Simultaneous and converse approximation theorems in weighted lebesgue spaces

Tam metin

(1)See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/260861163. M athematical I nequalities & A pplications SIMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS IN WEIGHTED LEBESGUE SPACES Article in Mathematical Inequalities and Applications · March 2014 CITATIONS. READS. 2. 92. 2 authors: Yunus Emre Yıldırır. Daniyal M. Israfilov. Balikesir University. Balikesir University. 23 PUBLICATIONS 81 CITATIONS . 83 PUBLICATIONS 612 CITATIONS . SEE PROFILE. Some of the authors of this publication are also working on these related projects:. Approximation in Variable Lebesgue and Smirnov spaces View project. Approximation in Variable Lebesgue and Smirnov Spaces View project. All content following this page was uploaded by Yunus Emre Yıldırır on 18 March 2014.. The user has requested enhancement of the downloaded file.. SEE PROFILE.

(2) M athematical I nequalities & A pplications Volume 14, Number 2 (2011), 359–371. SIMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS IN WEIGHTED LEBESGUE SPACES. Y UNUS E. Y ILDIRIR AND DANIYAL M. I SRAFILOV (Communicated by J. Marshall Ash) Abstract. In this paper we deal with the simultaneous and converse approximation by trigonometric polynomials of the functions in the Lebesgue spaces with weights satisfying so called Muckenhoupt’s A p condition.. 1. Introduction and the main results Let T :=[−π , π ]. A positive almost everywhere (a.e.), integrable function w : T → [0, ∞] is called as a weight function. With any given weight w we associate the wweighted Lebesgue space Lwp (T) consisting of all measurable functions f on T such that f Lwp (T) = f wL p (T) < ∞. Let 1 < p < ∞ and 1/p + 1/q = 1. A weight function w belongs to the Muckenhoupt class A p (T) if ⎛ ⎝1 |I|. . ⎞1/p ⎛ w p (x)dx⎠. I. ⎝1 |I|. . ⎞1/q w−q (x)dx⎠. c. I. with a finite constant c independent of I, where I is any subinterval of T and |I| denotes the length of I . For formulation of the new results we will begin with some required informations. Let ∞ ∞ a0 + ∑ (ak cos kx + bk sin kx) f (x) ∼ ∑ ck eikx = (1) 2 k=1 k=−∞ and f˜(x) ∼. ∞. ∑ (ak sin kx − bk cos kx). k=1. Mathematics subject classification (2010): 41A10, 42A10. Keywords and phrases: Best approximation,weighted Lebesgue space, mean modulus of smoothness, fractional derivative. c , Zagreb Paper MIA-14-29. 359.

(3) 360. Y. E. Y ILDIRIR AND D. M. I SRAFILOV. be the Fourier and the conjugate Fourier series of f ∈ L1 (T), respectively. In addition, we put Sn (x, f ) :=. n. ∑ ck eikx =. k=−n. n a0 + ∑ (ak cos kx + bk sin kx), 2 k=1. n = 1, 2, ..... By L10 (T) we denote the class of L1 (T) functions f for which the constant term c0 in (1) equals zero. If α > 0, then α -th integral of f ∈ L10 (T) is defined as Iα (x, f ) :=. ∑∗ ck (ik)−α eikx ,. k∈Z. where (ik)−α :=|k|−α e(−1/2)π iα sign k and Z∗ := {±1, ±2, ±3, ...}. For α ∈ (0, 1) let d f (α ) (x) := I1−α (x, f ), dx (r) d r+1 = r+1 I1−α (x, f ) f (α +r) (x) := f (α ) (x) dx if the right hand sides exist, where r ∈ Z+ := {1, 2, 3, ...} [14, p. 347] . By c, c(α , ...) we denote the absolute constants or the constants whose values depend only on the parameters given in their brackets. Let x,t ∈ R := (−∞, ∞), r ∈ R+ := (0, ∞) and let tr f (x) :=. ∞. ∑ (−1)k [Ckr ] f (x + (r − k)t),. f ∈ L1 (T),. (2). k=0. where [Ckr ] := r(r−1)...(r−k+1) for k > 1, [Ckr ] := r for k = 1 and [Ckr ] := 1 for k = 0. k! Since [14, p. 14]. r(r − 1)...(r − k + 1) c(r) + |[Ckr ]| = kr+1 , k ∈ Z k! we have C(r) :=. ∞. ∑ |[Ckr ]| < ∞,. k=0. and therefore tr f (x) is defined a.e. on R. Furthermore, the series in (2) converges absolutely a.e. and tr f (x) is measurable [16]. If r ∈ Z+ , then the fractional difference tr f (x) coincides with usual forward difference. Now let. σδr. 1 f (x) := δ. δ. 0. |tr f (x)| dt, 1 < p < ∞.

(4) 361. S IMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS. for f ∈ Lwp (T), w ∈ A p (T). Since the series in (2) converges absolutely a.e., we have σδr f (x) < ∞ a.e. and using boundedness of the Hardy-Littlewood Maximal function [13] in Lwp (T), w ∈ A p (T), we get. r. σ f (x) p c(p, r) f p < ∞. (3) δ Lw L w. Hence, if r ∈ R+ and w ∈ A p (T), 1 < p < ∞, we can define the r-th mean modulus p of smoothness of a function f ∈ Lw (T) as. Ωr ( f , h)Lwp := sup σδr f (x) L p . |δ |h. w. If r ∈ Z+ , then Ωr ( f , h)Lwp coincides with Ky’s mean modulus of smoothness, defined in [9]. R EMARK 1. Let f , f1 , f2 ∈ Lwp (T), w ∈ A p (T), 1 < p < ∞. The r -th mean modulus of smoothness Ωr ( f , h)Lwp , r ∈ R+ , has the following properties: ( i) Ωr ( f , h)Lwp is non-negative and non-decreasing function of h 0. ( ii) Ωr ( f1 + f2 , ·)Lwp Ωr ( f1 , ·)Lwp + Ωr ( f2 , ·)Lwp . ( iii) lim Ωr ( f , h)Lwp = 0. h→0. The best approximation of f ∈ Lwp (T) in the class Πn of trigonometric polynomials of degree not exceeding n is defined by

(5). En ( f )Lwp = inf f − Tn Lwp : Tn ∈ Πn . A polynomial Tn (x, f ) := Tn (x) of degree n is said to be a near best approximant of f if f − Tn Lwp c(p)En ( f )Lwp , n = 0, 1, 2, .... p. p. α (T), α > 0, be the class of functions f ∈ L (T) such that f (α ) ∈ L (T). Let Wp,w w w 1 < p < ∞, α > 0, becomes a Banach space with the norm. (α ). f Wp,w. p. α (T) := f L p + f w. α (T), Wp,w. Lw. In this paper we deal with the simultaneous and converse approximation by trigonometric polynomials of the functions in the Lebesgue spaces with weights satisfying Muckenhoupt’s A p condition. Our new results are the following. α (T), α ∈ R+ := [0, ∞], 1 < p < ∞, and w ∈ A (T). T HEOREM 1. Let f ∈ Wp,w p 0 If Tn ∈ Πn is a near best approximant of f , then. (α ) (α ). f − Tn p cEn ( f (α ) )Lwp , n = 0, 1, 2, .... Lw.

(6) 362. Y. E. Y ILDIRIR AND D. M. I SRAFILOV. with a constant c = c(p, α ) > 0. This simultaneous approximation theorem in case of α ∈ Z+ for Lebesgue spaces L (T), 1 p ∞, was proved in [3]. Detailed information on simultaneous weighted approximation can be found in the book [4]. p. r (T), r ∈ R+ , 1 < p < ∞, and w ∈ A (T), then T HEOREM 2. If f ∈ Wp,w p. Ωr ( f , h)Lwp chr f (r) p , 0 < h π Lw. with a constant c = c(p, r) > 0. In case of r ∈ Z+ , for the usual nonweighted modulus of smoothness defined in the Lebesgue spaces L p (T), 1 p ∞, this inequality was proved in [11] and for the general case r ∈ R+ was obtained in [2] (See also [16]). In case of r ∈ Z+ , w ∈ A p (T), 1 < p < ∞, this inequality in the weighted Lebesgue spaces Lwp (T) was proved in [9]. T HEOREM 3. Let f ∈ Lwp (T), 1 < p < ∞, and w ∈ A p (T). Then for a given r ∈ R+ , and γ = min {2, p}. 1/γ n c γ r γ −1 Ωr ( f , π /(n + 1))Lwp ∑ (k + 1) Ek ( f )Lwp (n + 1)r k=0 with a constant c independent of n and f . In the space L p (T), 1 p ∞, this inequality was proved in [16] without γ . In case of r ∈ Z+ in the spaces Lwp (T), w ∈ A p (T), 1 < p < ∞, this theorem was proved in [9] without γ . For the positive and even integer r this theorem in spaces Lwp (T), w ∈ A p (T), by using Butzer-Wehrens’s type modulus of smoothness was obtained in [5]. The analogues of some classical theorems for best polynomial approximation in weighted spaces with doubling weights were proved in [12]. T HEOREM 4. Let f ∈ Lwp (T), 1 < p < ∞, and w ∈ A p (T). If ∞. γ. ∑ kαγ −1 Ek ( f )Lwp < ∞. k=1. α (T) and the estimate for α ∈ (0, ∞) and γ = min {2, p} , then f ∈ Wp,w ⎧. 1/γ ⎫ ⎨ ⎬ ∞ γ En ( f (α ) )Lwp c nα En ( f )Lwp + kαγ −1 Ek ( f )Lwp ∑ ⎩ ⎭ k=n+1. (4). holds with a constant c independent of n and f . In the space L p (T), 1 p ∞, this inequality for α ∈ Z+ was proved without γ in [15]. In case of α ∈ Z+ , in Lwp (T), w ∈ A p (T), 1 < p < ∞, an inequality of type (4) was proved in [7]..

(7) 363. S IMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS. C OROLLARY 1. Let f ∈ Lwp (T), 1 < p < ∞, and w ∈ A p (T) and r > 0. If ∞. γ. ∑ kαγ −1 Ek ( f )Lwp < ∞. k=1. α and for n = 0, 1, 2, ... for α ∈ (0, ∞) and γ = min {2, p} , then f ∈ Wp,w. Ωr ( f (α ) , π /(n + 1))Lwp ⎧ 1/γ 1/γ ⎫ ⎨ n ⎬ ∞ c γ γ (α +r)γ −1 αγ −1 p p k E ( f ) + k E ( f ) ∑ ∑ L L k−1 k w w ⎭ (n + 1)r ⎩ k=1 k=n+1 with a constant c independent of n and f . In cases of α , r ∈ Z+ and α , r ∈ R+ , this corollary in the spaces L p (T), 1 p ∞, was proved without γ in [18] (See also [15]) and in [17], respectively. For the weighted Lebesgue spaces Lwp (T), 1 < p < ∞, when w ∈ A p (T), and α , r ∈ Z+ , similar type inequality was obtained using generalized modulus of continuity for the derivatives of f ∈ Lwp (T) in [7]. 2. Auxiliary results L EMMA 1. Let w ∈ A p (T) and r ∈ R+ , 1 < p < ∞. If Tn ∈ Πn , n 1, then there exists a constant c > 0 depends only on r and p such that. (r). Ωr (Tn , h)Lwp chr Tn p , 0 < h π /n. Lw. Proof. Since t r [r] tr Tn x − t = ∑ 2i sin ν cν eiν x , 2 2 ν ∈Z∗ n. [r] (r−[r]) t Tn. . [r] x− t 2. =. ∑∗. ν ∈Zn. t [r] (iν )r−[r] cν eiν x 2i sin ν 2. with Z∗n := {±1, ±2, ±3, ...}, [r] ≡ integer part of r, putting 2 t r−[r] t [r] sin z ϕ (z) := 2i sin z (iz)r−[r] , g(z) := , − n z n, g(0) := t r−[r] , 2 z 2 we get [r] (r−[r]). t Tn. [r] [r] x − t = ∑ ϕ (ν )cν eiν x , tr Tn x − t = ∑ ϕ (ν )g(ν )cν eiν x . 2 2 ν ∈Z∗ ν ∈Z∗ n. n.

(8) 364. Y. E. Y ILDIRIR AND D. M. I SRAFILOV. Taking into account the fact that [16] ∞. ∑. g(z) =. dk eikπ z/n. k=−∞. uniformly in [−n, n], with d0 > 0, (−1)k+1 dk 0, d−k = dk (k = 1, 2, ...) , we have ∞ kπ r − [r] [r] (r−[r]) r t Tn (·) = ∑ dk t Tn + t . ·+ n 2 k=−∞ Consequently we get. δ. 1. r. | T (·)| dt t n. δ. 0. p. Lw. δ. 1 =. δ. 0. . ∞ kπ r − [r] . [r] (r−[r]) + t dt. ·+ ∑ dk t Tn . k=−∞ n 2. δ ∞. 1 ∑ |dk |. δ. 0 k=−∞. p. Lw. . . [r] (r−[r]) r − [r] k π t Tn + t dt. ·+. n 2. p. Lw. and since [19, p. 103] [r] (r−[r]) t Tn (·). =. t 0. t. (r). ... Tn (· + t1 + ...t[r] )dt1 ...dt[r] 0. we find. δ . . 1 [r] (r−[r]). kπ r − [r] . + t dt. Ωr (Tn , h)Lwp sup ∑ |dk |. ·+. δ t Tn n 2 |δ |hk=−∞. p. 0 Lw. . . δ t t ∞ . 1 (r) r − [r] k π. + t + t1 + ...t[r] dt1 ...dt[r] dt. = sup ∑ |dk |. ... Tn ·+. n 2 |δ |hk=−∞ p. δ 0 0 0 Lw. δ δ δ. ∞. 1 1 Tn(r) · + kπ + r − [r] t + t1 + ...t[r] dt1 ...dt[r] dt. h[r] sup ∑ |dk |. .... δ δ [r]. n 2 |δ |hk=−∞. p. 0 0 0 Lw ⎫ ⎧. δ δ δ. ⎨ ⎬ ∞. 1 1 r− [r] π k (r) [r]. Tn t+t1 +...t[r] dt dt1 ...dt[r]. h sup ∑ |dk | [r] ... ·+ +. ⎭ n 2 |δ |hk=−∞. p. δ 0 0 ⎩δ 0 Lw. δ . ∞. 1 (r) kπ r − [r] . + t dt. c(p, r)h[r] sup ∑ |dk |. ·+. δ Tn. n 2 |δ |hk=−∞. ∞. p. 0. kπ r−[r]. . 1 ·+ n + 2 δ ∞ (r) . [r] c(p, r)h sup ∑ |dk | r−[r] Tn (u) du. |δ |hk=−∞. 2 δ k π ·+ n. Lw. . p. Lw.

(9) 365. S IMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS. On the other hand [16] ∞. ∑. |dk | < 2g(0) = 2t r−[r] , 0 < t π /n. k=−∞. and for 0 < t < δ < h π /n we have ∞. ∑. |dk | < 2g(0) = 2hr−[r] .. k=−∞ p. Therefore the boundedness of Hardy-Littlewood maximal function in Lw (T) implies that. (r). Ωr (Tn , h)Lwp c(p, r)hr Tn p . Lw. By similar way for 0 < −h π /n, the same inequality also holds and the proof of Lemma 1 is completed. 3. Proof of the main results Proof of Theorem 1. We set Wn ( f ) := Wn (x, f ) := Since we have. (α ). (α ). f (·) − Tn (·, f ). 1 2n Sν (x, f ), n = 0, 1, 2, .... n + 1 ν∑ =n (α ). Wn (·, f (α ) ) = Wn (·, f ),. p. Lw. (α ). (α ) f (α ) (·) − Wn(·, f (α ) ) p + Tn (·,Wn ( f )) − Tn (·, f ) p Lw Lw. (α ). (α ) + Wn (·, f ) − Tn (·,Wn ( f )) p =: I1 + I2 + I3 . Lw. We denote by Tn∗ (x, f ) the best approximating polynomial of degree at most n to f in Lwp (T). In this case, from the boundedness of in Lwp (T) we have. I1 f (α ) (·) − Tn∗ (·, f (α ) ) p + Tn∗ (·, f (α ) ) − Wn (·, f (α ) ) p Lw Lw. (α ) p ∗ (α ) (α ). c(p)En ( f )Lw + Wn (·, Tn ( f ) − f ) p c(p, α )En ( f (α ) )Lwp . Lw. By [10, Theorem 1] I2 c(p, α )nα Tn (·,Wn ( f )) − Tn (·, f )Lwp.

(10) 366. Y. E. Y ILDIRIR AND D. M. I SRAFILOV. and I3 c(p, α )(2n)α Wn (·, f ) − Tn (·,Wn ( f ))Lwp c(p, α )(2n)α En (Wn ( f ))Lwp .. Now we have Tn (·,Wn ( f )) − Tn (·, f )Lwp Tn (·,Wn ( f )) − Wn (·, f )Lwp + Wn (·, f ) − f (·)Lwp + f (·) − Tn (·, f )Lwp c(p)En (Wn ( f ))Lwp + c(p)En( f )Lwp + c(p)En( f )Lwp . Since En (Wn ( f ))Lwp c(p)En ( f )Lwp , we get. (α ) (α ). f (·) − Tn (·, f ). c(p, α )En ( f. (α ). p. Lw. )Lwp + c(p)nα En (Wn ( f ))Lwp + c(p)nα En ( f )Lwp. +c(p, α )(2n)α En (Wn ( f ))Lwp. c(p, α )En ( f (α ) )Lwp + c(p, α )nα En ( f )Lwp . Since [1]. c(p, α ) En ( f (α ) )Lwp , (n + 1)α. (α ) (α ). f (·) − Tn (·) p cEn ( f (α ) )Lwp En ( f )Lwp . we obtain. (5). Lw. and the proof is completed.. . Proof of Theorem 2. Let Tn ∈ Πn be the trigonometric polynomial of the best approximation of f in Lwp (T) metric. By Remark 1 (ii), Lemma 1 and (3) we get Ωr ( f , h)Lwp Ωr (Tn , h)Lwp + Ωr ( f − Tn , h)Lwp. (r). c(p, r)hr Tn p + c(p, r)En ( f )Lwp , 0 < h < π /n. Lw. Using (5), the direct inequality in [9, Theorem 2] and the inequality. l Ωl ( f , h)Lwp chl f (l) p , f ∈ Wp,w (T), l = 1, 2, 3, ..., Lw. given in [9, Theorem 1], we have c(p, r) E ( f (r−[r]) )Lwp Ω[r] r−[r] n (n + 1) (n + 1)r−[r] . 2π [r]. c(p, r). (r). f. p. Lw (n + 1)r−[r] n + 1. En ( f )Lwp . c(p, r). . f (r−[r]) ,. 2π n+1. p. Lw.

(11) 367. S IMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS. On the other hand, by Theorem 1 we find. (r). (r). Tn p Tn − f (r) p + f (r) p Lw Lw L. w. c(p, r)En ( f (r) )Lwp + f (r) p c(p, r) f (r). Lw. p. Lw. .. Choosing h with π /(n + 1) < h π /n, (n = 1, 2, 3, ...) , we obtain. Ωr ( f , h)Lwp c(p, r)hr f (r) p Lw. and we are done.. . Proof of Theorem 3. Let Sn be the n − th partial sum of the Fourier series of f ∈ Lwp (T), w ∈ A p (T) and let m ∈ Z+ . Thanks to the Theorem of Hunt-MuckenhouptWheeden [6], we obtain that the best approximation by trigonometric polynomials in Lwp (T) with w ∈ A p (T) has the same order as deviation by the partial sum of the Fourier series. It means that for ϕ ∈ Lwp ϕ − Sn(ϕ )Lwp cEn (ϕ )Lwp with a positive constant c independent on ϕ and n. By Remark 1 ( ii) and (3) Ωr ( f , π /(n + 1))Lwp Ωr ( f − S2m , π /(n + 1))Lwp + Ωr (S2m , π /(n + 1))Lwp c(p, r)E2m ( f )Lwp + Ωr (S2m , π /(n + 1))Lwp. and by Lemma 1,. . Ωr (S2m , π /(n + 1)). p Lw. π c(p, r) n+1. Since (r). (r). S2m (x) = S1 (x) + we have . π Ωr (S2m , π /(n + 1))Lwp c(p, r) n+1. r. (r). S2m. p. Lw. , n + 1 2m .. m−1 . ∑. ν =0.

(12) (r) (r) S2ν +1 (x) − S2ν (x) ,. ⎧. ⎫. r ⎨. m−1 ⎬. (r). (r) (r). S2ν +1 − S2ν. . (6). S +. p⎭ ⎩ 1 Lwp ν∑ =0 Lw. Applying the weighted version of Littlewood-Paley’s theorem [8] and following the method used in [7], we obtain for 1 < p 2. m−1 . m−1 2ν +1. . (r) (r). ∑ S2ν +1 (x) − S2ν (x) = ∑ ∑ Bk,r (x). ν =0. p. ν =0 k=2ν +1. p Lw. Lw. 2 ⎞ 12. m−1 2ν +1. c ⎝ ∑ ∑ kr Bk,r (x) ⎠ p. ν =0 k=2ν +1 ⎛. Lw.

(13) 368. Y. E. Y ILDIRIR AND D. M. I SRAFILOV. p ⎞ 1p. c ⎝ ∑ ∑ kr Bk,r (x) ⎠. p ν ν =0 k=2 +1 ⎛. m−1 2ν +1. Lw. ∑. =c. p. (r) (r). S2ν +1 (x) − S2ν (x) p. m−1. ν =0. 1. p. Lw. where Bk,r (x) is the r − th derivative of (ak cos kx + bk sin kx), and for p > 2. m−1 . m−1 2ν +1. . (r) (r). ∑ S2ν +1 (x) − S2ν (x) = ∑ ∑ kr Bk,r (x). ν =0. p. ν =0 k=2ν +1. p Lw. Lw. ⎛. 2 ⎞ 12. m−1 2ν +1. c ⎝ ∑ ∑ kr Bk,r (x) ⎠. p ν ν =0 k=2 +1 Lw. 2 ⎞ 12. m−1 2ν +1. c ⎝ ∑ ∑ kr Bk,r (x) ⎠. p ν ν =0 k=2 +1 ⎛. Lw. ∑. =c. 2. (r) (r). S2ν +1 (x) − S2ν (x) p. m−1. ν =0. 1 2. Lw. .. Consequently, we have. m−1 . (r) (r). ∑ S2ν +1 (x) − S2ν (x). ν =0. c. p Lw. γ. (r) (r). S2ν +1 (x) − S2ν (x) p. m−1. ∑. ν =0. 1 γ. Lw. , γ = min{p, 2}.. Hence, by [10, Theorem 1], we get. (r). (r). S2ν +1 (x) − S2ν (x) p c(p, r)2ν r S2ν +1 (x) − S2ν (x)Lwp c(p, r)2ν r+1 E2ν ( f )Lwp Lw. and. (r). S1. p Lw. (r) (r). = S1 − S0. p. Lw. c(p, r)E0 ( f )Lwp .. Then from (6) we have Ωr (S2m , π /(n + 1))Lwp c(p, r). π n+1. ⎧ r ⎨ ⎩. E0 ( f )Lwp +. m−1. γ. ∑ 2(ν +1)rγ E2ν ( f )Lwp. ν =0. 1γ ⎫ ⎬ ⎭. .. It is easily seen that γ. 2(ν +1)rγ E2ν ( f )Lwp c(r). 2ν. ∑. μ =2ν −1 +1. μ γ r−1 Eμγ ( f )Lwp , ν = 1, 2, 3, ..... (7).

(14) 369. S IMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS. Therefore, Ωr (S2m , π /(n + 1))Lwp ⎧ ⎞1 ⎫ ⎛ γ⎪ r ⎪ ν ⎨ ⎬ m−1 2 π γ r−1 γ ⎠ p c(p, r) μ E ( f ) E0 ( f )Lwp + 2r E1 ( f )Lwp + c(r) ⎝ ∑ μ ∑ Lw ⎪ n+1 ⎪ ⎩ ⎭ ν =0 μ =2ν −1 +1 ⎧ ⎫ m 1γ r ⎨ ⎬ 2 π γ γ r−1 c(p, r) E0 ( f )Lwp + ∑ μ Eμ ( f )Lwp ⎭ n+1 ⎩ μ =1 . π c(p, r) n+1. 1γ r 2m −1 γ r−1 γ ∑ (ν + 1) Eν ( f )Lwp . ν =0. If we choose 2m n + 1 2m+1 , then c(p, r) Ωr (S2m , π /(n + 1))Lwp (n + 1)r. n. ∑ (ν. ν =0. γ + 1)γ r−1 Eν ( f )Lwp. 1γ .. Taking also the relation E2m ( f )Lwp E2m−1 ( f )Lwp. c(p, r) (n + 1)r. n. ∑ (ν. ν =0. 1 γ. γ + 1)γ r−1 Eν ( f )Lwp. . into account we obtain the required inequality of Theorem 3.. Proof of Theorem 4. If Tn is the best approximating polynomial of f , then by [10, Theorem 1]. (α ) (α ). T2m+1 − T2m p c(p, α )2(m+1)α E2m ( f )Lwp Lw. and hence by this inequality, (7) and hypothesis of Theorem 4 we have ∞. ∞. ∞. m=1. m=1. m=1. (α ). ∞. c(p, α ) ∑ 2(m+1)α E2m ( f )Lwp m=1 ∞. c(p, α ) ∑. 2m. ∑. m=1 j=2m−1 +1 ∞. jα −1 E j ( f )Lwp. c(p, α ) ∑ jα −1 E j ( f )Lwp < ∞. j=2. (α ). α (T) = ∑ T2m+1 − T2m L p + ∑ T m+1 − T2m ∑ T2m+1 − T2m Wp,w 2 w. p. Lw.

(15) 370. Y. E. Y ILDIRIR AND D. M. I SRAFILOV. Therefore. ∞. α (T) < ∞, ∑ T2m+1 − T2m Wp,w. m=1. α (T). Since T m → f in the which implies that {T2m } is a Cauchy sequence in Wp,w 2 p α Banach space Lw (T ), we have f ∈ Wp,w (T). It is clear that. En ( f (α ) )Lwp f (α ) − Sn f (α ) p Lw. ∞ . . (α ) (α ). (α ) (α ). S2m+2 f − Sn f p + ∑ S2k+1 f − S2k f. .. p. k=m+2 Lw Lw. By [10, Theorem 1]. S2m+2 f (α ) − Sn f (α ). p. Lw. c(p, α )2(m+2)α En ( f )Lwp c(p, α )(n + 1)α En ( f )Lwp. for 2m < n < 2m+1 . On the other hand, following the method given in the proof of Theorem 3, we get. ∞ . ∑ S2k+1 f (α ) −S2k f (α ). k=m+2. c p. Lw. ∞. ∑. k=m+2. γ. (α ). (α ). S2k+1 (x)−S2k (x) p. Lw. 1 γ. , γ = min{p, 2}. Since by [10, Theorem 1]. (α ). (α ). S2k+1 (x) − S2k (x). p. Lw. c(p, α )2kα S2k+1 (x) − S2k (x)Lwp c(p, α )2kα +1 E2k ( f )Lwp ,. we get. ∞ . (α ) (α ). ∑ S2k+1 f − S2k f. k=m+2. ∞. ∑. c p. k=m+2. Lw. 1 γ 2γ kα +1 E2k ( f )Lwp. γ. .. Therefore, we have. ∞ . ∑ S2k+1 f (α ) − S2k f (α ). k=m+2. c p. Lw. for 2m < n < 2m+1 . This completes the proof.. . ∞. γ kγα −1 Ek ( f )Lwp k=n+1. ∑. 1 γ.

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