ON THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS Strong Approximation of Integrable Functions WŁODZIMIERZ ŁENSKI University of Zielona Góra Faculty of Mathematics, Computer Science and Econometrics 65-516 Zielona Góra, ul. Szafrana 4a, Poland EMail: W.Lenski@wmie.uz.zgora.pl Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents Received: 20 December, 2006 Accepted: 25 July, 2007 Communicated by: L. Leindler 2000 AMS Sub. Class.: 42A24, 41A25. Key words: Degree of strong approximation, Special sequences. Abstract: We show the strong approximation version of some results of L. Leindler [3] connected with the theorems of P. Chandra [1, 2]. JJ II J I Page 1 of 23 Go Back Full Screen Close Contents 1 Introduction 3 2 Statement of the Results 5 3 Auxiliary Results 9 4 Proofs of the Results 11 Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 2 of 23 Go Back Full Screen Close 1. Introduction Let Lp (1 < p < ∞) [C] be the class of all 2π–periodic real–valued functions integrable in the Lebesgue sense with p–th power [continuous] over Q = [−π, π] and let X p = Lp when 1 < p < ∞ or X p = C when p = ∞. Let us define the norm of f ∈ X p as 1 R |f (x)|p dx p when 1 < p < ∞, Q kf kX p = kf (·)kX p = supx∈Q |f (x)| when p = ∞, Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 and consider its trigonometric Fourier series Title Page ∞ ao (f ) X Sf (x) = + (aν (f ) cos νx + bν (f ) sin νx) 2 ν=0 with the partial sums Sk f . Let A := (an,k ) (k, n = 0, 1, 2, . . . ) be a lower triangular infinite matrix of real numbers and let the A−transforms of (Sk f ) be given by n X Tn,A f (x) := an,k Sk f (x) − f (x) (n = 0, 1, 2, . . . ) k=0 JJ II J I Page 3 of 23 Go Back Full Screen Close and q Hn,A f Contents (x) := ( n X ) 1q q an,k |Sk f (x) − f (x)| (q > 0, n = 0, 1, 2, . . . ) . k=0 As a measure of approximation, by the above quantities we use the pointwise characteristic wx f (δ)Lp Z δ p1 1 p := |ϕx (t)| dt , δ 0 where ϕx (t) := f (x + t) + f (x − t) − 2f (x) . wx f (δ)Lp is constructed based on the definition of Lebesgue points (Lp −points), and the modulus of continuity for f in the space X p defined by the formula Strong Approximation of Integrable Functions Włodzimierz Łenski ωf (δ)X p := sup kϕ · (h)kX p . vol. 8, iss. 3, art. 70, 2007 0≤|h|≤δ We can observe that with pe ≥ p, for f ∈ X pe, by the Minkowski inequality Title Page kw · f (δ)p kX pe ≤ ωf (δ)X pe . The deviation Tn,A f was estimated by P. Chandra [1, 2] in the norm of f ∈ C and for monotonic sequences an = (an,k ). These results were generalized by L. Leindler [3] who considered the sequences of bounded variation instead of monotonic ones. q In this note we shall consider the strong means Hn,A f and the integrable functions. We shall also give some results on norm approximation. By K we shall designate either an absolute constant or a constant depending on some parameters, not necessarily the same of each occurrence. Contents JJ II J I Page 4 of 23 Go Back Full Screen Close 2. Statement of the Results Let us consider a function wx of modulus of continuity type on the interval [0, +∞), i.e., a nondecreasing continuous function having the following properties: wx (0) = 0, wx (δ1 + δ2 ) ≤ wx (δ1 ) + wx (δ2 ) for any 0 ≤ δ1 ≤ δ2 ≤ δ1 + δ2 and let Lp (wx ) = {f ∈ Lp : wx f (δ)Lp ≤ wx (δ)} . We can now formulate our main results. To start with, we formulate the results on pointwise approximation. an,m ≥ 0, n X Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Theorem 2.1. Let an = (an,m ) satisfy the following conditions: (2.1) Strong Approximation of Integrable Functions an,k = 1 Title Page k=0 Contents and m−1 X (2.2) |an,k − an,k+1 | ≤ Kan,m , JJ II J I k=0 Page 5 of 23 where m = 0, 1, . . . , n and n = 0, 1, 2, . . . X−1 k=0 =0 . Suppose wx is such that p1 Z π p p (w (t)) x (2.3) uq dt = O (uHx (u)) as u → 0+, p t1+ q u where Hx (u) ≥ 0, 1 < p ≤ q and Z t (2.4) Hx (u) du = O (tHx (t)) as t → 0 + . 0 Go Back Full Screen Close If f ∈ Lp (wx ) , then 0 q Hn,A f (x) = O (an,n Hx (an,n )) with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q. Theorem 2.2. Let (2.1), (2.2) and (2.3) hold. If f ∈ Lp (wx ) then π π q0 Hn,A f (x) = O wx + O an,n Hx n+1 n+1 Strong Approximation of Integrable Functions Włodzimierz Łenski and if, in addition, (2.4) holds then vol. 8, iss. 3, art. 70, 2007 0 q Hn,A f (x) = O an,n Hx π n+1 Title Page Contents with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q. Theorem 2.3. Let (2.1) , (2.3) , (2.4) and ∞ X (2.5) |an,k − an,k+1 | ≤ Kan,m , JJ II J I Page 6 of 23 k=m Go Back where m = 0, 1, . . . , n and n = 0, 1, 2, . . . hold. If f ∈ Lp (wx ) then Full Screen Close 0 q f (x) = O (an,0 Hx (an,0 )) Hn,A with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q. Theorem 2.4. Let us assume that (2.1), (2.3) and (2.5) hold. If f ∈ Lp (wx ) , then π π q0 Hn,A f (x) = O wx ( ) + O an,0 Hx . n+1 n+1 If, in addition, (2.4) holds then q0 Hn,A f (x) = O an,0 Hx π n+1 Strong Approximation of Integrable Functions Włodzimierz Łenski 0 with q ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q. vol. 8, iss. 3, art. 70, 2007 Consequently, we formulate the results on norm approximation. Theorem 2.5. Let an = (an,m ) satisfy the conditions (2.1) and (2.2). Suppose ωf (·)X pe is such that (2.6) Z p uq u π (ωf (t)X pe )p p t1+ q p1 dt = O (uH (u)) as u → 0+ holds, with 1 < p ≤ q and pe ≥ p, where H (≥ 0) instead of Hx satisfies the condition (2.4). If f ∈ X pe then 0 q Hn,A f (·) = O (an,n H (an,n )) e Xp with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q. Theorem 2.6. Let (2.1) , (2.2) and (2.6) hold. If f ∈ X pe then 0 π π q + O an,n H . Hn,A f (·) = O ωf n + 1 X pe n+1 e Xp Title Page Contents JJ II J I Page 7 of 23 Go Back Full Screen Close If, in addition, H (≥ 0) instead of Hx satisfies the condition (2.4) then 0 π q Hn,A f (·) = O an,n H n+1 e Xp with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q. Theorem 2.7. Let (2.1), (2.4) with a function H (≥ 0) instead of Hx , (2.5) and (2.6) hold. If f ∈ X pe then 0 q Hn,A f (·) = O (an,0 H (an,0 )) e Xp Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q. Theorem 2.8. Let (2.1), (2.5) and (2.6) hold. If f ∈ X pe then 0 π π q + O an,0 H . Hn,A f (·) = O ωf n + 1 X pe n+1 e Xp If, in addition, H (≥ 0) instead of Hx satisfies the condition (2.4) , then 0 π q Hn,A f (·) = O an,0 H n+1 e Xp with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q. Title Page Contents JJ II J I Page 8 of 23 Go Back Full Screen Close 3. Auxiliary Results In tis section we denote by ω a function of modulus of continuity type. Lemma 3.1. If (2.3) with 0 < p ≤ q and (2.4) with functions ω and H (≥ 0) instead of wx and Hx , respectively, hold then Z u ω (t) (3.1) dt = O (uH (u)) (u → 0+) . t 0 Proof. Integrating by parts in the above integral we obtain Z π Z u Z u ω (t) d ω (s) dt = t ds dt t dt s2 0 0 t Z π u Z u Z π ω (s) ω (s) = −t ds + ds dt s2 s2 t 0 t 0 Z π Z u Z π ω (s) ω (s) ≤u ds + ds dt s2 s2 u 0 t Z π Z u Z π ω (s) ω (s) =u ds + ds dt 1+ pq +1− pq 1+ pq +1− pq u s 0 t s p1 Z u Z π Z π p p p 1 ω (s) (ω (s)) ds + tq ds dt, ≤ uq 1+p/q s1+p/q 0 t u s t since 1 − p q ≥ 0. Using our assumptions we have Z u Z u ω (t) 1 dt = O (uH (u)) + O (tH (t)) dt = O (uH (u)) t 0 0 t and thus the proof is completed. Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 9 of 23 Go Back Full Screen Close Lemma 3.2 ([4, Theorem 5.20 II, Ch. XII]). Suppose that 1 < q (q − 1) ≤ p ≤ q and ξ = 1/p + 1/q − 1. If t−ξ g (t) ∈ Lp then ( (3.2) ) 1q Z π p1 ∞ −ξ p |ao (g)|q X q q + (|ak (g)| + |bk (g)| ) ≤K t g (t) dt . 2 −π k=0 Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 10 of 23 Go Back Full Screen Close 4. Proofs of the Results q Since Hn,A f is the monotonic function of q we shall consider, in all our proofs, the q q0 quantity Hn,A f instead of Hn,A f. Proof of Theorem 2.1. Let Z ( n q ) 1q 1 π 1 X sin k + t q 2 Hn,A f (x) = an,k ϕx (t) dt 1 π 0 2 sin t 2 k=0 Z ( n q ) 1q 1 an,n 1 X sin k + t 2 ≤ an,k dt ϕx (t) π 0 2 sin 12 t k=0 ( n q ) 1q 1 Z π 1 X sin k + t 2 dt + an,k ϕx (t) π an,n 2 sin 12 t k=0 = I (an,n ) + J (an,n ) and, by (2.1) , integrating by parts, we obtain, Z an,n |ϕx (t)| I (an,n ) ≤ dt 2t 0 Z t Z an,n 1 d = |ϕx (s)| ds dt 2t dt 0 0 Z an,n Z an,n wx f (t)1 1 = |ϕx (t)| dt + dt 2an,n 0 2t 0 Z an,n wx f (t)1 1 = wx f (an,n )1 + dt 2 t 0 Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 11 of 23 Go Back Full Screen Close π Z =K an,n an,n Z ≤K π an,n an,n ap/q n,n ≤K Z π an,n Z wx f (t)1 dt + t2 an,n 0 ! wx f (t)1 dt t ! p1 (wx f (t)L1 )p dt t2 Z an,n wx f (t)L1 dt t an,n wx f (t)Lp dt . t +K 0 ! p1 (wx f (t)Lp )p dt t1+p/q Z +K 0 Since f ∈ Lp (wx ) and (2.4) holds, Lemma 3.1 and (2.3) give I (an,n ) = O (an,n Hx (an,n )) . Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page The Abel transformation shows that q k Z π X sin ν + 12 t 1 q dt (J (an,n )) = (an,k − an,k+1 ) ϕ (t) π an,n x 2 sin 12 t ν=0 k=0 Z q n π X sin ν + 12 t 1 + an,n ϕx (t) dt , π an,n 2 sin 12 t ν=0 n−1 X Contents JJ II J I Page 12 of 23 Go Back whence, by (2.2), q n Z π 1 X sin ν + t 1 2 (J (an,n ))q ≤ (K + 1) an,n ϕx (t) dt . 1 π an,n 2 sin t 2 ν=0 Using inequality (3.2), we obtain J (an,n ) ≤ K (an,n ) 1 q (Z π an,n ) p1 |ϕx (t)|p dt . t1+p/q Full Screen Close Integrating by parts, we have J (an,n ) ≤ K (an,n ) 1 q ( 1 p p + 1+ q ( ≤ K (an,n ) (wx f (t)Lp ) tp/q 1 q π t=an,n Z π an,n ) p1 (wx f (t)Lp )p dt t1+p/q Z π (wx f (π)Lp ) + an,n ) p1 (wx f (t)Lp )p . dt t1+p/q Since f ∈ Lp (wx ) , by (2.3), 1 J (an,n ) ≤ K (an,n ) q Z p Z π (an,n ) q an,n p π ) p1 (wx (t)) dt 1+p/q an,n t ) p1 p (wx (t)) dt t1+p/q Contents and I J I Full Screen Proof of Theorem 2.2. Let, as before, π π q Hn,A f (x) ≤ I +J n+1 n+1 n+1 ≤ π II Go Back Thus our result is proved. JJ Page 13 of 23 = O (an,n Hx (an,n )) . π n+1 vol. 8, iss. 3, art. 70, 2007 (wx (π)) + ( Włodzimierz Łenski Title Page ( ≤K Strong Approximation of Integrable Functions Z 0 π n+1 |ϕx (t)| dt = wx π n+1 Close . L1 π In the estimate of J n+1 we again use the Abel transformation and (2.2). Thus q q n Z π 1 X t sin ν + π 1 2 J ≤ (K + 1) an,n ϕx (t) dt 1 π n+1 π 2 sin t 2 ν=0 n+1 and, by inequality (3.2), J π n+1 ≤ K (an,n ) 1 q (Z π π n+1 ) p1 |ϕx (t)|p dt . t1+p/q Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Integrating (2.3) by parts, with the assumption f ∈ Lp (wx ) , we obtain ) p1 ( pq Z π p 1 (w (t)) π π x dt J ≤ K ((n + 1) an,n ) q 1+p/q π t n+1 n+1 n+1 1 π π = O ((n + 1) an,n ) q Hx n+1 n+1 π as in the previous proof, with n+1 instead of an,n . Finally, arguing as in [3, p.110], we can see that, for j = 0, 1, . . . , n − 1, n−1 n−1 X X |an,j − an,n | ≤ (an,k − an,k+1 ) ≤ |an,k − an,k+1 | ≤ Kan,n , k=j Strong Approximation of Integrable Functions Title Page Contents JJ II J I Page 14 of 23 Go Back Full Screen k=0 Close whence an,j ≤ (K + 1) an,n and therefore (K + 1) (n + 1) an,n ≥ n X j=0 an,j = 1. This inequality implies that π π J = O an,n Hx n+1 n+1 and the proof of the first part of our statement is complete. π To prove of the second part of our assertion we have to estimate the term I n+1 once more. π Proceeding analogously to the proof of Theorem 2.1, with an,n replaced by n+1 , we obtain π π π I =O Hx . n+1 n+1 n+1 By the inequality from the first part of our proof, the relation (n + 1)−1 = O (an,n ) holds, whence the second statement follows. Proof of Theorem 2.3. As usual, let q Hn,A f (x) ≤ I (an,0 ) + J (an,0 ) . Since f ∈ Lp (wx ) ,by the same method as in the proof of Theorem 2.1, Lemma 3.1 and (2.3) yield I (an,0 ) = O (an,0 Hx (an,0 )) . By the Abel transformation q k Z π X sin ν + 12 t 1 q (J (an,0 )) ≤ |an,k − an,k+1 | dt ϕ (t) π an,0 x 2 sin 12 t ν=0 k=0 q n Z π X sin ν + 12 t 1 ϕ (t) + an,n dt π an,0 x 2 sin 12 t ν=0 n−1 X Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 15 of 23 Go Back Full Screen Close ≤ ∞ X q ∞ Z π 1 X sin ν + t 1 2 ϕx (t) dt . 1 π an,0 2 sin t 2 ν=0 ! |an,k − an,k+1 | + an,n k=0 Arguing as in [3, p.110], by (2.5), we have |an,n − an,0 | ≤ n−1 X |an,k − an,k+1 | ≤ k=0 ∞ X |an,k − an,k+1 | ≤ Kan,0 , k=0 whence an,n ≤ (K + 1) an,0 and therefore ∞ X Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 |an,k − an,k+1 | + an,n ≤ (2K + 1) an,0 k=0 and Strong Approximation of Integrable Functions Title Page q ∞ Z π 1 X sin ν + t 1 2 dt . (J (an,0 ))q ≤ (2K + 1) an,0 ϕx (t) 1 π an,0 2 sin 2 t ν=0 Finally, by (3.2), J (an,0 ) ≤ K (an,0 ) 1 q (Z π an,0 ) p1 |ϕx (t)|p dt t1+p/q Contents JJ II J I Page 16 of 23 Go Back Full Screen and, by (2.3), J (an,0 ) = O (an,0 Hx (an,0 )) . This completes of our proof. Proof of Theorem 2.4. We start as usual with the simple transformation π π q Hn,A f (x) ≤ I +J . n+1 n+1 Close Similarly, as in the previous proofs, by (2.1) we have π π I . ≤ wx f n+1 n + 1 L1 We estimate the term J in the following way q q X k Z π n−1 X sin ν + 12 t π 1 J |an,k − an,k+1 | ϕ (t) dt ≤ π π x n+1 2 sin 12 t ν=0 n+1 k=0 Z q n π X sin ν + 12 t 1 ϕx (t) dt + an,n π π 2 sin 12 t ν=0 n+1 Z ! q ∞ ∞ π X X sin ν + 12 t 1 dt . ϕx (t) ≤ |an,k − an,k+1 | + an,n π π 2 sin 12 t ν=0 n+1 k=0 From the assumption (2.5), arguing as before, we can see that q ! 1q ∞ Z π X sin ν + 12 t π 1 J ≤ K an,0 ϕx (t) dt 1 π n+1 π 2 sin t 2 ν=0 n+1 Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 17 of 23 Go Back and, by (3.2) , J π n+1 ≤ K (an,0 ) 1 q (Z π π n+1 ) p1 |ϕx (t)|p dt . t1+p/q From (2.5) , it follows that an,k ≤ (K + 1) an,0 for any k ≤ n, and therefore (K + 1) (n + 1) an,0 ≥ n X k=0 an,k = 1, Full Screen Close whence J π n+1 ( pq Z p π ) p1 |ϕx (t)| dt π t1+p/q n+1 ( ) p1 pq Z π p π |ϕx (t)| ≤ K (n + 1) an,0 dt . π n+1 t1+p/q n+1 ≤ K ((n + 1) an,0 ) 1 q π n+1 . Since f ∈ Lp (wx ), integrating by parts we obtain π π π J ≤ K (n + 1) an,0 Hx n+1 n+1 n+1 π ≤ Kan,0 Hx n+1 and the proof of the first part of our statement is complete. π To prove the second part, we have to estimate the term I n+1 once more. Proceeding analogously to the proof of Theorem 2.1 we obtain π π π =O Hx . I n+1 n+1 n+1 Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents JJ II J I Page 18 of 23 Go Back Full Screen From the start of our proof we have (n + 1) sertion also follows. −1 = O (an,0 ) , whence the second as- Proof of Theorem 2.5. We begin with the inequality q H f (·) ≤ kI (an,n )k + kJ (an,n )k n,A p e e Xp X e Xp . Close By (2.1) , Lemma 3.1 gives kI (an,n )k e Xp kϕ (t)k an,n Z ≤ e Xp dt 2t Z an,n ωf (t)X pe ≤ dt 2t 0 = O (an,n H (an,n )) . 0 Strong Approximation of Integrable Functions Włodzimierz Łenski As in the proof of Theorem 2.1, kJ (an,n )k e Xp vol. 8, iss. 3, art. 70, 2007 ( ) p1 Z π p 1 |ϕ (t)| ≤ K (an,n ) q dt 1+p/q an,n t Title Page e Xp 1 (Z kϕ (t)k π ≤ K (an,n ) q an,n 1 (Z e Xp t1+p/q dt ) p1 π ≤ K (an,n ) q an,n ωf (t)X pe dt t1+p/q e Xp JJ II J I Page 19 of 23 , Go Back whence, by (2.6) , kJ (an,n )k Contents ) p1 p Full Screen = O (an,n H (an,n )) Close holds and our result follows. Proof of Theorem 2.6. It is clear that q π H f (·) ≤ I n,A e n+1 Xp X π + J n + 1 p e e Xp and immediately π I n+1 e Xp π ≤ w · f n + 1 1 ≤ ωf e Xp π n+1 X pe and π J n+1 ≤ K (an,n ) 1 q (Z kϕ (t)kp pe π X π n+1 e Xp t1+p/q ) p1 Strong Approximation of Integrable Functions dt Włodzimierz Łenski ( Z ) p1 π ωf (t)X pe dt 1+p/q π t n+1 1 π π ≤ K ((n + 1) an,n ) q H n+1 n+1 π π ≤ K (n + 1) an,n H n+1 n+1 π . = O an,n H n+1 1 ≤ K ((n + 1) an,n ) q π n+1 Thus our first statement holds. The second one follows on using a similar process to that in the proof of Theorem 2.1. We have to only use the estimates obtained in the π proof of Theorem 2.5, with n+1 instead of an,n ,and the relation Proof of Theorem 2.7. As in the proof of Theorem 2.5, we have q H f (·) ≤ kI (an,0 )k + kJ (an,0 )k e Xp e Xp Title Page Contents JJ II J I Page 20 of 23 Go Back Full Screen Close (n + 1)−1 = O (an,n ) . n,A vol. 8, iss. 3, art. 70, 2007 e Xp and kI (an,0 )k = O (an,0 H (an,0 )) . e Xp Also, from the proof of Theorem 2.3, kJ (an,0 )k e Xp ( ) p1 Z π p 1 |ϕ (t)| · ≤ K (an,0 ) q dt 1+p/q an,0 t Strong Approximation of Integrable Functions e Xp ≤ K (an,0 ) 1 q (Z π an,0 ) p1 ωf (t)X pe dt t1+p/q Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 and, by (2.6) , kJ (an,0 )k Title Page = O (an,0 H (an,0 )) . e Xp Contents Thus our result is proved. Proof of Theorem 2.8. We recall, as in the previous proof, that q π π H f (·) ≤ I + J n,A e n+1 n + 1 Xp p e and π I n+1 e Xp ≤ ωf π n+1 II J I Page 21 of 23 e Xp X JJ Go Back . Full Screen X pe We apply a similar method as πthat used in the proof of Theorem 2.4 to obtain an estimate for the quantity J n+1 , e Xp ( ) p1 pq Z π p π π |ϕx (t)| J ≤ K (n + 1) an,0 dt 1+p/q π n + 1 pe n + 1 t n+1 X e Xp Close ( ≤ K (n + 1) an,0 ( ≤ K (n + 1) an,0 π n+1 pq Z π n+1 pq Z π kϕ· (t)kp pe X t1+p/q π n+1 π π n+1 ) p1 dt ) p1 ωf (t)X pe dt t1+p/q Strong Approximation of Integrable Functions and, by (2.6) , π J n+1 e Xp π ≤ K (n + 1) an,0 H n+1 π ≤ Kan,0 H . n+1 π n+1 Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 Title Page Contents Thus the proof of the first part of our statement is complete. To prove of the second part, we follow the line of the proof of Theorem 2.6. JJ II J I Page 22 of 23 Go Back Full Screen Close References [1] P. CHANDRA, On the degree of approximation of a class of functions by means of Fourier series, Acta Math. Hungar., 52 (1988), 199–205. [2] P. CHANDRA, A note on the degree of approximation of continuous functions, Acta Math. Hungar., 62 (1993), 21–23. [3] L. LEINDLER, On the degree of approximation of continuous functions, Acta Math. Hungar., 104 (2004), 105–113. Strong Approximation of Integrable Functions Włodzimierz Łenski vol. 8, iss. 3, art. 70, 2007 [4] A. ZYGMUND, Trigonometric Series, Cambridge, 2002. Title Page Contents JJ II J I Page 23 of 23 Go Back Full Screen Close