APPROXIMATION BY INTERPOLATING POLYNOMIALS IN SMIRNOV ...

J. Korean Math. Soc. 43 (2006), No. 2, pp. 413–424

APPROXIMATION BY INTERPOLATING POLYNOMIALS IN SMIRNOV-ORLICZ CLASS ¨ n and Daniyal M. Israfilov Ramazan Akgu Abstract. Let Γ be a bounded rotation (BR) curve without cusps in the complex plane C and let G := int Γ. We prove that the rate of convergence of the interpolating polynomials based on the zeros of the Faber polynomials Fn for G to the function of the reflexive Smirnov-Orlicz class EM (G) is equivalent to the best approximating polynomial rate in EM (G).

1. Introduction and main results Let Γ be a closed rectifiable Jordan curve in the complex plane C. The curve Γ separates the plane into two domains G := int Γ and G− := ext Γ. We denote D := {z ∈ C : |z| < 1} , T := ∂D and D− := ext T. Let w = φ (z) be the conformal map of G− onto D− normalized by the conditions φ (z) φ (∞) = ∞, lim > 0, z→∞ z and let ψ := φ−1 be its inverse mapping. When |z| is sufficiently large, φ has the Laurent expansion φ (z) = dz + d0 +

d1 + ··· z

and hence we have [φ (z)]n = dn z n +

n−1 X

X

k=0

k 0, and let γr be the image of the circle |w| = r, 0 < r < 1, under the mapping φ0 . We say that a function f analytic in G, belongs to the Smirnov class E p (G), 0 < p < ∞, if for any r ∈ (0, 1) the inequality Z |f (z)|p |dz| ≤ c < ∞ γr

holds. Every function in E p (G), 1 < p < ∞, has nontangential boundary values almost everywhere (a. e.) on Γ and the boundary function belongs to Lp (Γ) . For p > 1, E p (G) is a Banach space with respect to the norm  1 p Z p   kf kE p (G) := kf kLp (Γ) := |f (z)| |dz| . Γ

A continuous and convex function M : [0, ∞) → [0, ∞) which satisfies the conditions M (0) = 0, M (x) > 0

for x > 0,

M (x) M (x) = 0, lim = ∞, x→∞ x→0 x x is called an N -function. The complementary N -function to M is defined by lim

N (y) := max (xy − M (x)) , y ≥ 0. x≥0

Approximation by interpolating polynomials

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We denote by LM (Γ) the linear space of Lebesgue measurable functions f : Γ → C satisfying the condition Z M [α |f (z)|] |dz| < ∞ Γ

for some α > 0. The space LM (Γ) becomes a Banach space with the norm ½Z ¾ kf kLM (Γ) := sup |f (z) g (z)| |dz| : g ∈ LN (Γ) ; ρ (g; N ) ≤ 1 , Γ

where N is the complementary N -function to M and Z ρ (g; N ) := N [|g (z)|] |dz| . Γ

This norm is called Orlicz norm and the Banach space LM (Γ) is called Orlicz space. We note that (see, for example, [12, p.51]) LM (Γ) ⊂ L1 (Γ) . An N -function M satisfies the ∆2 -condition if lim sup x→∞

M (2x) < ∞. M (x)

The Orlicz space LM (Γ) is reflexive if and only if the N -function M and its complementary function N both satisfy the ∆2 -condition [12, p.113]. Let Γr be the image of the circle {w ∈ C : |w| = r, 0 < r < 1} under some conformal map of D onto G and let M be an N -function. Definition 1. The class of functions which are analytic in G and satisfy the condition Z M [|f (z)|] |dz| < ∞ Γr

uniformly in r is called the Smirnov-Orlicz class and denoted by EM (G). The Smirnov-Orlicz class is a generalization of the familiar Smirnov class E p (G). In particular, if M (x) := xp , 1 < p < ∞, then the SmirnovOrlicz class EM (G) determined by M coincides with the Smirnov class E p (G) .

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Ramazan Akg¨ un and Daniyal M. Israfilov

Since (see [10]) EM (G) ⊂ E 1 (G), every function in the class EM (G) has the nontangential boundary values a.e. on Γ and the boundary value function belongs to LM (Γ). Hence the EM (G) norm can be defined as: (1)

kf kEM (G) := kf kLM (Γ) ,

f ∈ EM (G) .

Let γ be an oriented rectifiable curve. For z ∈ γ, δ > 0 we denote by s+ (z, δ) (respectively s− (z, δ)) the subarc of γ in the positive (respectively negative) orientation of γ with the z starting point and arc length from z to each point less than δ. If γ is a smooth curve and ½ Z ¾ Z lim |dς arg (ς − z)| + |dς arg (ς − z)| = 0 δ→0

s− (z,δ)

s+ (z,δ)

holds uniformly for z ∈ γ, then it’s said [16] that γ is of vanishing rotation (V R). As follows from this definition, the V R condition is stronger than smoothness. In [16] L. Zhong and L. Zhu also proved that there exists a smooth curve which is not of V R. On the other hand, if the angle of inclination θ (s) of tangent to γ as a function of the arclength s along γ satisfies the condition Zδ 0

ω (t) dt < ∞, t

where ω (t) is the modulus of continuity of θ (s) , then [16] γ is V R. Approximation properties of the Faber and generalized Faber polynomials in the different functional spaces are well known (see for example: [1]-[2], [4]-[8] and also [5, Chapter 1, pp. 42–57], [15]). In this work we investigate the convergence property of the interpolating polynomials based on the zeros of the Faber polynomials in the reflexive SmirnovOrlicz class. This problem isn’t new. It was studied by several authors. In their work [13] under the assumption Γ ∈ C (2, α), 0 < α < 1, X. C. Shen and L. Zhong obtain a series of interpolation nodes in G and show that interpolating polynomials and the best approximating polynomial have the same order of convergence in E p (G), 1 < p < ∞. In [17] considering Γ ∈ C (1, α) and choosing the interpolation nodes as the zeros of the Faber polynomials L. Y. Zhu obtain similar result. In the above cited works Γ does not admit corners. Many domains in the complex plane may have corners or cusps. When Γ is a piecewise V R curve without cusps, L. Zhong and L. Zhu [16] showed that the


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interpolating polynomials based on the zeros of the Faber polynomials converge in the Smirnov class E p (G), 1 < p < ∞. In this work we investigate the convergence property of the interpolating polynomials based on the zeros of the Faber polynomials in the reflexive Smirnov-Orlicz class under the assumption that Γ is a BR curve without cusps. Definition 2. [6] Let γ be a rectifiable Jordan curve with length L and let z = z (t) be its parametric representation with arclenght t ∈ [0, L]. If β (t) := arg z 0 (t) can be defined on [0, L] to become a function of bounded variation, then γ is called of bounded rotation (γ ∈ BR) R and Γ |dβ (t)| is called total rotation of γ. If γ ∈ BR, then there are two half tangents at each point of γ. The class of bounded rotation curves is sufficiently wide. For example, a curve which is made up of finitely many convex arcs (corners are permitted), is bounded rotation [5, p.45]. It is easily seen that every V R curve and also a piecewise V R curve considered in [16] is BR curve. Since a BR curve may have cusps or corners, there exists a BR curve which is not a VR curve (for example, a rectangle in the plane). In the case that all of the zeros of the nth Faber polynomial Fn (z) are in G, we denote by Ln (f, z) the (n − 1) th interpolating polynomial to f (z) ∈ EM (G) based on the zeros of the Faber polynomials Fn . For f ∈ EM (G), we denote by (2) n o EnM (f, G) := inf kf − pn kEM (G) : pn is a polynomial of degree ≤ n ½ ½Z ¾¾ = inf sup |(f (ς) − pn (ς)) g (ς)| |dς| ; ρ (g; N ) ≤ 1 Γ

the minimal error of approximation of f by polynomials of degree at most n. The main results of this work are the following. Theorem. Let Γ be a BR curve without cusps. Then for sufficiently large natural number n, the roots of the Faber polynomials are in G and for every f which belongs to reflexive Smirnov-Orlicz class EM (G), M kf − Ln (f, ·)kEM (G) ≤ c · En−1 (f, G)

with a positive constant c depending only on Γ and M . In particular, when M (x) := xp , 1 < p < ∞, we have the following result.

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Corollary. Let Γ be a BR curve without cusps. Then for sufficiently large natural number n, the roots of the Faber polynomials are in G and for every f which belongs to Smirnov class E p (G), 1 < p < ∞, kf − Ln (f, ·)kE p (G) ≤ c · En−1 (f, G)p , where the constant c > 0 depend only on p and G. When Γ is a piecewise V R curve without cups, this corollary was proved in [16]. We use c, c1 , c2 , . . . to denote constants (which may, in general, differ in different relations) depending only on numbers that are not important for the question of our interest. 2. Auxiliary results Let Γ be a BR curve without cusps. Then (see, for example, Pommerenke [11]) Z 1 Fn (z) = [φ (ς)]n dς arg (ς − z) , z ∈ Γ, π Γ

where the jump of arg (ς − z) at ς = z equals to the exterior angle αz π. Hence we have Z 1 n (3) Fn (z) − [φ (z)] = [φ (ς)]n dς arg (ς − z) + (αz − 1) [φ (z)]n , π Γ\{z}

and (4)

0 ≤ max |αz − 1| < 1. z∈Γ

Lemma 1. [3] Let Γ be a BR curve. For any ² > 0 and θ , there exists a δ > 0 such that (5) θ+δ Zθ− ¯ ¯ ³ ¡ ¢ ³ ´´¯ Z ³ ¡ ¢ ³ ´´¯ ¯ ¯ ¯ ¯ it iθ ¯dt arg ψ e − ψ e ¯+ ¯dt arg ψ eit − ψ eiθ ¯ < ², θ−δ

θ+

and for any η ∈ (θ − δ, θ + δ) different from θ Zη η−δ

¯ ¡ ¡ ¢ ¡ ¢¢¯ ¯dt arg ψ eit − ψ eiη ¯ +

η+δ Z ¯ ¡ ¡ ¢ ¡ ¢¢¯ ¯dt arg ψ eit − ψ eiη ¯ η


419

< ² + |αz − 1| π,

¡ ¢ where αz π is the external angle to Γ at z = ψ eiθ . Lemma 2. If Γ is a BR curve, then for every ² > 0, there exists a δ > 0 such that Z Z (6) |dς arg (ς − z)| + |dς arg (ς − z)| < ², z ∈ Γ. s− (z,δ)\{z}

s+ (z,δ)\{z}

Proof. ¡ ¢We take arbitrary z ∈ Γ and fix it. By the change of variable ς = ψ eit we get Z s− (z,δ)\{z}

Zθ− ¯ ³ ¡ ¢ ³ ´´¯ ¯ ¯ |dς arg (ς − z)| = ¯dψ(eit ) arg ψ eit − ψ eiθ ¯ θ−δ

Zθ− ¯ ³ ¡ ¢ ³ ´´¯ ¯ ¯ = ¯dt arg ψ eit − ψ eiθ ¯ θ−δ

and similarly Z s+ (z,δ)\{z}

θ+δ Z ¯ ³ ¡ ¢ ³ ´´¯ ¯ ¯ |dς arg (ς − z)| = ¯dψ(eit ) arg ψ eit − ψ eiθ ¯ θ+ θ+δ Z ¯ ³ ¡ ¢ ³ ´´¯ ¯ ¯ = ¯dt arg ψ eit − ψ eiθ ¯ . θ+

For any ² > 0, using (5) we have (6). we denote by Iθ,δ , the image of the set ¡ any ¡ iθ ¢δ >¢ 0, θ ¡∈ [0, ¡ iθ2π], ¢ ¢ª © For s− ψ e , δ ∪ s+ ψ e , δ under φ and let ( eit ψ0 eit ( ) eit ∈ / Iθ,δ , ψ(eit )−ψ (eiθ ) v (t, θ; δ) := 0 eit ∈ Iθ,δ . The following lemma was proved by L. Zhong and L. Zhu [16] in the case of a domain bounded by a piecewise V R curve without cusps. When the boundary of the domain is a BR curve, the proof goes similarly. Lemma 3. For any ² > 0, δ > 0, there exists a natural number k such that for θ ∈ [0, 2π] there exists a trigonometric polynomial Tθ (t)

420


of t with degree at most k satisfying Z2π (7)

|v (t, θ; δ) − Tθ (t)| dt < ². 0

Lemma 4. Let Γ be a BR curve without cusps. Then for arbitrary ² > 0, there exists a positive integer n0 such that |Fn (z) − [φ (z)]n | < |αz − 1| + ²,

(8)

z∈Γ

holds for n > n0 . Proof. For any ² > 0, there exists a δ > 0 such that (6) holds. Let s (z) := {s− (z, δ) ∪ s+ (z, δ)} , z ∈ Γ. Hence by Lemma 3, for given ² and δ ¡there ¢ is a positive integer n0 such that (7) is valid. By (3) for z = ψ eiθ we have Fn (z) − [φ (z)]n ½ Z 1 = + π s+ (z,δ)\{z}

+ 1 = π

Z

1 π ½

1 = π

¾ [φ (ς)]n dς arg (ς − z)

s− (z,δ)\{z}

[φ (ς)]n dς arg (ς − z) + (αz − 1) einθ Γ\s(z)

Z

Z


+ s+ (z,δ)\{z}

+

Z

Z

1 π ½

s− (z,δ)\{z}

³ ¡ ¢ ³ ´´ eint dt arg ψ eit − ψ eiθ + (αz − 1) einθ

eit ∈I / θ,δ

Z

Z +

s+ (z,δ)\{z}

1 + π


s− (z,δ)\{z}

Z2π eint Im [iv (t, θ; δ)] dt + (αz − 1) einθ . 0

Since eint is orthogonal to Tθ (t) as n > n0 , we get ¾ ½ Z Z 1 n [φ (ς)]n dς arg (ς − z) Fn (z) − [φ (z)] = + π s+ (z,δ)\{z}

s− (z,δ)\{z}


1 + π

421

Z2π eint Im [iv (t, θ; δ) − iTθ (t)] dt + (αz − 1) einθ , 0

and hence 1 |Fn (z) − [φ (z)] | ≤ π

½

Z

¾

Z

n

+ s+ (z,δ)\{z}

1 + π

|dς arg (ς − z)|+|αz − 1|

s− (z,δ)\{z}

Z2π |v (t, θ; δ) − Tθ (t)| dt. 0

If z is not a corner of Γ, then |αz − 1| = 0 and by (6) and (7) our assumption follows. If z is a corner of Γ, then 0 ≤ |αz − 1| < 1 and hence by (6) and (7) we have (8) again. For z ∈ Γ and ² > 0 let Γ (z, ²) denote the portion of Γ which is inside the open disk of radius ² centered at z, i.e., Γ (z, ²):={t ∈ Γ : |t − z| < ²} . Further, let |Γ (z, ²)| denote the length of Γ (z, ²) . A rectifiable Jordan curve Γ is called a Carleson curve if 1 sup sup |Γ (z, ²)| < ∞. ²>0 z∈Γ ² We consider the Cauchy-type integral Z f (ς) 1 dς, (Hf ) (z) := 2πi ς −z

z∈G

Γ

and Cauchy’s singular integral of f ∈ L1 (Γ) defined as Z 1 f (ς) SΓ f (z) := lim dς, z ∈ Γ. ε→0 2πi ς −z Γ\Γ(z,²)

The linear operator SΓ : f → SΓ f is called the Cauchy singular operator. Lemma 5. [9] Let Γ be a rectifiable Jordan curve and let LM (Γ) be a reflexive Orlicz space on Γ. Then the singular operator SΓ is bounded on LM (Γ) , i.e., kSΓ f kLM (Γ) ≤ c1 kf kLM (Γ) ,

f ∈ LM (Γ) ,

for some constant c1 > 0 if and only if Γ is a Carleson curve.

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3. Proof of Theorem We proof firstly that for sufficiently large n, all zeros of the Faber polynomials Fn are in G. Let κ := max |αz − 1| , z∈Γ

z ∈ Γ.

Then by (4) we have 0 ≤ κ < 1. Setting ² := sufficiently large n we get |Fn (z) − [φ (z)]n |