# On the approximation by trigonometric polynomials in weighted lorentz spaces

Tam metin

(2) 68. Trigonometric polynomials in weighted Lorentz spaces. In inequality (1.1), En (f ) denotes the best approximation of a 2π periodic continuous function f by trigonometric polynomials of degree ≤ n, i.e., En (f ) = inf max |f (x) − Tk (x)| , x∈[0,2π]. where the inﬁmum is taken with respect to all trigonometric polynomials of degree k ≤ n, and ω(f, δ) = sup. max |f (x + h) − f (x)|. |h|≤δ x∈[0,2π]. denotes the modulus of continuity of f. The analog of Jackson’s inequality is valid also for the integral metrics and moduli of continuity of higher orders (see, e.g., [21, Section 5.3.1]). Yet by the year 1912, S. Bernstein obtained the estimate inverse to Jackson’s inequality in the space of continuous functions for some special cases [3]. Later, Quade [17], brothers A. and M. Timan [22], S. B. Stechkin [19], M. Timan [20], etc. proved such inverse estimates, including the case of of the spaces Lp , 1 < p < ∞. Inequalities of this type played an important role in the investigation of properties of the conjugate functions [1], in the study of absolutely convergent Fourier series [18], and in related problems. In the case of Lebesgue spaces, the inverse inequalities for classical moduli of smoothness and the best approximation theorems were obtained in papers [20], [4]. In [10], this result was extended to reﬂexive Orlicz spaces. For the study of the approximation problems in weighted Lebesgue and Orlicz spaces we refer to [7], [13], [9], [12], [23]. The order of the modulus of smoothness, as it has been shown in [20] and [4], depends not only on the rate of the best approximation but also on the metric of the spaces. In the present paper we reveal that the similar inﬂuence in weighted Lorentz spaces Lps is expressed not only in terms of the ”leading” parameter p, but also in terms of the second parameter s. In the role of structural characteristic we consider the general modulus of continuity deﬁned by the Steklov means. It is caused by the failure of the shift operator continuity in the weighted Lebesgue spaces. The generalized shift operator suits well for the spaces mentioned above. Let T = [−π, π) and w : T → R1 be an almost everywhere positive, integrable function. Let fw∗ (t) be a nondecreasing rearrangement of f : T → R1 with respect to the Borel measurew(e) = w(x)dx, i.e., e. fw∗ (t) = inf {τ ≥ 0 : w (x ∈ T : |f (x)| > τ ) ≤ t} ..

(3) V. Kokilashvili, Y. E. Yildirir. 69. Let 1 < p, s < ∞ and let Lps w (T) be a weighted Lorentz space, i.e., the set of all measurable functions for which ⎞1/s ⎛ s dt s f Lps = ⎝ (f ∗∗ (t)) t p ⎠ < ∞, w t T. where f ∗∗ (t) =. 1 t. t 0. fw∗ (u)du.. By En (f )Lps we denote the best approximation of f ∈ Lps w (T) by w trigonometric polynomials of degree ≤ n, i.e., = inf f − Tk Lps , En (f )Lps w w where the inﬁmum is taken with respect to all trigonometric polynomials of degree k ≤ n. The generalized modulus of smoothness of a function f ∈ Lps w (T) is deﬁned as. l. ps , δ > 0, Ωl (f, δ)Lw = sup Π (I − Ahi ) f. 0<hi <δ. i=1. Lps w. where I is the identity operator and 1 (Ahi f ) (x) := 2hi. x+h i. f (u)du. x−hi. The weights w used in the paper are those which belong to the Muckenhoupt class Ap (T), i.e., they satisfy the condition. sup. 1 |I|. I. ⎛ w(x)dx ⎝. 1 |I|. I. ⎞p−1 . w1−p (x)dx⎠. < ∞,. p =. p p−1. where the supremum is taken with respect to all the intervals I with length ≤ 2π and |I| denotes the length of I. Whenever w ∈ Ap (T), 1 < p, s < ∞, the Hardy-Littlewood maximal function of every f ∈ Lps w (T), and therefore the average Ahi f belong (T) ([5, Theorem 3]). Thus Ωl (f, δ)Lps makes sense for every to Lps w w w ∈ Ap (T). We use the convention that c denotes a generic constant, i.e. a constant whose values can change even between diﬀerent occurrences in a chain of inequalities..

(4) 70. Trigonometric polynomials in weighted Lorentz spaces. 2.. Main results. In the present paper we prove the following results. Theorem 1. Let 1 < p < ∞ and 1 < s ≤ 2 or p > 2 and s ≥ 2. Let w ∈ Ap (T). Then there exists a positive constant c such that (2.1). n 1/γ

(5) c 1 2lγ−1 γ ≤ 2l k Ek−1 (f )Lps Ωl f, w n Lps n w k=1. for arbitrary f ∈ Lps w (T) and natural n, where γ = min(s, 2). Theorem 2. Let 1 < p < 2 < s < ∞ and let w ∈ Ap (T). Then for arbitrary p0 , 1 < p0 < p, there exists a positive constant c such that n 1/p0

(6) c 1 2lp0 −1 p0 ≤ 2l k Ek−1 (f )Lps Ωl f, w n Lps n w k=1. for arbitrary f ∈ Lps w (T) and natural n. Theorem 3. Let 1 < p < ∞ and 1 < s ≤ 2 or p > 2 and s ≥ 2. Let w ∈ Ap (T) and f ∈ Lps w (T). Assume that (2.2). ∞

(7). k rγ−1 Ekγ (f )Lps <∞ w. k=1. for some natural number r and γ = min(s, 2). Then there exists the absolutely continuous (r − 1)th order derivative f (r−1) (x) such that f (r) ∈ Lps w (T) and ⎧ ∞ 1/γ ⎫ ⎬ ⎨

(8) r rγ−1 γ ps + ps (2.3) En (f (r) )Lps ≤ c E (f ) k E (f ) n n L L k w w w ⎭ ⎩ k=n+1. for arbitrary natural n, where γ = min(s, 2) and the constant c does not depend on f and n.. Theorem 4. Let 1 < p < ∞ and 1 < s ≤ 2 or p > 2 and s ≥ 2 . Assume that (2.2) is fulﬁlled for some natural number r and γ = min(s, 2). Then.

(9) V. Kokilashvili, Y. E. Yildirir. 71. there exists a positive constant c such that. (2.4). n 1/γ

(10) c (r) 1 (r+2l)γ−1 γ ps Ωl f , ≤ 2l k Ek−1 (f )Lw n Lps n w k=1 ∞ 1/γ

(11) rγ−1 γ +c k Ek (f )Lps w k=n+1. for arbitrary f ∈ Lps w (T) and natural n, where γ = min(s, 2). Corollary. Let 1 < p < ∞ and 1 < s ≤ 2 or p > 2 and s ≥ 2 . Assume 1 for some integer r ≥ 1 and l ≥ 1. Then = O nr+2l that En (f )Lps w (2.5). (ln n)1/γ (r) 1 Ωl f , =O n Lps n2l w. where γ = min(s, 2). Let {αn } be a monotonic sequence of positive numbers convergent to zero. ps ps Let Φps w (αn ) be the set of functions f ∈ Lw for which c1 αn ≤ En (f )Lw ≤ c2 αn for some constants c1 and c2 independent of f . When s, p > 2 the sharpness of (2.1) is shown by the following theorem. Theorem 5. For each αn ↓ 0 there exists f0 ∈ Φps w (αn ) satisfying the inequality (2.6). n 1/2 1 c

(12) 3 2 Ω1 f 0 , ≥ 2 k αk−1 n Lps n w k=1. with a constant c > 0 independent of n.. 3.. Auxiliary results. In this section we present some known results in weighted Lorentz spaces. Proposition 3.1. Let 1 < p, s < ∞. Then there exists a positive constant c such that for arbitrary f ∈ Lps w −1 (3.1) c f Lps ≤ sup f (x)g(x)w(x)dx ≤ c f Lps , w w T.

(13) 72. Trigonometric polynomials in weighted Lorentz spaces. where the supremum is taken with respect to all those functions g for which gLp s ≤ 1 (see [5], also [11, Proposition 5.1.2]). Here p = p/(p − 1). w. Proposition 3.2. Let 1 < p, s < ∞ and let ϕ be a measurable function of two variables. Then. . . ϕ(x, ·)dx. ≤ c ϕ(x, ·)Lps dx.. w. ps T. T. Lw. Proof. By proposition 3.1, Fubini’s theorem and the H¨ older’s inequality we obtain. ⎛ ⎞. . . ϕ(x, ·)dx. ⎝ |ϕ(x, y)| dx⎠ |g(y)| w(y)dy ≤ c sup. g p s ≤1. ps Lw T T T Lw ⎛ ⎞ ⎝ |ϕ(x, y)| |g(y)| w(y)dy ⎠ dx = c sup ≤ c. g. p s Lw. g. p s Lw. ≤1. T. ϕ(x, ·)Lps gLp s dx w. sup . ≤ c. ≤1. T. . w. T. ϕ(x, ·)Lps dx. w T. Proposition 3.3. Let 1 < p, s < ∞ and let w ∈ Ap (T). The trigonometric Fourier series of any f ∈ Lps w (T) converges in the norm and almost everywhere to f (x). Proof. The norm convergence follows in the standard way from the boundedness of conjugate functions in Lps w with 1 < p, s < ∞ and w ∈ Ap (T) (see [11, Theorem 6.6.2]). p0 for some When f ∈ Lps w with w ∈ Ap (T) (1 < p, s < ∞), then f ∈ L p1 ps p0 > 1. Indeed, from the inclusion Lw ⊂ Lw , 1 < p1 < p and the openness p , 1 < p0 < p1 < p such that of Ap it follows that there exist p0 and 1 f ∈ Lpw1 (T) and w ∈ Ap1 /p0 . Thus w. 1−. p1 p0. . ∈ A p1 and w. ⎛ |f (x)|. T. p0. dx ≤ ⎝. ⎞p0 /p1 ⎛. |f (x)| T. p1. p1 p0. p0. By the H¨older inequality we have . 1−. w(x)dx⎠. ⎝. w T. p − p0 1. . p1 p0. . . ∈ L1 .. ⎞ p1p−p0 1. (x)dx⎠. ..

(14) V. Kokilashvili, Y. E. Yildirir But − pp01. . p1 p0. . 0 = − p1p−p = 1− 0. . p1 p0. . 73. . Therefore the right-hand side. of the last inequality is ﬁnite and f ∈ Lp0 (T). Using the Hunt almost everywhere convergence theorem for the trigonometric Fourier series of f ∈ Lp0 , 1 < p0 < ∞, (see [8, Theorem 1]) we obtain the desired result. Proposition 3.4. Let 1 < p, s < ∞ and let w ∈ Ap (T). Then there ≤ cEn (f )Lps for each exists a positive constant c such that f − Sn (f )Lps w w ps f ∈ Lw and n ≥ 1, where Sn (f ) stands for the n- th partial sum of trigonometric Fourier series of f. Proof. The last inequality is obtained in the standard way as a consequence of the boundedness of the conjugate function in Lps w (T) with ≤ w ∈ Ap (T) which, (see [24, Chapter VI]) implies that Sn (f )Lps w . Indeed, let T be a trigonometric polynomial of the best c f Lps n w approximation. Then we have f − Sn (f )Lps w. ≤ f − Tn Lps + Tn − Sn (f )Lps w w = f − Tn Lps + Sn (Tn − f )Lps w w ≤ cEn (f )Lps . w . The following theorem is a weighted version of the Littlewood-Paley decomposition for trigonometric Fourier series (see [16], [24, Chapter XV, Theorem 4.24]). Theorem A. Let 1 < p, s < ∞ and let w ∈ Ap (T). Suppose that f (x) ∼. ∞

(15). (an cos nx + bn sin nx) .. n=0. Then there exist positive constants c1 and c2 independent of f such that. 1/2. ∞.

(16) 2. ps c1 f Lw ≤. Γm (x) ≤ c2 f Lps ,. w. m=0 ps Lw. where Γm (x) =. m 2

(17) −1. (an cos nx + bn sin nx) ,. Γ0 = a 0 .. n=2m−1. One can derive this result by means of interpolation arguments for Lorentz spaces from its Lpw -version (see [14],[15]) and openness of Ap . Indeed, let.

(18) 74. Trigonometric polynomials in weighted Lorentz spaces. w ∈ Ap (T). It is well known that there exists p1 , 1 < p1 < p < ∞, such that w ∈ Ap1 (T) and w ∈ Ap2 (T) for arbitrary p2 > p. According to Theorem 4.1 in [14], we have. 1/2. ∞.

(19) 2. cj f Lpwj ≤. Γ (x) ≤ cj f Lpwj , j = 1, 2. m. m=0 pj Lw. Applying the interpolation theorem for Lorentz spaces (see [2, Theorem 5.5]), we get the desired result.. 4.. Proofs of the main results. Theorem A is basic for our proofs. We need also some further auxiliary (k) statements. In the sequel, we say that f ∈ Wps,w if the derivative f (k−1) (x) (k) ps ∈ Lw (T). is absolutely continuous and f (2l). Lemma 4.1. Let f ∈ Wps,w . Then. 2l (l). ≤ cδ Ωl (f (2l) , δ)Lps. f. w. (4.1). Lps w. ,. where the positive constant c is independent of f and δ. Proof. It is suﬃcient to prove that Ωl (f, δ) ≤ cδ 2 Ωl−1 (f , δ). Let l. β(x) = Π (I − Ahi ) f (x). i=2. Then l. Π (I − Ahi ) f (x). i=1. =. =. =. =. 1 β(x) − 2h1 1 2h1. h1 β(x + t)dt −h1. h1 [β(x) − β(x + t)] dt −h1. 1 − 4h1 1 − 8h1. h1 [β(x + t) − 2β(x) + β(x − t)] dt −h1. h1 t y 0 0 −y. β (x + z)dzdydt..

(20) V. Kokilashvili, Y. E. Yildirir. 75. Applying Proposition 3.2 we obtain. l. Π (I − Ahi ) f. i=1. ≤. Lps w. =. h1 t y. β (· + z)dz. dydt. ps 0 0 −y Lw. y h t 1. . 1. 1 . 2y. β (· + z)dz. dydt.. 8h1. 2y. 1 8h1. −y. 0 0. Lps w. Using the uniform boundedness of Ahi (f ) in Lps w with respect to h we get. l. Π (I − Ahi ) f. i=1. Lps w. h1 t. c ≤ 8h1. Ay β Lps dydt ≤ ch21 β Lps . w w. 0 0. From the last inequality we conclude that Ωl (f, δ)Lps w. ≤. c sup h21 β Lps w. =. l. . cδ sup Π (I − Ahi ) f. 0<h <δ i=2. 0<hi <δ 1≤i≤l 2. Lps w. i. 2≤i≤l. = cδ 2 Ωl−1 (f , δ). . Lemma 4.2. Let 1 < p < ∞ and 1 < s ≤ 2. Then for an arbitrary m system of functions {ϕj (x)}j=1 , ϕj ∈ Lps w we have. (4.2). ⎛ ⎞1/2. m.

(21). ⎝. 2⎠ ϕj. j=1. Lps w. ⎞1/s ⎛ m

(22) s ≤ c⎝ ϕj ps ⎠ Lw. j=1. with a constant c independent of ϕj and m. Proof. We shall use the following well-known relations (f ∗ )α = (f α )∗ and (f + g)∗∗ ≤ (f ∗∗ + g ∗∗ ) [2, pp. 41 and 54]. By the Hardy inequality (see [2, pp. 129]), we get. ⎛ ⎞1/2. m.

(23). ⎝. 2⎠ ϕj I =. j=1. Lps w.

(24) 76. Trigonometric polynomials in weighted Lorentz spaces. ⎛ ⎜ ≤ c⎝ ⎛ ⎜ = c⎝ ⎛ ≤ c⎝. T. T. T. ⎛⎛⎛ ⎞1/s ⎞1/2 ⎞∗ ⎞s m s ⎜⎜⎝

(25) 2 ⎠ ⎟ ⎟ −1 ⎟ ϕj ⎝⎝ ⎠ ⎠ (t)t p dt⎠ j=1. ⎛⎛. ⎞1/s ⎞s/2 ⎞∗ m

(26) s ⎜⎝ ⎟ ⎟ ϕj 2 ⎠ ⎠ (t)t p −1 dt⎠ ⎝ j=1. ⎛ ⎝. m

(27). ⎞∗. ⎞1/s. ϕj s ⎠ (t)t. s p −1. dt⎠. .. j=1. Thus. I. ≤. ⎞∗∗ ⎛ ⎛ ⎞1/s

(28) m s c ⎝ ⎝ ϕj s ⎠ (t)t p −1 dt⎠. ≤. ⎛ ⎛ ⎞ ⎞1/s

(29) m s ∗∗ c⎝ ⎝ (ϕj s ) (t)⎠ t p −1 dt⎠. =. ⎛ ⎞1/s m

(30) s ∗∗ c⎝ (ϕj s ) (t)t p −1 dt⎠ .. T. j=1. T. j=1. j=1 T. Applying again the Hardy inequality we have ⎛ I. ≤ c⎝ ⎛ ≤ c⎝. m

(31). ⎞1/s s. (ϕj s )∗ (t)t p −1 dt⎠. j=1 T m

(32) j=1 T. ⎛ ≤ c⎝. ⎞1/s (ϕj ∗∗ )s (t)t. s p −1. dt⎠. ⎞1/s. m

(33). s. (ϕj ∗ )s (t)t p −1 dt⎠. j=1 T. ⎛ = c⎝. m

(34) j=1. ⎞1/s. ⎠ ϕj sLps w. . .

(35) V. Kokilashvili, Y. E. Yildirir. 77. Lemma 4.3. Let 2 < p < ∞ and s ≥ 2. For an arbitrary system ps {ϕj (x)}m j=1 , ϕj ∈ Lw , we have. ⎛ ⎞1/2. m.

(36). ⎝. 2⎠ ϕj. j=1. (4.3). ⎞1/2 ⎛ m

(37) 2 ≤ c⎝ ϕj ps ⎠ Lps w. j=1. Lw. with a constant c independent of ϕj and m.. Proof. By the deﬁnition. ⎛ ⎞1/2. m.

(38). ⎝. 2⎠ I :=. ϕj. j=1. ⎛ ⎜ =⎝. . ⎛⎛⎛ ⎜⎜⎝ ⎝⎝. T. Lps w. m

(39). ⎞1/s ⎞1/2 ⎞∗∗ ⎞s ⎟ ⎟ s ⎟ ϕj 2 ⎠ ⎠ (t)⎠ t p −1 dt⎠ .. j=1 p s. According to the Hardy inequality and taking into account that Lw2 2 is a normed space in the current situation, we have. I. ≤. ⎞1/s ⎛ ⎛⎛⎛ ⎞1/2 ⎞∗ ⎞s m ⎟ ⎟ s ⎟ ⎜ ⎜⎜

(40) c ⎝ ⎝⎝⎝ ϕj 2 ⎠ ⎠ (t)⎠ t p −1 dt⎠ j=1. T. =. ⎛ ⎛⎛ ⎞1/s ⎞∗ ⎞s/2 m

(41) s ⎜ ⎟ c ⎝ ⎝⎝ ϕj 2 ⎠ (t)⎠ t p −1 dt⎠ j=1. T. ≤. ⎛ ⎞ 2s 12 m

(42) 2 ∗∗ s/2 s −1 c⎝ (t) t p dt⎠ . ϕj j=1 T. If we use the Hardy inequality once more inside the sum, we get ⎛. I. ⎞ 2s 12 m

(43) 2 ∗ s/2 s −1 ≤ c⎝ ϕj (t) t p dt⎠ ⎛. j=1 T. ⎞ 2s 12 m

(44) ∗ s s −1 ϕj (t) t p dt⎠ ≤ c⎝ j=1 T.

(45) 78. Trigonometric polynomials in weighted Lorentz spaces. ≤. ⎛ ⎞ 2s 12 m

(46) ∗∗ s s −1 c⎝ ϕj (t) t p dt⎠. ≤. ⎛ ⎛ ⎞ 2s ⎞ 12 m ⎜

(47) ⎝ ∗∗ s ps −1 ⎠ ⎟ ϕj (t) t c⎝ dt ⎠. =. ⎞1/2 ⎛ m

(48) 2 c⎝ ϕj ps ⎠ .. j=1 T. j=1. T. Lw. j=1. Lemma 4.4. Let 1 < p0 < p < 2 < s < ∞. Then for an arbitrary m system of functions {ϕj (x)}j=1 , ϕj ∈ Lps w , we have. ⎛ ⎞1/2. m.

(49). ⎝. ϕj 2 ⎠. j=1. Lps w. ⎛ ≤ c⎝. m

(50) j=1. ⎞ p1. 0. p ⎠ ϕj L0ps w. with a constant c independent of ϕj and m. Proof. Using the arguments of claim that. ⎛ ⎞ 12. m.

(51). ⎝ 2⎠. ϕj =. ps. j=1 Lw. ≤. the proof of Lemmas 4.2 and 4.3, we. ⎛ ⎞ p20 p1. m. 0.

(52). ⎝. 2⎠ ϕj. j=1. ps Lw. ⎛ ⎞ p1. m. 0.

(53). ⎝. |ϕj | p0 ⎠. j=1. ps Lw. ≤. p1. 0. m.

(54) p0. c. |ϕj |. pp0 ps0. j=1 Lw. ≤. ⎛ m

(55) p c⎝ |ϕj | 0 j=1. p s p p Lw0 0. ⎞ p1. 0. ⎠.

(56) V. Kokilashvili, Y. E. Yildirir ⎛ ≤ c⎝. m

(57) j=1. 79. ⎞ p1. 0. p ⎠ ϕj L0ps w. . . ps Lemma 4.5. Let 1 < p, s < ∞, Lw (T ) and w ∈ Ap (T ). If f ∈ π π Bk,μ (x) = ak cos k + μ 2 x +bk sin k + μ 2 x, where ak , bk are Fourier coeﬃcients of f, then. i+1. 2

(58). μ. (4.4) k B ≤ c2iμ E2i (f )Lps , k,μ. w. ps. k=2i +1 Lw. where the constant c is independent of f and i. Proof. Let us introduce the notation τj,μ (x) :=. j k=1. Bk,μ (x). By means of. the Abel transformation we obtain i+1 2

(59). μ. k Bk,μ (x). =. n=2i +1. i+1 2

(60). [k μ − (k + 1)μ ] τk,μ (x) − τ2i ,μ (x). n=2i +1 i+1. +2. τ2i+1 ,μ (x) − τ2i ,μ (x) .. But by Proposition 3.4 we deduce that. i+1. 2i+1. 2

(61).

(62) μ. k Bk,μ. ≤ c [(k + 1)μ − k μ ] E2i (f )Lps + w. k=2i +1. ps k=2i +1 Lw. ≤ c2iμ E2i (f )Lps . +c2iμ E2i (f )Lps w w Proof of Theorem 1. Let 2m < n ≤ 2m+1 and δ = partial sum of Fourier series of f. Then we have (4.5). 1 n.. Let Sn (f ) be a. ≤ Ωl ((f − S2m+1 (f )) , δ)Lps + Ωl (S2m+1 , δ)Lps . Ωl (f, δ)Lps w w w. By the uniform boundedness of the averaging operator Ah in Lps w we obtain (4.6). Ωl ((f − S2m+1 (f )) , δ)Lps ≤ c f − S2m+1 Lps ≤ cEn (f )Lps . w w w. Then according to Lemma 4.1 we have. 2l (2l). Ωl (S2m+1 , δ)Lps ≤ cδ. S. m+1 w 2. Lps w. ..

(63) 80. Trigonometric polynomials in weighted Lorentz spaces. Then (4.7) Ωl (S2m+1 , δ)Lps w. ⎧. m .

(64). ⎨. . (2l) (2l) (2l) (2l) ≤ cδ 2l S1 − S0 ps +. S2i+1 − S2i. ⎩ Lw. ⎫ ⎬. Lps w. i=0. ⎭. .. For the ﬁrst term on the right side of (4.7) we have. (2l) (2l). . (4.8). S1 − S0 ps ≤ c (|a1 | + |b1 |) ≤ cE0 (f )Lps w Lw. Applying Theorem A to the second term, we get. i+1.

(65). m m 2

(66).

(67) . (2l) (2l). 2l. S2i+1 − S2i. =. k B (x). k,2l. ps. i=0 k=2i +1. ps i=0 Lw Lw. ⎛ . 2 ⎞1/2. . m 2i+1 ⎟. ⎜

(68)

(69) 2l . ≤ c ⎝ k Bk,2l (x) ⎠. . . i=0 k=2i +1. .. Lps w. Now with the aid of Lemmas 4.2 and 4.3 we conclude that. γ ⎞1/γ ⎛. i+1. m m 2

(70).

(71) .

(72). (2l) (2l). 2l. ⎠ , ⎝ S2i+1 − S2i. ≤c k Bk,2l (x). ps. i=0 i=1 k=2i +1 Lw where γ = min(s, 2). Then by Lemma 4.5 we have (4.9). m.

(73) . (2l) (2l). S2i+1 − S2i. ≤c. Lps w. i=0. m

(74) i=1. 1/γ 22γli E2γi (f )Lps w. .. Thus from (4.5), (4.6), (4.8) and (4.9) we derive the estimate ⎡ 1/γ ⎤ m

(75) ⎦. Ωl (f, δ)Lps ≤ cδ 2l ⎣E0 (f )Lps + En (f )Lps + 22γli E2γi (f )Lps w w w w i=1. Since Ek (f )Lps is monotonically decreasing, we conclude that w Ωl (f, δ)Lps w. c ≤ 2l n. n

(76). 1/γ k. 2lγ−1. γ Ek−1 (f )Lps w. .. k=1. .

(77) V. Kokilashvili, Y. E. Yildirir. 81. Proof of Theorem 2. We can repeat the proof of Theorem 1 just using Lemma 4.4 instead of Lemma 4.3 for a system of functions i+1 2

(78). ϕj (x) =. k 2l Bk,2l (x).. k=2i +1. Proof of Theorem 5. Let {αn } be a decreasing sequence convergent to 0. Deﬁne the function f with lacunary Fourier expansion (4.10). f (x) =. ∞ "

(79) α22n − α22n+1 sin 2n x.. n=0. Then (I − Ah )f (x) =. ∞ "

(80) sin 2n h α22n − α22n+1 1 − sin 2n x. nh 2 n=0. As it was shown in the proof of Proposition 3.3, there exists p0 > 1 such that (I − Ah )f Lps ≥ c (I − Ah )f Lp0 .. (4.11). Since the series is lacunary (see [24, Vol. 2, pp. 132]), we get (I − Ah )f Lp0 ≥ c (I − Ah )f L2. (4.12) and then. 2. (I − Ah )f L2 ≥ c. ∞

(81) 2 4 α2k − α22k+1 2k h . k=0. 1 n. m. Take h = and m such that 2 ≤ n < 2m+1 . Then the right-hand side of (4.12) is not less than (4.13). m+1 m+1

(82) 15 4k c

(83) 2 c 2 2 4k 2 4(m+1) 2 α2k − 2 α2k − α2k+1 2 = 4 α1 + α2m+2 . n4 n 16 k=0. k=1. m+1 4 4(m+1) Since 2 n4 α2m+2 = 2 2 α4n ≤ 24 α4n −→ 0 we can write that the last expression in (4.13) is more than (4.14). c n4. α21. +. m+1

(84) k=1. k 3 k 2 2 2 α2k. m+1. 2 n c

(85) 3 2 c

(86) 3 2 k αk . k αk ≥ 4 ≥ 4 n n k=1. k=1.

(87) 82. Trigonometric polynomials in weighted Lorentz spaces . Then inequality (2.6) follows from (4.11)-(4.14).. Theorem 5 shows that estimation 2.1 cannot be improved when 2 < p, s < ∞. In order to prove Theorems 3 and 4 we need several Lemmas. Lemma 4.6 Let {fn } be a sequence of absolutely continuous functions and let w ∈ Ap (T) . If {fn } converges to a function f in Lps w (T) , 1 < p, s < ∞, and the sequence of ﬁrst derivatives {fn } converges to a function g in Lps w (T) , then f is absolutely continuous and f (x) = g(x) almost everywhere. → 0, there exists p0 , 1 < p0 < p, such that Proof. Since fn − f Lps w fn − f Lp0 → 0. Thus there exists a subsequence {fnk } of the sequence {fn } such that fnk (x) → f (x) almost everywhere. Let x0 be a point of convergence. By H¨older’s inequality for Lorentz spaces we get x x. . fn (t)dt − g(t)dt ≤ fn − g ps w−1 p s . k k Lw Lw x0. x0. Since w ∈ Ap (T), we have w−1 Lp s < ∞ (see [5]). Thus we get w. x x lim fn k (t)dt − g(t)dt = 0. k−→∞ x0. x0. Therefore, we obtain x. x g(t)dt = lim. n→∞. x0. fn k (t)dt = lim (fnk (x) − fnk (x0 )) = f (x) − f (x0 ). k→∞. x0. . This completes the proof. Proof of Theorem 3. Let 2m < n < 2m+1 . We have. (r). f − Sn (f (r) ). Lps w. (4.15). ≤ S2m+2 (f (r) ) − Sn (f (r) ). Lps w. +. ∞.

(88). S2k+1 (f (r) ) − S2k (f (r) ). k=m+2. Lps w. ..

(89) V. Kokilashvili, Y. E. Yildirir. 83. By the weighted version of Bernstein’s inequality (see [6, Theorem 4.1]), we have. S2m+2 (f (r) ) − Sn (f (r) ) ps = S2rm+2 (f ) − Snr (f )Lps w Lw. ≤ c2(m+2)r S2m+2 (f ) − Sn (f ) ≤ c2(m+2)r E2m+2 (f )Lps w ≤ cnr En (f )Lps . w. (4.16). Applying Theorem A and Lemma 4.5, we obtain. ∞.

(90) . S2k+1 (f (r) ) − S2k (f (r) ). k=m+2. Lps w. γ ⎞ γ1. 2k+1 ∞

(91)

(92). r. ⎝ ⎠ ≤c μ Bμ,r (x). k k=m+2 μ=2 +1 ps ⎛. ≤c. ∞

(93). k=m+2. (4.17). ≤c. ∞

(94). 2krγ E2γk −1 (f )Lps w. k rγ−1 Ekγ (f )Lps w. γ1. Lw. γ1 .. k=n+1. Gathering formulas (4.15), (4.16) and (4.17), we deduce the desired inequality. Proof of Theorem 4. Let 2m < n ≤ 2m+1 and δ = n1 . Using Lemma 4.6, we conclude that under condition (2.3) there exists the absolutely continuous (r − 1)th order derivative f (r−1) (x) and f (r) ∈ Lps w . Then (r) (r) (4.18) Ωl (f (r) , δ) ≤ Ωl f (r) − S2m+1 , δ + Ωl (S2m+1 , δ). But. . . (r) (r). (4.19) Ωl f (r) − S2m+1 , δ ≤ c f (r) − S2m+1. Lps w. ≤ cEn (f (r) )Lps . w. On the other hand. (r+2l). (r) Ωl (S2m+1 , δ) ≤ cδ 2l S2m+1 ps Lw ⎧. m .

(95). ⎨. . (r+2l) (r+2l) (r+2l) (r+2l) S2i+1 − S2i ≤ cδ 2l S1 − S0. ps +. ⎩ Lw i=0. Consequently, applying Theorem A, we have. Lps w. ⎫ ⎬ ⎭. ..

(96) 84. Trigonometric polynomials in weighted Lorentz spaces. (r). Ωl (S2m+1 , δ) ≤ cδ 2l. ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩. E0 (f )Lps w. ⎛. 2 ⎞1/2. m 2i+1 . ⎜

(97)

(98) r+2l ⎟. + ⎝ k Bk,r+2l (x) ⎠. . i=0 k=2i +1 . Lps w. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭. Then by Lemmas 4.2 and 4.3 (4.20) ⎧. i+1. γ ⎞1/γ ⎫ ⎛. ⎪ ⎪ m 2 ⎨ ⎬

(99)

(100). (r) 2l r+2l. ⎠ ⎝ Ωl (S2m+1 , δ) ≤ cδ + k B E0 (f )Lps . k,r+2l w. ⎪ ⎪. ps ⎩ ⎭ i=0 k=2i +1 Lw. Using the arguments of the proof of Theorem 1, according to Lemma 4.5 and Theorem 3, we obtain the assertion from (4.18), (4.19) and (4.20). Note that using the standard method of proving inverse inequalities (see [21, Section 6.1], and [9, Theorem 4]), on the base of Proposition 3.2 and 3.4 and Lemma 4.1, we can establish the following statement. Proposition 4.1 Let 1 < p, s < ∞ and let w ∈ Ap (T). Then there exists a positive constant c such that n 1 c

(101) 2l−1 k Ek−1 (f )Lps Ωl f, ≤ 2l w n n. (4.21). k=1. for an arbitrary f ∈ Lps w (T) and every natural n. On the other hand, for arbitrary β, 1 < β < ∞, natural number μ and sequence αn ↓ 0 the inequality. n

(102). 1/β. k βμ−1 αβk−1 k=1. ≤c. n

(103). k. μ−1. αk−1. k=1. holds, where the constant c does not depend on αk and n (see, for example, [20]). Thus Theorems 1 and 2 improve the estimation (4.21). Acknowledgement. The authors were supported by the grants of INTAS Nr 06-1000017-8792 and GNSF/ST07/3-169 during the above research.. References [1] N.K. Bary and S.B. Stechkin, Best approximation and diﬀerential properties of two conjugate functions (Russian), Trudy Moskov. Mat. Obshch., 5 (1956), 483–522.. ..

(104) V. Kokilashvili, Y. E. Yildirir. 85. [2] C. Bennet and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, MA, 1988. [3] S. Bernstein, Sur l’ordre de la meilleure approximation des fonctiones continues par des polynomes de degr`e donn`e, Mem. Acad. Roy. Belgigue, 2 me s`erie, 4 (1912), 1–104. [4] O.V. Besov, On some conditions belonging of derivatives of periodic functions to Lp , (Russian) Nauchnie Doklady Vishei Skolj, 1 (1959), 13–17. [5] H.M. Chang, R.A. Hunt and D.S. Kurtz, The Hardy-Littlewood maximal functions on L(p, q) with weights, Indiana Univ. Math. J., 31 (1982), 109–120. [6] A. Guven and V. Kokilashvili, On the mean summability by Cesaro method of Fourier trigonometric series in two-weighted setting, J. Inequal. Appl., 2006, ??? [7] E.A. Haciyeva, Investigation of properties of functions with quasimonotone Fourier coeﬃcients in generalized Nikolskii-Besov spaces (Russian), Summary of Candidate Dissertation, Tbilisi State University, 1986. [8] R.A. Hunt, On the convergence of Fourier series. Orthogonal expansions and their continuous analogous,. Proc. Conf. Edurandsville, Ill., (1964) Southern Illinois Univ. Press, Carbondale, 1968, 235–255. [9] D.M. Israﬁlov and A. Guven, Approximation in weighted Smirnov classes, East J. on App., 11 (2005), 91–102. [10] V. Kokilashvili, On inverse theorem of the constructive theory of functions in Orlicz spaces, (Russian) Bull. Acad. Sri. Georgian SSR, 36 (1965), 263–270. [11] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientiﬁc Publishing Co. Inc. River Edge, NJ, 1991. [12] V. Kokilashvili and Y.E. Yildirir, On the approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst., 143 (2007) 103–113. [13] N.X. Ky, On approximation by trigonometric polynomials in Lpu -spaces, Studia Sci. Math. Hungar., 28 (1993), 183–188. [14] D.S. Kurtz, Littlewood-Paley and multiplier theorems on weighted Lp spaces, Ph. D. Dissertation, Rutgers University, 1968. [15] D.S. Kurtz, Littlewood-Paley and multiplier theorems in weighted Lp spaces, Trans. Amer. Math. Soc., 259 (1980), 235–254..

(105) 86. Trigonometric polynomials in weighted Lorentz spaces. [16] J.E. Littlewood and R.E. Paley, Theorems on Fourier series and power series II, Proc. London Math. Soc., 42 (1986), 52–89. [17] E.S. Quade, Trigonometric approximation in the mean, Duke Math. J. 3 (1937), 523–543. [18] S.B. Stechkin, On the absolute convergence of Fourier series, (Russian) Izv. Akad. Nauk. SSSR. Ser. Mat., 17 (1953), 499–512. [19] S.B. Stechkin, On the order of the best approximation of continuous functions, (Russian) Izv. Akad. Nauk. SSSR, Ser. Math., 15 (1954), 219–242. [20] M.F. Timan, Best approximation and modulus of smoothness of functions prescribed on the entire real axis, (Russian), Izv. Vyssh. Uchebn. Zaved Matematika, 25 (1961), 108–120. [21] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, (1963). [22] A.F. Timan and M.F. Timan, Generalized modulus of continuity and best approximation in the mean, (Russian) Doklady Akad. Nauk. SSSR, 71 (1950), 17–20. [23] G. Mastroianni, V. Totik, Jackson type inequalities for doubling and AP weights, Proceedings of the 3rd international conference on functional analysis and approximation theory, Acquafredda di Maratea (Potenza), Italy, September 23–28, 1996. Vols. I and II. Palermo: Circolo Matem` atico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 52 (1998), 83–99. [24] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1968. A. Razmadze Mathematical Institute Aleksidze St. 1, Tbilisi 0193 International Black Sea University, Agmashenebeli Alley 2, Tbilisi 0131 Georgia (E-mail : [email protected]) Balikesir University, Faculty of Education Department of Mathematics, Balikesir, 10100 Turkey (E-mail : [email protected] ) (Received : October 2008 ).

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