrals are obviously O(𝜌𝑛) for some 𝜌<1, so that typical integrals 𝐼𝑗=𝑠𝑗||||Φ(𝜁)𝑛||Φ(||𝜁)1/𝑝||||𝑑𝜁(3.3) extended over the line segments 𝑠𝑗 (and ğ‘ î…žğ‘—) remain to be estimated. If 𝑠 is arc length of 𝑠𝑗 measured from 𝑧𝑗, by using (2.4) and (2.5), ||||Φ(𝜁)≤1−𝑐𝑠1/𝜆𝑗<exp−𝑐𝑠1/𝜆𝑗,||||Φ′(𝜁)≤𝑐𝑠1/𝜆𝑗−1,(3.4) and, therefore, 𝐼𝑗<ğ‘âˆž0exp−𝑐𝑛𝑠1/𝜆𝑗𝑠(1/𝜆𝑗−1)/𝑝=𝑐𝑑𝑠𝑛(1−𝜆𝑗)/𝑝+ğœ†ğ‘—î€œâˆž0𝑒−𝑡𝑡(1/𝑝−1)(1−𝜆𝑗)𝑑𝑠.(3.5) If 1/2â‰¤ğ‘â‰¤âˆž, then the integral in the last line is convergent; that is, 𝐼𝑗<𝑐𝑛(1−𝜆𝑗)/𝑝+𝜆𝑗.(3.6) From this we obtain the Theorem 1.1.
In the case 0<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 Remark 1.3. We now show that 𝑂 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)  (ğ‘›â†’âˆž) for some fixed 𝑧∈𝐺.
(a) If 0<𝜆<1 and 𝑝>1, we consider for|𝑤|>1Ψ𝑔(𝑤)=(𝑤)1−1/𝑝=Ψ(𝑤)âˆ’ğ‘§âˆžî“ğ‘›=0𝐹𝑛,𝑝(𝑧)𝑤𝑛+1.(3.7) Since âˆžî“ğ‘›=1𝑥𝑛𝑛𝛼∼Γ(1−𝛼)(1−𝑥)1−𝛼,when𝑥→1−(3.8) for 𝛼<1, where Γ is Gamma Function (see [9, page 225]), then ||||𝑔(𝑤)=ğ‘œâˆžî“ğ‘›=11𝑛1−𝜂𝑤𝑛1=𝑜(1)(𝑤−1)𝜂,as𝑤→1+.(3.9) This contradicts (2.7).
(b) If 1<𝜆<2, then, by using (3.8) and ğ‘”î…ž(𝑤)=âˆ’âˆžî“ğ‘›=0(𝑛+1)𝐹𝑛,𝑝(𝑧)𝑤𝑛+2,(3.10) we get ||||𝑔′(𝑤)=ğ‘œâˆžî“ğ‘›=1𝑛𝑛1−𝜂𝑤𝑛1=𝑜(1)(𝑤−1)𝜂+1,as𝑤→1+(3.11) which is contradiction to (2.8).

Proof of Theorem 1.5. The proof goes along similar lines as those taken in the proof of Theorem 1.1 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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Copyright © 2011 T. Tunc and M. Kucukaslan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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