Approximating polynomials for functions of weighted smirnov-orlicz spaces

Tam metin

(1)Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 982360, 41 pages doi:10.1155/2012/982360. Research Article Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces ¨ Ramazan Akgun Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145 Balikesir, Turkey Correspondence should be addressed to Ramazan Akgun, ¨ rakgun@balikesir.edu.tr Received 9 July 2009; Accepted 13 December 2009 Academic Editor: V. M. Kokilashvili Copyright q 2012 Ramazan Akgun. ¨ 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. Let G0 and G∞ be, respectively, bounded and unbounded components of a plane curve Γ satisfying Dini’s smoothness condition. In addition to partial sum of Faber series of f belonging to weighted Smirnov-Orlicz space EM,ω G0 , we prove that interpolating polynomials and Poisson polynomials are near best approximant for f. Also considering a weighted fractional moduli of smoothness, we obtain direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems, we prove direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces EM,ω G0 and EM,ω G∞ .. 1. Introduction Let G0 and G∞ be, respectively, bounded and unbounded components of a closed rectifiable curve Γ of complex plane C. Without loss of generality we may suppose that 0 ∈ G0 . By Riemann conformal mapping theorem 1, page 26, if Γ is connected Jordan curve that consists of more than one point, there exists a conformal mapping ϕ0 : D → G0 of complex unit disc D : {w ∈ C : |w| 1} onto G0 . Let γr : ϕ0 {w ∈ C : |w| r} for a given r ∈ 0, 1. We denote by Ep G0 , 1 ≤ p ≤ ∞, Smirnov’s classes of analytic functions f : G0 → C satisfying . fzp |dz| < c,. sup r∈0,1. if 1 ≤ p < ∞,. γr. maxfz < C, z∈G0. where positive constant c is independent of r.. if p ∞,. 1.1.

(2) 2. Journal of Function Spaces and Applications. It is well known that Ep G0 ⊂ E1 G0 for every 1 ≤ p < ∞ and every function f ∈ E G0 has a nontangential boundary values a.e. on Γ, the boundary function belongs to Lebesgue space L1 Γ on Γ. If 1 ≤ p < ∞, then Ep G0 is a Banach space with the norm 1. f : p,Γ. . 1 2π. Γ. 1/p fzp |dz| .. 1.2. Smirnov classes Ep G∞ , 1 ≤ p ≤ ∞, of analytic functions f : G∞ → C can be defined similarly and Ep G∞ are fulfilling the same above properties to that of Ep G0 . A smooth Jordan curve Γ will be called Dini-smooth, if the function θs, the angle between the tangent line and the positive real axis expressed as a function of arclength s, has modulus of continuity Ωθ, s satisfying the Dini condition δ 0. Ωθ, s ds < ∞, s. δ > 0.. 1.3. A Jordan curve Γ will be called Radon curve, if θs has bounded variation and it does not contain cusp point. Main approximation problems in the spaces Ep G0 , 1 ≤ p ≤ ∞, were dealt with by several mathematicians so far. Walsh and Russell gave 2 results in Ep G0 , 1 < p < ∞, for algebraic polynomial approximation orders in case of analytic boundary. Al’per proved 3 direct and converse approximation theorems by algebraic polynomials in Ep G0 , 1 < p < ∞, for Dini-smooth boundary. Kokilashvili improved 4 to Al’per’s direct and converse results of algebraic polynomial approximation, and then considering Regular curves that Cauchy’s Singular Integral Operator is bounded corners are permitted, he obtained 5 improved direct and converse approximation theorems in Smirnov spaces Ep G0 , 1 < p < ∞. Andersson proved 6 that Kokilashvili’s results also holds in E1 G0 . When the boundary is a regular curve, approximation of functions of Ep G0 , 1 < p < ∞, by partial sum of Faber series was obtained by Israfilov in 7, 8. These results are generalized to Muckenhoupt weighted Smirnov’s spaces in 9–12. Approximation properties of Faber series in so-called weighted and unweighted Smirnov-Orlicz spaces are investigated in 13– 20. Most of the above results use the partial sums of Faber series as approximation tool. Interpolating polynomials 16 and Poisson polynomials 21 can be also considered as an approximating polynomial. In the present paper we obtain that in addition to partial sums of Faber series of f belonging to weighted Smirnov-Orlicz space EM,ω G0 , interpolating polynomials and Poisson polynomials are near best approximant for f. Also considering a weighted fractional moduli of smoothness, we obtain in Section 2 direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems we prove in Section 3 direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces EM,ω G0 and EM,ω G∞ . Throughout the work, we will denote by c, C, the constants that are different in different places..

(3) Journal of Function Spaces and Applications. 3. 2. Approximation Theorems in Weighted Orlicz Space A function Φ is called Young function if Φ is even, continuous, nonnegative in R, increasing on 0, ∞ such that Φ0 0,. lim Φx ∞.. x→∞. 2.1. A Young function Φ is said to satisfy Δ2 condition Φ ∈ Δ2 if there is a constant c > 0 such that Φ2x ≤ cΦx. 2.2. for all x ∈ R. Two Young functions Φ and Φ1 are said to be equivalent if there are c, C > 0 such that Φ1 cx ≤ Φx ≤ Φ1 Cx,. ∀x > 0.. 2.3. A function M : 0, ∞ → 0, ∞ is said to be quasiconvex if there exist a convex Young function Φ and a constant c ≥ 1 such that Φx ≤ Mx ≤ Φcx,. ∀x ≥ 0,. 2.4. holds. A nonnegative function ω defined on T : 0, 2π will be called weight if ω is M,ω T measurable and a.e. positive. Let M be a quasiconvex Young function. We denote by L the class of Lebesgue measurable functions f : T → R satisfying the condition T. M fx ωxdx < ∞.. 2.5. M,ω T, denoted by LM,ω T, becomes a normed The linear span of the weighted Orlicz class L space with the Orlicz norm f . M,ω. : sup. T. fxgxωxdx :. T. . g ωxdx ≤ 1 , M. 2.6. where My : supx≥0 xy − Mx, y ≥ 0, is the complementary function of M.. is its complementary function, then Young’s inequality holds If M is quasiconvex and M . y , xy ≤ Mx

(4) M. x, y ≥ 0.. 2.7.

(5) 4. Journal of Function Spaces and Applications. For a quasiconvex function M we define the indice pM of M as.

(6) 1 : inf p : p > 0, Mp is quasiconvex , pM pM p M : . pM − 1. 2.8. The indice pM was first defined and used by Gogatishvili and Kokilashvili in 22 to obtain weighted inequalities for maximal function. We note that the indice pM is much more convenient than Gustavsson and Peetre’s lower index and Boyd’s upper index. If ω ∈ ApM , then it can be easily seen that LM,ω T ⊂ L1 T and LM,ω T becomes a Banach space with the Orlicz norm. The Banach space LM,ω T is called weighted Orlicz space. We define the Luxemburg functional as fx ωxdx ≤ 1 . : inf τ > 0 : M τ T. f . M,ω. . 2.9. There exist 23, page 23 constants c, C > 0 such that cf M,ω ≤ f M,ω ≤ Cf M,ω .. 2.10. For a weight ω we denote by Lp T, ω the class of measurable functions on T such that ω1/p f belongs to Lebesgue space Lp T on T. We set f p,ω : ω1/p f p for f ∈ Lp T, ω. A 2π-periodic weight function ω belongs to the Muckenhoupt class Ap , 1 < p < ∞, if . 1 |J|. . ωxdx J. 1 |J|. p−1. ω. −1/p−1. xdx. ≤c. 2.11. J. with a finite constant c independent of J, where J is any subinterval of T and |J| denotes the length of J. We will denote by QC2θ 0, 1 a class of functions g satisfying Δ2 condition such that g θ is quasiconvex for some θ ∈ 0, 1. In the present section we consider the trigonometric polynomial approximation problems for functions and its fractional derivatives in the spaces LM,ω T, ω ∈ ApM , where M ∈ QC2θ 0, 1. We prove a Jackson type direct theorem and a converse theorem of trigonometric approximation with respect to the fractional order moduli of smoothness in weighted Orlicz spaces with Muckenhoupt weights. In the particular case, we obtain a constructive characterization of Lipschitz class in these spaces. In weighted Lebesgue and Orlicz spaces with Muckenhoupt weights, these results were investigated in 24–29. For more general doubling weights, some of these problems were investigated in 30. Jackson and converse inequalities were proved for Lebesgue spaces with Freud weight in 31. For a general discussion of weighted polynomial approximation, we can refer to the books 32, 33..

(7) Journal of Function Spaces and Applications. 5. Let b0 0, ak , bk ∈ R, ck ak − ibk /2, c−k ak

(8) ibk /2, c0 a0 /2 fx ∼. ∞ . ck eikx . k−∞. fx ∼. ∞ . ak cos kx

(9) bk sin kx,. 2.12. k0 ∞ ak sin kx − bk cos kx. 2.13. k1. be the Fourier and the conjugate Fourier series of f ∈ L1 T, respectively. Putting Ak x : ck eikx in 2.12, we define for n 0, 1, 2, . . . n n . a0

(10) ak cos kx

(11) bk sin kx, Sn f : Sn x, f : Ak x

(12) A−k x 2 k1 k0 α n . k

(13) α 1− Rn f, x : Ak x

(14) A−k x, α ∈ R

(15) , n

(16) 1 k0

(17) α. Θm : . 1 1

(18) α

(19) α R − Rm , 1 − m

(20) 1/2m

(21) 1r 2m 2m

(22) 1/m

(23) 1r − 1 m 1, 2, 3, . . . . 2.14. For a given f ∈ L1 T, assuming T. fxdx 0,. 2.15. we define αth fractional α ∈ R

(24) integral of f as 34, v.2, page 134 . Iα x, f : ck ik−α eikx ,. 2.16. k∈Z∗. where ik−α : |k|−α e−1/2πiα sign k. 2.17. as principal value. Let α ∈ R

(25) be given. We define fractional derivative of a function f ∈ L1 T, satisfying 2.15, as f α x : provided the right hand side exists.. . dα

(26) 1 I1

(27) α−α x, f α

(28) 1 dx. 2.18.

(29) 6. Journal of Function Spaces and Applications Setting x, t ∈ T, r ∈ R

(30) , M ∈ QC2θ 0, 1, ω ∈ ApM , and f ∈ LM,ω T, we define σtr fx : I − σt r fx t t ∞ 1 r · · · fx

(31) u1

(32) · · · uk du1 · · · duk , −1k k 2tk −t −t k0. 2.19. where kr : rr − 1 · · · r − k

(33) 1/k! for k ≥ 1 and 0r : 1 are Binomial coefficients, t σt fx : 1/2t −t fx

(34) udu is Steklov’s mean operator, and I is identity operator. Theorem A see 23, page 278, Theorem 6.7.1. One suppose that L is anyone of the operators If M ∈ QCθ 0, 1, ω ∈ ApM , and f ∈ LM,ω T, then there exists a constant c > 0 Sn , σh , and f. 2 such that T. M Lft ωtdt ≤ c. T. M ft ωtdt. 2.20. holds. Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from 2.20 that Lf ≤ cf M,ω M,ω. 2.21. with a constant c > 0 independent of f. By 35, page 14, 1.51, there exists a constant c depending only on r such that r c k ≤ kr

(35) 1 ,. k 1, 2, . . . ,. 2.22. we have ∞ r k <∞. 2.23. r σ f ≤ cf M,ω < ∞ t M,ω. 2.24. k0. and therefore. provided f ∈ LM,ω T, ω ∈ ApM , where M ∈ QC2θ 0, 1..

(36) Journal of Function Spaces and Applications. 7. Let M ∈ QC2θ 0, 1. For r ∈ R

(37) , we define the fractional modulus of smoothness of index r for f ∈ LM,ω T, ω ∈ ApM as. ΩrM,ω. r r−r f f, δ : sup I − σhi I − σt 0<hi , t<δ i1. . ,. 2.25. M,ω. where x denotes the integer part of a real number x. Since the operator σt is bounded in LM,ω T, ω ∈ ApM , where M ∈ QC2θ 0, 1, we have by 2.24 that . ΩrM,ω f, δ ≤ cf M,ω ,. 2.26. where the constant c > 0, dependent only on r and M. Remark 2.1. The modulus of smoothness ΩrM,ω f, δ, where r ∈ R

(38) , M ∈ QC2θ 0, 1, ω ∈ ApM , f ∈ LM,ω T has the following properties: i ΩrM,ω f, δ is nonnegative, nondecreasing function of δ ≥ 0 and subadditive, ii limδ → 0 ΩrM,ω f, δ 0. For formulations of our results, we need several lemmas. Lemma A see 36. For α ∈ R

(39) , we suppose that i a1

(40) a2

(41) · · ·

(42) an

(43) · · · , ii a1

(44) 2α a2

(45) · · ·

(46) nα an

(47) · · · , be two series in a Banach space B, · . Let

(48) α Rn

(49) α ∗ Rn. α n k : 1− ak , n

(50) 1 k0 α n k : 1− kα ak n

(51) 1 k0. 2.27. for n 1, 2, . . . . Then,

(52) α ∗ Rn ≤ c,. n 1, 2, . . .. 2.28. for some c > 0 if and only if there exists a R ∈ B such that C

(53) α Rn − R ≤ α , n where c and C are constants depending only on one another.. 2.29.

(54) 8. Journal of Function Spaces and Applications. If M ∈ QC2θ 0, 1, ω ∈ ApM , and f ∈ LM,ω T, then from Theorem Aii and Abel’s transformation we get

(55) α Rn f, · . M,ω. ≤ cf M,ω ,. n 1, 2, 3, . . . , x ∈ T. 2.30. ≤ cf M,ω ,. n 1, 2, 3, . . . , x ∈ T.. 2.31. and therefore from 2.14 and2.30

(56) α Θn f, · . M,ω. From the property 2m . 1 k

(57) 1α − kα Sk x, f , α α km

(58) 1 k

(59) 1 − k k m

(60) 1.

(61) α . Θm. f x 2m. x ∈ T, f ∈ L1 T 2.32. it is known that

(62) α. Θm Tm Tm. 2.33. for Tm ∈ Tm , m 1, 2, 3, . . . . Lemma 2.2. Let Tn ∈ Tn , n 1, 2, 3, . . . , M ∈ QC2θ 0, 1, and ω ∈ ApM . If α ∈ R

(63) , then there exists a constant c > 0 independent of n such that α Tn . M,ω. ≤ cnα Tn M,ω. 2.34. holds. Proof. Without loss of generality one can assume that Tn M,ω 1. Since Tn . n . Ak x

(64) A−k x,. k0. α n Tn α Ak x − A−k x k nα inα k1. n Tn Ak x − A−k x , nα k1 nα. 2.35. we have by 2.30 and Theorem Aiii that

(65) α Tn Rm nα . M,ω. ≤. c c c T ≤ α Tn M,ω α n M,ω nα n n. 2.36.

(66) Journal of Function Spaces and Applications. 9. and from Lemma A α

(67) α Tn Rm α in . ≤ c.. 2.37. M,ω. Hence from 2.33 and 2.31, we find α Tn . M,ω. α Tn

(68) α nα Θm α in . ≤ cnα Tn M,ω .. 2.38. M,ω. General case follows immediately from this. α Let M ∈ QC2θ 0, 1. We denote by WM T, ω, α > 0, ω ∈ ApM , the linear space of 2π-periodic real valued functions f ∈ LM,ω T such that f α ∈ LM,ω T. α Lemma 2.3. Let M ∈ QC2θ 0, 1. If f ∈ WM T, ω with ω ∈ ApM and α ≥ 0, then for n 0, 1, 2, . . ., there is a constant c > 0 dependent only on α and M such that. α α f · − Sn ·, f . M,ω. ≤ cEn f α. 2.39. M,ω. holds. Proof. If α 0, then from boundedness see 2.21 of the operator Sn we get that f − Sn f ≤ cEn f M,ω . M,ω Let Wn f : Wn x, f : 1/n

(69) 1. 2n. νn. 2.40. Sν x, f, n 0, 1, 2, . . . . Since. . α Wn ·, f α Wn ·, f ,. 2.41. we have α α f · − Sn ·, f . M,ω. α α ≤ f α · − Wn ·, f α

(70) Sn ·, Wn f − Sn ·, f M,ω M,ω α α

(71) Wn ·, f − Sn ·, Wn f : I1

(72) I2

(73) I3 . M,ω. 2.42. From 2.21 we get the boundedness of Wn in LM,ω T and we have I1 ≤ f α · − Sn ·, f α

(74) Sn ·, f α − Wn ·, f α M,ω M,ω α α α −f ≤ cEn f

(75) Wn ·, Sn f ≤ cEn f α M,ω. M,ω. M,ω. .. 2.43.

(76) 10. Journal of Function Spaces and Applications. From Lemma 2.2 we get I2 ≤ cnα Sn ·, Wn f − Sn ·, f M,ω , I3 ≤ c2nα Wn ·, f − Sn ·, Wn f M,ω ≤ c2nα En Wn f M,ω .. 2.44. Now we have Sn ·, Wn f − Sn ·, f . M,ω. ≤ Sn ·, Wn f − Wn ·, f M,ω

(77) Wn ·, f − f·M,ω

(78) f· − Sn ·, f M,ω. ≤ cEn Wn f M,ω

(79) cEn f M,ω .. 2.45. Since. En Wn f M,ω ≤ cEn f M,ω ,. 2.46. we get α α f · − Tn ·, f. M,ω. ≤ cEn f α. M,ω.

(80) cnα En Wn f M,ω

(81) cnα En f M,ω. .

(82) c2nα En Wn f M,ω ≤ cEn f α. M,ω.

(83) Cnα En f M,ω .. 2.47. Now we show that c α . α En f M,ω n

(84) 1. 2.48. . Ak x, f : ak cos kx

(85) bk sin kx.. 2.49. En f M,ω ≤ For this we set. For given f ∈ LM,ω T and ε > 0, by Lemma 3 of 37, there exists a trigonometric polynomial T such that 2.50 M fx − T x ωxdx < ε T. which by 2.7 this implies that f − T < ε, M,ω. 2.51. and hence we obtain En f M,ω −→ 0. as n −→ ∞.. 2.52.

(86) Journal of Function Spaces and Applications. 11. In this case from 2.40 we have. fx . ∞ . . Ak x, f. 2.53. k0. in · M,ω norm. If k 1, 2, 3, . . ., then . απ απ απ απ , f cos

(87) Ak x

(88) , f sin , Ak x, f Ak x

(89) 2k 2 2k 2 απ α α Ak x, f ,f . k Ak x

(90) 2k. 2.54. Hence, ∞ ∞ ∞ . . απ απ απ απ , f

(91) sin ,f Ak x, f A0 x, f

(92) cos Ak x

(93) Ak x

(94) 2 k1 2k 2 k1 2k k0 ∞ ∞ απ απ A0 x, f

(95) cos k−α Ak x, f α

(96) sin k−α Ak x, fα . 2 k1 2 k1. . 2.55. Therefore, ∞ ∞ . απ 1 1 απ fx − Sn x, f cos Ak x, f α

(97) sin Ak x, fα . α α 2 kn

(98) 1 k 2 kn

(99) 1 k. 2.56. Since ∞ . ∞ k−α Ak x, f α k−α Sk ·, f α − f α · − Sk−1 ·, f α − f α ·. kn

(100) 1. kn

(101) 1. . ∞ . k−α − k

(102) 1−α. Sk ·, f α − f α ·. . kn

(103) 1. − n

(104) 1−α Sn ·, f α − f α · ,. ∞ . ∞ −α k−α Ak x, fα k − k

(105) 1−α Sk ·, fα − fα ·. kn

(106) 1. kn

(107) 1. − n

(108) 1−α Sn ·, fα − fα · ,. 2.57.

(109) 12. Journal of Function Spaces and Applications. we obtain ∞ −α −α α α f· − Sn ·, f ·, f − f ≤ −

(110) 1 k k · S k M,ω kn

(111) 1. M,ω.

(112) n

(113) 1−α Sn ·, f α − f α ·. M,ω. −α k − k

(114) 1−α Sk ·, fα − fα ·. ∞ .

(115). M,ω. kn

(116) 1.

(117) n

(118) 1−α Sn ·, fα − fα · ≤c. M,ω. ∞ . k−α − k

(119) 1−α Ek f M,ω

(120) n

(121) 1−α En f α. . kn

(122) 1. .

(123) C. M,ω. k−α − k

(124) 1−α Ek f M,ω

(125) n

(126) 1−α En fα. ∞ . kn

(127) 1. M,ω. . 2.58. Consequently, α fx − Sn x, f f ≤ cE k M,ω

(128) cEn . M,ω. . fα. ≤ cEn f α. ∞ . k−α − k

(129) 1−α

(130) n

(131) 1−α. kn

(132) 1. . M,ω. . M,ω. ∞ . k−α − k

(133) 1−α

(134) n

(135) 1−α 2.59. kn

(136) 1 ∞ . . k−α − k

(137) 1. −α .

(138) n

(139) 1−α. kn

(140) 1. c α ≤ , α En f M,ω n

(141) 1 and 2.48 holds. Now 2.47 and 2.48 imply the result. Lemma 2.4. Let Tn ∈ Tn , n 0, 1, 2, . . ., M ∈ QC2θ 0, 1, and ω ∈ ApM . If α ∈ R

(142) , then ΩαM,ω. π c α Tn , ≤ α Tn M,ω n

(143) 1 n

(144) 1. 2.60. hold, where the constant c > 0 is dependent only on α and M. Proof. First we prove that if 0 < α < β, then β. ΩM,ω Tn , · ≤ cΩαM,ω Tn , ·.. 2.61.

(145) Journal of Function Spaces and Applications. 13. It is easily seen that if α ≤ β, α, β ∈ Z

(146) , then 2.61 holds. Now, we assume 0 < α < β ≤ 1. In this case, putting Kx : σtα Tn x, we have. β−α. σt. t t ∞ . 1 β−α · · · K x

(147) u1

(148) · · · uj du1 · · · duj −1j j j 2t −t −t j0 ∞ 1 β−α −1j j 2tj j0 t t ∞ 1 α × ··· −1k k 2tk −t −t k0 t t . × · · · Tn x

(149) u1

(150) · · · uj

(151) uj

(152) 1

(153) · · · uj

(154) k duj

(155) 1 · · · duj

(156) k du1 · · · duj. Kx . −t. −t. t t ∞ ∞ . 1 α j

(157) k β − α · · · Tn x

(158) u1

(159) · · · uj

(160) k du1 · · · duj

(161) k −1 j

(162) k j k 2t −t −t j0 k0 ⎧ ⎫ t t ∞ ⎨ υ ⎬ 1 β − α α ··· Tn {x

(163) u1

(164) · · · uυ }du1 · · · duυ −1υ−μ υ ⎩ ⎭ υ−μ μ 2t −t −t υ0 μ0 t t ∞ υ 1 β−α α · · · T

(165) . . . u · · · du

(166) u −1υ x du n 1 υ 1 υ υ υ − μ μ 2t −t −t υ0 μ0 t t ∞ 1 β β ··· Tn x

(167) u1

(168) . . . uυ du1 · · · duυ σt Tn x. −1υ υ υ 2t −t −t υ0 2.62. Then, β σt Tn . M,ω. β−α σt K . M,ω. ≤ cσtα Tn M,ω ,. 2.63. and hence 2.61 holds. We note that if r1 , r2 ∈ Z

(169) , α1 , β1 ∈ 0, 1 taking α : r1

(170) α1 , β : r2

(171) β1 for the remaining cases r1 r2 , α1 < β1 or r1 < r2 , α1 < β1 or r1 < r2 , α1 > β1 , it can easily be obtained from the last inequality that the required inequality 2.61 holds. Now we will show 2r that if ∈ WM T, ω, r 1, 2, 3, . . ., then . ΩrM,ω , δ ≤ cδ2r 2r . M,ω. .. 2.64. Putting. gx :. r I − σhi x, i2. 2.65.

(172) 14. Journal of Function Spaces and Applications. we have I − σh1 gx r . 1 2h1. I − σhi x . i1. h1 −h1. r I − σhi x, i1. . 1 gx − gx

(173) t dt − 8h1. h1 t u 0. 0. −u. g x

(174) sds du dt. 2.66. Therefore, r I − σhi x i1 M,ω h t u 1 1 . sup g x

(175) sds du dt|vx|ωxdx : M|vx|ωxdx ≤1 8h1 0 −u T 0 T u 1 1 h1 t 2u g du dt

(176) sds ≤ x 8h1 0 0 2u −u M,ω c h1 t ≤ 2ug M,ω du dt ch21 g M,ω . 8h1 0 0 2.67 Since g x . r . I − σhi x,. 2.68. i2. we obtain that ΩrM,ω. . , δ ≤ sup. 0<hi ≤δ. . . ch21 g M,ω. i1,2,...,r. r cδ I − σhi x i2 2. r cδ2 sup I − σhi x 0<hi ≤δ i2 i2,...,r. M,ω. cδ2 Ωr−1 M,ω , δ. M,ω. 4 2r 2r ≤ cδ4 Ωr−2 , δ ≤ · · · ≤ Cδ M,ω . . 2.69. M,ω. .. Using 2.61, 2.64, and Lemma 2.2, we get 2α π π π 2α α ΩαM,ω Tn , ≤ cΩM,ω Tn , ≤c Tn M,ω n

(177) 1 n

(178) 1 n

(179) 1 c c α α−α−α α ≤

(180) 1 n T T n n M,ω M,ω n

(181) 1α n

(182) 12α. 2.70.

(183) Journal of Function Spaces and Applications. 15. which is the required result 2.60 for α ≥ 1. On the other hand in case of 0 < α < 1 the inequality 2.60 can be obtained by Marcinkiewicz Multiplier Theorem for LM,ω T where M ∈ QC2θ 0, 1 and ω ∈ ApM . Definition 2.5. For f ∈ LM,ω T, δ > 0, and r 1, 2, 3, . . ., the PeetreK-functional is defined as K. . α δ, f; LM,ω T, WM T, ω. :. inf. α g∈WM T,ω. r f − g

(184) δ g M,ω. M,ω. .. 2.71. Proposition 2.6. Let M ∈ QC2θ 0, 1, ω ∈ ApM , and f ∈ LM,ω T. Then the K-functional 2r T, ω in 2.71 and the modulus ΩrM,ω f, δ, r 1, 2, 3, . . ., are equivalent. Kδ2r , f; LM,ω T, WM 2r Proof. If h ∈ WM T, ω, then we have. . ΩrM,ω f, δ ≤ cf − hM,ω

(185) cδ2r h2r . M,ω. 2r ≤ cK δ2r , f; LM,ω T, WM T, ω .. 2.72. Putting . Lδ f x : 3δ. −3. δ u t 0. 0. −t. fx

(186) sds dt du,. x ∈ T,. 2.73. we have d2 c Lδ f 2 I − σδ f, 2 dx δ. 2.74. and hence d2r r c L f 2r I − σδ r , dx2r δ δ. r 1, 2, 3, . . . .. 2.75. On the other hand, we find Lδ f ≤ 3δ−3 M,ω. δ u 0. 0. 2tσt f M,ω dt du ≤ cf M,ω .. 2.76. 2r T, ω and Now, let Arδ : I − I − Lrδ r . Then Arδ f ∈ WM. d2r r 2r Aδ f dx. M,ω. d2r r ≤ c 2r Lδ f dx . M,ω. . c c I − σδ r ≤ 2r ΩrM,ω f, δ . M,ω 2r δ δ. 2.77. Since I − Lrδ I − Lδ . r−1 j Lδ , j0. 2.78.

(187) 16. Journal of Function Spaces and Applications. we get I − Lr g δ. M,ω. ≤ cI − Lδ g M,ω ≤ 3cδ−3 ≤ c sup I − σt g M,ω .. δ u 0. 0. 2tI − σt g M,ω dt du. 2.79. 0<t≤δ. Taking into account r f − Ar f I − Lrδ f M,ω δ M,ω. 2.80. by a recursive procedure, we obtain r−1 f − Ar f ≤ c sup I − σt1 I − Lrδ f δ M,ω. M,ω. 0<t1 ≤δ. r−2 ≤ c sup sup I − σt1 I − σt2 I − Lrδ f. 2.81. M,ω. 0<t1 ≤δ 0<t2 ≤δ. r ≤ · · · ≤ c sup I − σti fx 0<ti ≤δ i1. . cΩrM,ω f, δ .. M,ω. i1,2,...,r. Now we can formulate the results. Theorem 2.7. Let M ∈ QC2θ 0, 1 and r ∈ R

(188) . If f ∈ LM,ω T with ω ∈ ApM , then there is a constant c > 0 dependent only on r and M such that for n 0, 1, 2, 3, . . . En f M,ω ≤ cΩrM,ω f,. 1 n

(189) 1. 2.82. holds. Proof. We put k − 1 < r ≤ k, k ∈ Z

(190) . From Remark 2.1i, 2.64, 2.71, Proposition 2.6, and 2k T and n 0, 1, 2, 3, . . . 2.61, we get for every g ∈ WM,ω . En f M,ω ≤ En f − g M,ω

(191) En g M,ω ≤ c f − g M,ω

(192) n

(193) 1−2k g 2k . . . 2k ≤ cK n

(194) 1−2k , f; LM,ω T, WM T, ω ≤ cΩkM,ω f,. ≤ cΩrM,ω f,. 1 . n

(195) 1. 1 n

(196) 1. . M,ω. 2.83.

(197) Journal of Function Spaces and Applications. 17. Theorem 2.8. Let M ∈ QC2θ 0, 1 and r ∈ R

(198) . If f ∈ LM,ω T with ω ∈ ApM , then there is a constant c > 0 dependent only on r and M such that for n 0, 1, 2, 3, . . . n π c ΩrM,ω f, ≤ ν

(199) 1r−1 Eν f M,ω r n

(200) 1 n

(201) 1 ν0. 2.84. holds. Proof. Let Tn ∈ Tn be the best approximating polynomial of f ∈ LM,ω T and let m ∈ Z

(202) . Then, ΩrM,ω. π f, n

(203) 1. . π π r ≤

(204) ΩM,ω T2m , ≤ cE2m f M,ω f − T2m , n

(205) 1 n

(206) 1 π .

(207) ΩrM,ω T2m , n

(208) 1 . ΩrM,ω. 2.85. By Lemma 2.4 we have ΩrM,ω T2m ,. π n

(209) 1. . ≤c. 1 n

(210) 1. r r T2m . M,ω. .. 2.86. Since r. r. T2m x T1 x

(211). ( r r T2ν

(212) 1 x − T2ν x ,. m−1 ' ν0. 2.87. we get ΩrM,ω. π T2m , n

(213) 1. . m−1 c r r r . ≤

(214) T1 T2ν

(215) 1 − T2ν M,ω M,ω n

(216) 1r ν0. 2.88. Fractional Bernstein inequality of Lemma 2.2 gives r r ≤ c2νr T2ν

(217) 1 − T2ν M,ω ≤ c2νr

(218) 1 E2ν f M,ω , T2ν

(219) 1 − T2ν M,ω r r r T1 − T0 ≤ cE0 f M,ω . T1 M,ω. 2.89. M,ω. Hence, ΩrM,ω. T2m ,. π n

(220) 1. . m−1 c ν

(221) 1r E0 f M,ω

(222) ≤ 2 E2ν f M,ω . n

(223) 1r ν0. 2.90.

(224) 18. Journal of Function Spaces and Applications. It is easily seen that 2ν μr−1 Eμ f M,ω , 2ν

(225) 1r E2ν f M,ω ≤ c∗. ν 1, 2, 3, . . . ,. 2.91. μ2ν−1

(226) 1. where. c∗ . ⎧ ⎨2r

(227) 1 , 0 < r < 1, ⎩22r ,. r ≥ 1.. 2.92. Therefore, ΩrM,ω T2m ,. π n

(228) 1. . ⎧ ⎫ 2ν m ⎨ ⎬ . . c r r−1 E

(229) 2 E

(230) C μ E ≤ f f f 0 1 μ M,ω M,ω M,ω ⎭ n

(231) 1r ⎩ ν1 μ2ν−1

(232) 1 ⎧ ⎫ 2m ⎨ ⎬ . c ≤ μr−1 Eμ f M,ω r ⎩E0 f M,ω

(233) ⎭ n

(234) 1 μ1 −1 2 c ν

(235) 1r−1 Eν f M,ω . r n

(236) 1 ν0 m. ≤. 2.93. If we choose 2m ≤ n

(237) 1 ≤ 2m

(238) 1 , then ΩrM,ω. π T2m , n

(239) 1. ≤. n c ν

(240) 1r−1 Eν f M,ω , r n

(241) 1 ν0. E2m f M,ω ≤ E2m−1 f M,ω ≤. n c ν

(242) 1r−1 Eν f M,ω . r n

(243) 1 ν0. 2.94. Last two inequalities complete the proof. From Theorems 2.7 and 2.8 we have the following corollaries. Corollary 2.9. Let M ∈ QC2θ 0, 1 and r ∈ R

(244) . If f ∈ LM,ω T with ω ∈ ApM and . En f M,ω O n−σ ,. σ > 0, n 1, 2, . . . ,. 2.95.

(245) Journal of Function Spaces and Applications. 19. then ⎧ Oδσ ; ⎪ ⎪ ⎪ ⎪ ⎨ 1 ΩrM,ω f, δ O δσ log ; ⎪ δ ⎪ ⎪ ⎪ ⎩ Oδα ;. r > σ, 2.96. r σ, r < σ,. hold. Definition 2.10. Let M ∈ QC2θ 0, 1 and r ∈ R

(246) . If f ∈ LM,ω T and ω ∈ ApM then for 0 < σ < r we set Lipσr, M, ω : {f ∈ LM,ω T : ΩrM,ω f, δ Oδσ , δ > 0}. Corollary 2.11. Let M ∈ QC2θ 0, 1 and r ∈ R

(247) . If f ∈ LM,ω T, ω ∈ ApM , 0 < σ < r and En fM,ω On−σ , n 1, 2, . . ., then f ∈ Lipσr, M, ω. Corollary 2.12. Let 0 < σ < r and let f ∈ LM,ω T, ω ∈ ApM , where M ∈ QC2θ 0, 1. Then the following conditions are equivalent: a f ∈ Lipσr, M, ω, . b En f M,ω O n−σ ,. 2.97. n 1, 2, . . . .. Theorem 2.13. Let f ∈ LM,ω T, ω ∈ ApM , where M ∈ QC2θ 0, 1. If α ∈ 0, ∞ and ∞ να−1 Eν f M,ω < ∞,. 2.98. ν1. then . En f. α. . . ∞ ≤ c n

(248) 1 En f M,ω

(249) να−1 Eν f M,ω. . α. M,ω. 2.99. νn

(250) 1. hold where the constant c > 0 is dependent only on α and M. Proof of Theorem 2.13. The condition 2.98 and Lemma 2.3 implies that f α exist and f α ∈ LM,ω T. Since α f − Sn f α . M,ω. ≤ S2m

(251) 2 f α − Sn f α . M,ω.

(252). ∞ S2k

(253) 1 f α − S2k f α km

(254) 2. M,ω. ,. 2.100. we have for 2m < n < 2m

(255) 1 S2m

(256) 2 f α − Sn f α . M,ω. ≤ c2m

(257) 2α En f M,ω ≤ Cn

(258) 1α En f M,ω .. 2.101.

(259) 20. Journal of Function Spaces and Applications. On the other hand, we find ∞ S2k

(260) 1 f α − S2k f α . M,ω. km

(261) 2. ∞ . ≤c. 2k

(262) 1α E2k f M,ω. km

(263) 2. ≤C. ∞ . 2k . μα−1 Eμ f M,ω. 2.102. km

(264) 2 μ2k−1

(265) 1. c. ∞ . ∞ να−1 Eν f M,ω ≤ c να−1 Eν f M,ω ,. ν2m

(266) 1

(267) 1. νn

(268) 1. and Theorem 2.13 is proved. As a corollary of Theorems 2.7, 2.8, and 2.13 we have the following. α Corollary 2.14. Let f ∈ WM T, ω, ω ∈ ApM , r ∈ 0, ∞, and ∞ ν1. να−1 Eν f M,ω < ∞. 2.103. for some α > 0. In this case for n 0, 1, 2, . . ., there exists a constant c > 0 dependent only on α, r, and M such that ΩrM,ω. ∞ n c π α ≤ f , να−1 Eν f M,ω ν

(269) 1α

(270) r−1 Eν f M,ω

(271) c r n

(272) 1 n

(273) 1 ν0 νn

(274) 1. 2.104. hold.. 3. Near Best Approximants in Weighted Smirnov-Orlicz Space Let w ϕz and w ϕ1 z be the conformal mappings of G∞ and G0 onto the complement D∞ of D, normalized by the conditions ϕ∞ ∞, ϕ1 0 ∞,. limz → ∞ ϕz/z > 0,. 3.1. lim z ϕ1 z > 0,. z→0. respectively. We denote by ψ and ψ1 the inverse mappings of ϕ and ϕ1 , respectively, and T : ∂D. These mappings ψ and ψ1 have in some deleted neighborhood of ∞ the representations. ψw αw

(275) α0

(276). ∞ αk k1 w. , k. α > 0,. ψ1 w . ∞ βl l1. wl. ,. β1 > 0.. 3.2.

(277) Journal of Function Spaces and Applications. 21. Therefore, the functions ψ w , ψw − z. ψ1 w , ψ1 w − z. z ∈ G0 ,. z ∈ G∞. 3.3. are analytic in D∞ and have, respectively, simple zero and zero of order 2 at ∞. Hence they have expansions ∞ ψ w Fk z , ψw − z k0 wk

(278) 1 ∞ ψ1 w Fk 1/z , ψ1 w − z k1 wk

(279) 1. z ∈ G0 , w ∈ D∞ , 3.4 z ∈ G∞ , w ∈ D∞ ,. where Fk z and Fk 1/z are, respectively, Faber Polynomials of degree k for continuums G0 and C \ G0 , with the integral representations 38, pp. 35, 255 1 Fk z 2πi T 1 1 Fk z 2πi T. wk ψ w dw, ψw − z. z ∈ G0 ,. wk ψ1 w dw, z ∈ G∞ , ψ1 w − z k ϕ ς 1 k dς, z ∈ G∞ , k 0, 1, 2, . . . , Fk z ϕ z

(280) 2πi Γ ς − z k ϕ1 ς 1 1 k dς, z ∈ G0 \ {0}. ϕ1 z − Fk z 2πi Γ ς − z. 3.5. 3.6 3.7. We put 1 ak : ak f : 2πi T 1 ak : ak f : 2πi T. f0 w dw, wk

(281) 1 f1 w dw, wk

(282) 1. k 0, 1, 2, . . . , 3.8 k 1, 2, . . .. and correspond the series ∞ ∞ 1 ak Fk z

(283) ak Fk z k0 k1. 3.9. with the function f ∈ L1 Γ, that is, fz ∼. ∞ ∞ 1 . ak Fk z

(284) ak Fk z k0 k1. 3.10.

(285) 22. Journal of Function Spaces and Applications. This series is called the Faber-Laurent series of the function f and the coefficients ak and ak are said to be the Faber-Laurent coefficients of f. For further information about the Faber polynomials and Faber Laurent series, we refer to monographs 39, Chapter I, Section 6, 40, Chapter II, and 38. It is well known that, using the Faber polynomials, approximating polynomials can be constructed 3. The interpolating polynomials can also be used for this aim. In their work 41 under the assumption Γ ∈ C2, α, 0 < α < 1, Shen and Zhong obtain a series of interpolation nodes in G0 and show that interpolating polynomials and best approximating polynomial in Ep G0 , 1 < p < ∞, have the same order of convergence. In 42 considering Γ ∈ C1, α and choosing the interpolation nodes as the zeros of the Faber polynomials, Zhu obtain similar result. In the above-cited works, Γ does not admit corners, whereas many domains in the complex plain may have corners. When Γ is a piecewise Vanishing Rotation curve 43 Zhong and Zhu show that the interpolating polynomials based on the zeros of the Faber polynomials converge to f in the Ep G0 , 1 < p < ∞ norm. A function ω : Γ → 0, ∞ is called a weight on Γ, if ω is measurable and ω−1 {0, ∞} has measure zero. We denote by LM,ω Γ the linear space of Lebesgue measurable functions f : Γ → C satisfying the condition Γ. M αfz ωz|dz| < ∞. 3.11. for some α > 0. The space LM,ω Γ becomes a Banach space with the Orlicz norm f : sup M,Γ,ω. Γ. . fzgzωz|dz| : g ∈ LN,ω Γ; ρ g; N ≤ 1 ,. 3.12. where N is the complementary function of M and . ρ g; N :. Γ. N gz ωz|dz|.. 3.13. The Banach space LM,ω Γ is called weighted Orlicz space on Γ. For z ∈ Γ and > 0 let Γz, : {t ∈ Γ : |t − z| < }. For fixed p ∈ 1, ∞, the set of all weights ω : Γ → 0, ∞ satisfying the relation p−1 1 1 −1/p−1 ωτ|dτ| ωτ < ∞, sup sup |dτ| Γz, Γz, z∈Γ >0 1 ωτ|dτ| ≤ cωz, ∀z ∈ Γ, if p 1 sup >0 Γz,. if p > 1, 3.14. is denoted by Ap Γ. We denote by Lp Γ, ω the set of all measurable functions f : Γ → C such that |f|ω1/p belongs to Lebesgue space Lp Γ, 1 ≤ p < ∞, on Γ..

(286) Journal of Function Spaces and Applications. 23. Definition 3.1. Let ω be a weight on Γ and let EM,ω G0 : {f ∈ E1 G0 : f ∈ LM,ω Γ}, EM,ω G∞ : {f ∈ E1 G∞ : f ∈ LM,ω Γ}, EM,ω G∞ : {f ∈ EM,ω G∞ : f∞ 0}. The classes of functions EM,ω G0 and EM,ω G∞ will be called weighted Smirnov-Orlicz classes with respect to domains G0 and G∞ , respectively. In this chapter, we prove that the convergence rate of the interpolating polynomials based on the zeros of the Fn is the same with the best approximating algebraic polynomials in the weighted Smirnov-Orlicz class EM,ω G0 under the assumption that Γ is a closed Radon curve. This means that interpolating polynomials based on the zeros of the Faber polynomials are near best approximant of f belonging to weighted Smirnov-Orlicz class EM,ω G0 . In the case that all of the zeros of the nth Faber polynomial Fn are in G0 , we denote by Ln f, · the n − 1th interpolating polynomial for f ∈ EM,ω G0 based on the zeros of Fn . Let f ∈ L1 Γ. Then the functions f

(287) and f − defined by 1 f z 2πi

(288). Γ. fς dς, ς−z. z ∈ G0 ,. 1 f z 2πi −. Γ. fς dς, ς−z. z ∈ G∞. 3.15. are analytic in G0 and G∞ , respectively, and f − ∞ 0. We denote by ( ' En f M,Γ,ω : inf f − pM,Γ,ω : p ∈ Pn. 3.16. the minimal error of approximation by polynomials of f, where Pn is the set of algebraic polynomials of degree not greater than n. Let Γ be a rectifiable Jordan curve, f ∈ L1 Γ, and let . 1 SΓ f t : lim ε → 0 2πi. Γ\Γt,ε. fς dς, ς−t. t∈Γ. 3.17. be Cauchy’s singular integral of f at the point t. The linear operator SΓ : f → SΓ f is called the Cauchy singular operator. If one of the functions f

(289) or f − has the nontangential limits a.e. on Γ, then SΓ fz exists a.e. on Γ and also the other one has the nontangential limits a.e. on Γ. Conversely, if SΓ fz exists a.e. on Γ, then both functions f

(290) and f − have the nontangential limits a.e. on Γ. In both cases, the formulae. fz , f

(291) z SΓ f z

(292) 2. fz f − z SΓ f z − 2. 3.18. hold, and hence f f

(293) − f− holds a.e. on Γ see, e.g., 1, page 431.. 3.19.

(294) 24. Journal of Function Spaces and Applications. Lemma 3.2. f Γ is a regular curve, M ∈ QC2θ 0, 1 and ω ∈ ApM Γ, then for every f ∈ EM,ω G0 one has SΓ f, · ≤ cf M,Γ,ω , M,Γ,ω. 3.20. where the constant c depends only on Γ and M. Proof. Assertion 3.20 immediately follows from modular inequality Γ. M SΓ f, t ωtdt ≤ c. Γ. M ft ωtdt. 3.21. given in 7.5.13 of 23. Theorem 3.3. If Γ is a closed Radon curve, M ∈ QC2θ 0, 1 and ω ∈ ApM Γ, then for every f ∈ EM,ω G0 one has f − Ln f, · ≤ cEn f M,Γ,ω , M,Γ,ω. 3.22. where the constant c depends only on Γ and M. Proof. First of all we know 16 that all zeros of the Faber polynomials are in G0 . Since interpolating operator Ln f, · is linear and corresponds f by a polynomial of degree not more than n − 1, we need only to show that, for large values of n, Ln f, · is uniformly bounded in weighted Smirnov-Orlicz class EM,ω G0 . We suppose that Pn−1 is the n − 1th best approximating algebraic polynomial for f in EM,ω G0 . In this case we have f − Ln f, · f − Pn−1 − Ln f − Pn−1 , · M,Γ,ω ≤ 1

(295) Ln f − Pn−1 M,Γ,ω . M,Γ,ω. 3.23. Since we assumed the interpolation nodes as the zeros of the Faber polynomials Fn , using 39, page 59, we have Fn z fz − Ln f, z 2πi. Γ. fς f dς Fn z SΓ z, Fn ςς − z Fn. z ∈ G0. 3.24. and consequently fz − Ln f, z Fn · SΓ f · M,Γ,ω F n. M,Γ,ω. ≤. f SΓ max|Fn z| . z∈Γ Fn M,Γ,ω 3.25. By Lemma 3.2, we get Fn z f f − Ln f, · f ≤ c max ≤ c max . z| |F n F M,Γ,ω M,Γ,ω z∈Γ z,ς∈Γ Fn ς n M,Γ,ω. 3.26.

(296) Journal of Function Spaces and Applications. 25. We set κ : maxz∈Γ |ϑz − 1|, where ϑz π is the exterior angle of the point z ∈ Γ. By the Radon assumption on Γ we get 0 ≤ κ < 1. Then one can find for z ∈ Γ 0, 5 − 0, 5 · κ < |Fn z| < 1, 5

(297) 0, 5 · κ,. 3.27. Fn z 3

(298) κ ≤ . max z,ς∈Γ Fn ς 1−κ. 3.28. 3

(299) κ f − Ln f, · f ≤c . M,Γ,ω M,Γ,ω 1−κ. 3.29. and therefore. From the last inequality we obtain. Since Ln f, · ≤ f M,Γ,ω

(300) f − Ln f, · M,Γ,ω ≤ M,Γ,ω. 3

(301) κ f 1

(302) c , M,Γ,ω 1−κ. 3.30. we obtain that Ln f, · is uniformly bounded in EM,ω G0 , namely,. Ln ≤ c.. 3.31. f − Ln f, · ≤ cf − Pn−1 M,Γ,ω cEn f M,Γ,ω M,Γ,ω. 3.32. Therefore, we conclude that. and interpolating polynomial Ln f, · is near best approximant for f. If Γ is Dini-smooth, then 44 there exist constants c and C such that 0 < c < ψ w < C < ∞,. |w| ≥ 1.. 3.33. Similar inequalities hold also for ψ1 and ϕ 1 , in case of |w| 1 and z ∈ Γ, respectively. We define Poisson polynomial for function f ∈ EM,ω G0 n 2n−1 k ck Fk z, 2− ck Fk z

(303) Vn f, z : n k0 kn

(304) 1. z ∈ G0 .. 3.34.

(305) 26. Journal of Function Spaces and Applications. Theorem 3.4. If Γ is a Dini-smooth curve, M ∈ QC2θ 0, 1 and ω ∈ ApM Γ, then for every f ∈ EM,ω G0 one has f − Vn f, · ≤ cEn f M,Γ,ω , M,Γ,ω. 3.35. where the constant c depends only on Γ and M. Proof. From 3.8 and 3.5, we have . 1 Vn f, z 2π. 2π 2n−1 k −ikt dt k−2n−1 λ|k| ϕ ςe it dς, f ψ e 2πi Γ ς−z 0. 3.36. where z ∈ G0 and. λ|k| :. ⎧ ⎨1, ⎩2 − k , n. 0 ≤ k ≤ n,. 3.37. n

(306) 1 ≤ k ≤ 2n − 1.. If Pn ∈ Pn is near best approximant for f ∈ EM,ω G0 , we get f − Vn f, · ≤ En f M,Γ,ω

(307) Pn − Vn f, · M,Γ,ω . M,Γ,ω. 3.38. Using 1 Pn z 2π. 2π 0. dt Pn ψ eit 2πi Γ. 2n−1 k−2n−1. λk|k| ϕςe−ikt. ς−z. dς,. z ∈ G0 ,. 3.39. we find . Pn z − Vn f, z 2n−1 k −ikt ( dt 1 2π ' it k−2n−1 λ|k| ϕςe it dς, Pn ψ e −f ψ e 2π 0 2πi Γ ς−z. z ∈ G0 . 3.40. Taking in the last inequality, the nontangential boundary values from inside of Γ, z → z0 ∈ Γ and using 3.18, we have . Pn z0 − Vn f, z0. 1 2π ⎡. 2π ' ( Pn ψ eit − f ψ eit dt 0. 1 2n−1 1 k −ikt ⎣ × λ ϕz0 e

(308) 2 k−2n−1 |k| 2πi Γ. 2n−1 k−2n−1. λk|k| ϕςe−ikt. ς − z0. ⎤ dς⎦. 3.41.

(309) Journal of Function Spaces and Applications Since ϕ−2n ς. 2n−1 k−2n−1. 27. λ|k| ϕk ςe−ikt is analytic in G∞ , we have. 1 2πi. 2n−1. k−2n−1. Γ. λ|k| ϕk ςe−ikt. ς − zϕ2n ς. dς 0,. z ∈ G0 ,. 3.42. and taking nontangential limit in 3.42 we get 2n ϕ z0 1 2n−1 1 k −ikt λ|k| ϕ z0 e − 2 k−2n−1 2πi Γ ϕ2n ς. 2n−1 k−2n−1. λ|k| ϕk ςe−ikt. ς − z0. dς,. 3.43. and hence by transformation z0 ψw0 we obtain . Pn z0 − Vn f, z0 2n−1 k −ikt ( dt w02n 1 2π ' it k−2n−1 λ|k| w e it 1 − 2n Pn ψ e −f ψ e ψ w dw. 2π 0 2πi T ψw − ψw0 w 3.44 Since one has 1 2πi. . w02n. 2n−1 k−2n−1. λ|k| wk e−ikt. ψ wdw ψw − ψw0 2n−1 w02n ψ w 1 1 k −ikt 1 − 2n − λ|k| w e dw 2πi T ψw − ψw0 w − w0 w k−2n−1 2n−1 w02n 1 1 1 − 2n

(310) λ|k| wk e−ikt , 2πi T w − w0 k−2n−1 w T. 1−. w2n . 3.45. we can write . Pn z0 − Vn f, z0 ( 1 2π ' it Pn ψ e − f ψ eit dt 2π 0 2n−1 w02n ψ w 1 1 1 − 2n × − λ|k| wk e−ikt dw 2πi T ψw − ψw0 w − w0 w k−2n−1 ( dt w02n 1 1 2π ' it it 1 − 2n Pn ψ e −f ψ e

(311) 2π 0 2πi T w − w0 w ×. 2n−1 . λ|k| wk e−ikt : I1

(312) I2 .. k−2n−1. 3.46.

(313) 28. Journal of Function Spaces and Applications. From equality 1 2πi. . w2n 1 − 02n w. T. . 1 w − w0. 2n−1 . λ|k| wk e−ikt dw . k−2n−1. 2n−1 . λ|k| w0k e−ikt ,. 3.47. k−2n−1. we have. I2 M,Γ,ω. 1 ≤ En f M,Γ,ω 2π. 2π 2n−1 k −ikt λ w e |k| 0 dt. 0 k−2n−1. 3.48. On the other hand, 1. I1 M,Γ,ω ≤ En f M,Γ,ω 2 4π. 2π 2n−1 w02n ψ w 1 k −ikt − λ|k| w e 1− 2n |dw|. T w ψw−ψw0 w−w0 0 k−2n−1 3.49. We denote by A a subarc of T with the center w0 such that it has arc lenght O1/n. In this case w02n ψ w 1 − |dw| 1 − 2n w ψw − ψw0 w − w0 A ψ w 1 2n − ≤ w − w02n |dw| ≤ c − w |w | ψw − ψw0 0 A. 3.50. and, by 1.3, w02n ψ w 1 1 − − |dw| w2n ψw − ψw0 w − w0 T\A ψ ww − w0 − ψw − ψw0 ≤2 |dw| ψw − ψw0 |w − w0 | T\A. . ≤c. 1 1/n. 3.51. . ω ψ , t dt ≤ c. t. Hence,. I1 M,Γ,ω. ≤ En f M,Γ,ω. 2π 2n−1 k −ikt λ w e |k| dt. 0 k−2n−1. 3.52.

(314) Journal of Function Spaces and Applications. 29. Inequalities 3.46, 3.48, and 3.52 imply that ⎧ ⎫ 2π 2n−1 ⎬ ⎨ 2π 2n−1 k −ikt k −ikt Pn − Vn f, ·

(315) ≤ E λ w e λ w e f n |k| |k| 0 ⎭. M,Γ,ω M,Γ,ω ⎩ 0 k−2n−1 0 k−2n−1 3.53 For every w ∈ T, one has 2π 2n−1 k −ikt λ w e |k| ≤ c, 0 k−2n−1. 3.54. and therefore we get the required inequality of Theorem 3.4. Theorem 3.4 signifies that Poisson polynomial is near best approximant for f. For g ∈ LM,ω T, we set 1 σh g w : 2h. h −h. g weit dt,. 0 < h < π, w ∈ T.. 3.55. If M ∈ QC2θ 0, 1 and ω ∈ ApM T, then by Theorem Aii we have σh g ≤ cg M,T,ω , M,T,ω. 3.56. and consequently σh g ∈ LM,ω T for any g ∈ LM,ω T. Definition 3.5. Let M ∈ QC2θ 0, 1, ω ∈ ApM T, and r > 0. The function. ΩrM,T,ω. r r−r g g, δ : sup I − σhi I − σt 0<hi ,t≤δ i1. . ,. δ>0. 3.57. M,T,ω. is called rth modulus of smoothness of g ∈ LM,ω T. It can easily be verified that the function ΩrM,T,ω g, · is continuous, nonnegative, subadditive and satisfy limδ → 0 ΩrM,T,ω g, δ 0 for g ∈ LM,ω T. Let Γ be a Dini-smooth curve and ω be a weight on Γ. We associate with ω the following two weights defined on T by ω0 : ω ◦ ψ,. ω1 : ω ◦ ψ1. 3.58.

(316) 30. Journal of Function Spaces and Applications. and let f0 : f ◦ ψ, f1 : f ◦ ψ1 for f ∈ LM,ω Γ. Then from 3.33, we have f0 ∈ LM,ω0 T and f1 ∈ LM,ω1 T for f ∈ LM,ω Γ. Using the nontangential boundary values of f0

(317) and f1

(318) on T, we define. . ΩrM,Γ,ω f, δ : ΩrM,T,ω0 f0

(319) , δ , .

(320) r r Ω M,Γ,ω f, δ : ΩM,T,ω1 f1 , δ. 3.59. for r, δ > 0. We set En f M,T,ω :. inf f − P M,T,ω ,. P ∈Pn D. En g M,Γ,ω : inf g − RM,Γ,ω , R∈Rn. 3.60. where f ∈ EM,ω D, g ∈ EM,ω G∞ , and Rn is the set of rational functions of the form n −k k0 ak z . Now we can give several applications of approximation theorems of Section 2. Theorem 3.6. Let Γ be a Dini-smooth curve, M ∈ QC2θ 0, 1 and f ∈ LM,ω Γ with ω ∈ A1 Γ. Then there is a constant c > 0 such that for any natural number n r f − Rn ·, f ≤ c Ω M,Γ,ω f, M,Γ,ω. 1 n

(321) 1. . r

(322) Ω M,Γ,ω f,. 1 n

(323) 1. ,. 3.61. where r > 0 and Rn ·, f is the nth partial sum of the Faber-Laurent series of f. Corollary 3.7. Let Γ be a Dini-smooth curve, M ∈ QC2θ 0, 1 and f ∈ EM,ω G0 with ω ∈ A1 Γ. Then there is a constant c > 0 such that for every natural number n r f − Pn ·, f ≤ c Ω M,Γ,ω f, M,Γ,ω. 1 , n

(324) 1. r > 0,. 3.62. where Pn ·, f is the nth partial sum of the Faber series of f. Corollary 3.8. Let Γ be a Dini-smooth curve, M ∈ QC2θ 0, 1 and f ∈ EM,ω G∞ with ω ∈ A1 Γ. Then there is a constant c > 0 such that for every natural number n r f − Rn ·, f ≤ c8 ΩM,Γ,ω f, M,Γ,ω. 1 , n

(325) 1. r > 0,. 3.63. where Rn ·, f is as in Theorem 3.6. Theorem 3.9. Let Γ be a Dini-smooth curve, M ∈ QC2θ 0, 1 and f ∈ EM,ω G0 with ω ∈ A1 Γ. Then for r > 0 there exists a constant c > 0 such that ΩrM,Γ,ω hold.. n c 1 r−1 ≤ r E0 f M,Γ,ω

(326) k Ek f M,Γ,ω f, n n k1. 3.64.

(327) Journal of Function Spaces and Applications. 31. Corollary 3.10. Under the conditions of Corollary 3.7, if En f M,Γ,ω O n−α ,. α > 0, n 1, 2, 3, . . . ,. 3.65. then for f ∈ EM,ω G0 and r > 0. ΩrM,Γ,ω. ⎧ ⎪ Oδα , ⎪ ⎪ ⎨ . 1 α f, δ O δ log , ⎪ δ ⎪ ⎪ ⎩Oδr ,. r > α, r α,. 3.66. r < α.. Definition 3.11. Let M ∈ QC2θ 0, 1 and α ∈ R

(328) . If f ∈ EM,ω G0 , then for 0 < σ < α we set ( ' . α α . Lipσα, M, Γ, ω : f ∈ EM,ω G∞ : Ω M,Γ,ω f, δ Oδ , ' ( . Lipσα, M, Γ, ω : f ∈ EM,ω G0 : ΩαM,Γ,ω f, δ Oδσ , δ > 0 .. 3.67. Corollary 3.12. Let M ∈ QC2θ 0, 1 and α ∈ R

(329) . If f ∈ EM,ω G0 , ω ∈ A1 Γ, 0 < σ < α and En fM,Γ,ω On−σ , n 1, 2, . . ., then f ∈ Lipσα, M, Γ, ω. By Corollaries 3.7 and 3.10 we have the constructive characterization of the class Lipσα, M, Γ, ω. Corollary 3.13. Let 0 < σ < α and f ∈ EM,ω G0 , ω ∈ A1 Γ, where M ∈ QC2θ 0, 1, be fulfilled. Then the following conditions are equivalent: a f ∈ Lipσα, M, Γ, ω. b En fM,Γ,ω On−σ , n 1, 2, . . . . The inverse theorem for unbounded domains has the following form. Theorem 3.14. Let Γ be a Dini-smooth curve, M ∈ QC2θ 0, 1 and f ∈ EM,ω G∞ with ω ∈ A1 Γ. Then there is a constant c > 0 such that for every natural number n r Ω M,Γ,ω. . 1 f, n. . n c r−1 ≤ r E0 f M,Γ,ω

(330) k Ek f M,Γ,ω , n k1. r>0. 3.68. holds. By the similar way to that of EM,ω G0 , we obtain the following corollaries. Corollary 3.15. Under the conditions of Corollary 3.8, if En f M,Γ,ω O n−α ,. α > 0, n 1, 2, 3, . . . ,. 3.69.