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Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 RESEARCH Open Access A study on degree of approximation by Karamata summability method Hare Krishna Nigam and Kusum Sharma* * Correspondence: kusum31sharma@rediffmail.com Department of Mathematics, Faculty of Engineering and Technology, Mody Institute of Technology and Science (Deemed University), Laxmangarh-332311, Sikar, Rajasthan, India Abstract Vuĉkoviĉ [Maths Zeitchr 89, 192 (1965)] and Kathal [Riv Math Univ Parma, Italy 10, 3338 (1969)] have studied summability of Fourier series by Karamata (Kl) summability method In present paper, for the first time, we study the degree of approximation of function f Ỵ Lip (a,r) and f Ỵ W(Lr,ξ(t)) by Kl-summability means of its Fourier series and ˜ ˜ conjugate of function f ∈ Lip(α, r) and f ∈ W(Lr, ξ (t)) by Kl-summability means of its conjugate Fourier series and establish four quite new theorems MSC: primary 42B05; 42B08; 42A42; 42A30; 42A50 Keywords: degree of approximation, Lip(α,r) class, W(Lr,ξ(t)) class of functions, Fourier series, conjugate Fourier series, Kλ-summability, Lebesgue integral Introduction The method Kl was first introduced by Karamata [1] and Lotosky [2] reintroduced the special case l = Only after the study of Agnew [3], an intensive study of these and similar cases took place Vuĉkoviĉ [4] applied this method for summability of Fourier series Kathal [5] extended the result of Vuĉkoviĉ [4] Working in the same direction, Ojha [6], Tripathi and Lal [7] have studied Kl-summability of Fourier series under different conditions The degree of approximation of a function f Î Lip a by Cesàro and Nörlund means of the Fourier series has been studied by Alexits [8], Sahney and Goel [9], Chandra [10], Qureshi [11], Qureshi and Neha [12], Rhoades [13], etc But nothing seems to have been done so far in the direction of present work Therefore, in present paper, we establish two new theorems on degree of approximation of function f belonging to Lip (a,r) (r ≥ 1) and to weighted class W(Lr, ξ (t))(r ≥ 1) by Kl-means on its Fourier series and two other new theorems on degree of approximation of function ˜ f , conjugate of a 2π-periodic function f belonging to Lip (a,r) (r > 1) and to weighted class W(Lr,ξ (t)) (r ≥ 1) by Kl-means on its conjugate Fourier series Definitions and notations Let us define, for n = 0, 1, 2, , the numbers n−1 n (x + ν) = v−0 m=0 n m x = m n , for ≤ m ≤ n, by m (x + n) (x) (2:1) = x(x + 1)(x + 2) (x + n − 1) © 2011 Nigam and Sharma; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 Page of 21 n are known as the absolute value of stirling number of first kind m The numbers Let {sn} be the sequence of partial sums of an infinite series ∑un, and let us write (λ) (λ + n) sλ = n n m=0 n λm sm m (2:2) to denote the nth Kl-mean of order l > If sλ → s as n ® ∞, where s is a fixed n finite number, then the sequence {sn} or the series ∑un is said to be summable by Karamata method (Kl) of order l > to the sum s, and we can write sλ → s K λ n as n → ∞ (2:3) Let f be a 2π-periodic function and integrable in the sense of Lebesgue The Fourier series associated with f at a point x is defined by a0 f (x) ∼ + ∞ ∞ (an cos nx + bn sin nx) ≡ n=1 An (x) (2:4) n=1 with nth partial sums sn(f;x) The conjugate series of Fourier series (2.4) is given by ∞ ∞ (an sin nx − bn cos nx) ≡ n=1 Bn (x) (2:5) n=1 s with nth partial sums ˜n (f ; x) Throughout this paper, we will call (2.5) as conjugate Fourier series of function f L∞-norm of a function f: R ® R is defined by f ∞ = sup{|f (x)| : x ∈ R} (2:6) Lr-norm is defined by 2π f r = |f (x)| dx r r , r ≥ (2:7) The degree of approximation of a function f: R ® R by a trigonometric polynomial tn of degree n under sup norm || ||∞ is defined by (Zygmund [14]) tn − f ∞ = sup{|tn (x) − f (x) | : x ∈ R} (2:8) and En (f) of a function f Ỵ Lr is given by En (f ) = tn tn − f r (2:9) This method of approximation is called trigonometric Fourier approximation A function f Ỵ Lip a if |f (x + t) − f (x) | = O(|t|α ) and for 0 0, l r + s = 1, ≤ r ≤ ∞, conditions (3.4) and (3.5) hold uniformly in x, sn is K -mean of Fourier series (2.4) 3.3 Theorem ˜ If a function f , conjugate to a 2π-periodic function f, belonging to Lip(a,r) then its degree of approximation by Kl-summability means on its conjugate Fourier series is given by ⎡ ⎤ log (n + 1) e 1 ˜ ˜n − f r = O ⎣ + s + 1⎦ , (3:6) (λ + n) (n + 1)2 (n + 1)α− r < α ≤ 1, n = 0, 1, 2, , where ˜n is Kl-mean of conjugate Fourier series (2.5) and s ˜ f (x) = − 2π π ψ (t) cot t dt 3.4 Theorem ˜ If a function f , conjugate to a 2π-periodic function f, belonging to W (Lr,ξ (t)) then its degree of approximation by Kl-summability means on its conjugate Fourier series is given by ˜ ˜n − f s r = O (n + 1)β+ r ξ log (n + 1) + + (n + 1) (n + 1)2 n+1 (λ + n) (3:7) provided that ξ (t) satisfies the conditions (3.3)-(3.5) in which δ is an arbitrary positive number such that s (1 - δ) - > 0, l r + s = 1,1≤r≤ ∞ Conditions (3.4) and (3.5) hold uniformly in x, ˜n is K -mean of conjugate Fourier series (2.5) and s ˜ f (x) = − 2π π ψ (t) cot t dt Lemmas For the proof of our theorems, following lemmas are required (3:8) Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 Page of 21 4.1 Lemma (Vuĉkoviĉ [14]) Let l > and < t < π , then λeit + n Im (λ cos t + n) sin = t | sin λ log (n + 1) sin t | sin t +O (1) as n → ∞ uniformly in t 4.2 Lemma Kn (t) = O λ log (n + 1) + O (1) Proof For < t < |Kn (t)| ≤ n+1 , (λ + n) t − cos t < t2 , sin nt ≤ nt and sin ≥ n m=0 n · λm m sin m + sin ⎡ ⎧ ⎫⎤ ⎨ it λeit + n ⎬ ⎥ ⎢ Im e ⎢ ⎩ ⎥ λeit ⎭ ⎥ ⎢ ⎢ ⎥ = O⎢ ⎥ t ⎢ ⎥ (λ + n) sin ⎢ ⎥ ⎣ ⎦ ⎡ t π t t by (2.1) ⎤ it Re λeit + n ⎢ Im λe + n ⎥ = O⎣ ⎦+O t (λ + n) (λ + n) sin ⎤ ⎡ ⎢ (λ cos t + n) = O⎣ · (λ + n) ⎡ ⎢ = O ⎣n−λ(1−cos t) · ⎡ λeit + n ⎥ t ⎦+O (λ cos t + n) sin ⎤2 Im (4:1) (λ cos t + n) (λ + n) λeit + n ⎥ −λ(1−cos t) t ⎦+O n (λ cos t + n) sin ⎤ Im ⎢ = O ⎣e−λ(1−cos t) log n · ⎡ λ2 ⎢ − t log(n+1) · = O ⎣e λeit + n ⎥ −λ(1−cos t) log n t ⎦+O e (λ cos t + n) sin ⎤ ⎡ λ ⎤ it + n − t log(n+1) Im λe ⎥ ⎣ ⎦ t ⎦+O e (λ cos t + n) sin Im Considering first part of (4.1) and using Lemma 1, λ Kn (t) = O e− t log(n+1) λ + O e− t λ = O e− t 2 · | sin λ log (n + 1) · sin t | sin t λ log(n+1) λ log(n+1) + O e− t log(n+1) log(n+1) = O λ log (n + 1) · | sin λ log (n + 1) · sin t | sin t | sin λ log (n + 1) · sin t | = O λ log (n + 1) + O (1) sin t + O e− t + O (1) Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 4.3 Lemma ⎡ ˜ Kn (t) =O λ −2t ⎣e log(n+1) t λ + O e− t Proof For < t < ⎤ ⎦ + O λ log (n + 1) | sin λ log (n + 1) · sin t | log(n+1) n+1 , (λ + n) |Kn (t)| ≤ Page of 21 · | sin t/2 | · t − cos t < t2 , sin nt ≤ nt and sin ≥ n n · λm m m=0 cos m + ⎡ ⎧ ⎫⎤ ⎨ it λeit + n ⎬ ⎥ ⎢ Re e ⎢ ⎩ ⎥ λeit ⎭ ⎥ ⎢ ⎢ ⎥ = O⎢ ⎥ t ⎢ ⎥ (λ + n) sin ⎢ ⎥ ⎣ ⎦ ⎡ sin t π t t by (2.1) ⎤ it Im λeit + n ⎢ Re λe + n ⎥ = O⎣ ⎦+O t (λ + n) (λ + n) sin 2⎤ ⎡ ⎢ (λ cos t + n) ⎥ = O⎣ t ⎦+O (λ + n) sin ⎤2 ⎡ −λ(1−cos t) ⎢n = O⎣ ⎡ t sin λeit + n ⎥ −λ(1−cos t) Im · ⎦+O n (λ cos t + n) ⎤ −λ(1−cos t) log n ⎢e = O⎣ ⎡ (λ cos t + n) Im λeit + n · (λ + n) (λ cos t + n) t sin λeit + n ⎥ −λ(1−cos t) log n Im · ⎦+O e (λ cos t + n) ⎤ λ2 ⎡ λ ⎤ t log n ⎢e ⎥ − t log n Im λeit + n ⎢ ⎥ + O ⎣e ⎦ = O⎣ · t ⎦ (λ cos t + n) sin ⎡ λ ⎤ ⎤ ⎡ λ − t log(n+1) ⎢e ⎥ − t log(n+1) Im λeit + n ⎢ ⎥ + O ⎣e ⎦ = O⎣ · ⎦ t (λ cos t + n) − Using Lemma 1, ⎡ Kn (t) = O λ −2t ⎣e log(n+1) t λ λ ⎦ + O e− t log(n+1) λ ⎤ · | sin λ log (n + 1) · sin t | + O e− t log(n+1) · | sin t/2 | ⎤ ⎡ λ e− t log(n+1) ⎦ + O λ log (n + 1) | sin λ log (n + 1) · sin t | = O⎣ t + O e− t □ log(n+1) · | sin t/2 | Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 Page of 21 4.4 Lemma (McFadden [15]), Lemma 5.40) If f(x) belongs to Lip(a,r) on [0,π], then (t) belongs to Lip(a,r) on [0,π] Proof of the theorems 5.1 Proof of Theorem Following Titchmarsh [16] and using Riemann-Lebesgue theorem, the mth partial sum sm(x) of series (2.4) at t = x is given by sm (x) − f (x) = 2π π φ (t) sin m + t dt t sin Therefore, (λ) (λ + n) n m=0 n λm sm (x) − f (x) = m 2π π φ (t) (λ) (λ + n) n m+0 n m sin m + t dt · λm t sin (λ) π sm (x) − f (x) = φ (t)Kn (t) dt 2π = (5:1) (λ) n+1 ∫0 + ∫π φ (t) Kn (t) dt 2π n+1 = O (I1.1 ) + O (I1.2 ) say Now we consider, n+1 I1.1 = ∫0 |φ (t) Kn (t) |dt Using Lemma 2, 1 n+1 n+1 I1.1 = O λ log (n + 1) ∫0 |φ (t) |dt + O ∫0 |φ (t) |dt Using Hölder’s inequality and Lemma 4, ⎡ ⎢ λ log (n + 1) + ⎣ I1.1 = O n+1 r tφ(t) tα ⎤ ⎡ r ⎥ ⎢ dt⎦ ⎣ ⎤ s n + tα−1 s dt⎥ ⎦ ⎤1 s ⎢ tαs−s+1 n +1⎥ ⎥ ⎢ ⎦ n + ⎣ αs − s + ⎡ λ log (n + 1) + =O log (n + 1) e =O (n + 1) ⎡ ⎢ log (n + 1) e = O⎢ ⎣ (n + 1) ⎡ ⎢ = O⎢ ⎣ 1 s (n + 1)αs−s+1 ⎤ ⎥ ⎥ 1⎦ α−1+ s (n + 1) ⎧ ⎫⎤ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬⎥ log (n + 1) e ⎥ ⎪⎦ ⎪ (n + 1) ⎪ ⎩ ⎭ α− ⎪ (n + 1) r (5:2) since 1 + = r s Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 Since, for Page of 21 t t ≤ t ≤ π , sin ≥ n+1 π t (λ + n) sin Kn (t) = O (5:3) (λ + n) t =O Next we consider, π |I1.2 | ≤ |φ (t) ||Kn (t) |dt n+1 Using Hölder’s inequality, (5.3) and Lemma 4, I1.2 = O =O ⎡ ⎡ ⎤1 ⎢ π r π r⎢ t−δ φ (t) ⎢ ⎣ dt ⎦ ⎢ ⎢ tα (λ + n) ⎣ n+1 n+1 ⎤1 ⎡ π s 1 ⎣ t(δ+α−1)s dt ⎦ (λ + n) (n + 1)−δ n+1 ⎤1 s ⎣ ⎦ (δ + α − 1) s + (n + 1)−δ n+1 1 s (n + 1)−δ (n + 1)(δ+α−1)s+1 ⎡ ⎤ ⎡ =O (λ + n) =O (λ + n) =O (λ + n) tδ+α t ⎤1 s ⎥ s ⎥ ⎥ dt ⎥ ⎥ ⎦ ⎢ ⎢ (n + 1)−δ ⎣ ⎡ π t(δ+α−1)s+1 (5:4) ⎥ ⎥ 1⎦ (δ+α−1)+ s (n + 1) ⎤ =O ⎢ ⎢ (λ + n) ⎣ =O ⎢ ⎢ (λ + n) ⎣ ⎥ ⎥ 1⎦ α− (n + 1) r ⎥ ⎥ 1⎦ α−1+ s (n + 1) ⎡ ⎤ Combining (5.1), (5.2) and (5.4), ⎡ ⎢ log (n + 1) e ⎜ ⎜ Sm − f (x) = O ⎢ ⎣ ⎝ (n + 1) ⎡ ⎢ = O⎢ ⎣ ⎞⎤ ⎛ 1 α− (n + 1) r ⎡ ⎢ ⎟⎥ ⎟⎥ + O ⎢ ⎣ ⎠⎦ α− (n + 1) r ⎤ log (n + 1) e + (n + 1) This completes the proof of Theorem ⎥ ⎥ (λ + n) ⎦ ⎛ ⎞⎤ ⎜ ⎜ (λ + n) ⎝ ⎟⎥ ⎟⎥ ⎠⎦ α− (n + 1) r Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 Page of 21 5.2 Proof of Theorem Following the proof of Theorem 1, ⎡ (λ) ⎢ n + Sm (x) − f (x) = + ⎣ 2π ⎤ π ⎥ ⎦ φ (t) Kn (t) dt n+1 = O (I2.1 ) + O (I2.2 ) (5:5) say We have |φ (x + t) − φ (x) | ≤ |f (u + x + t) − f (u + x) | + |f (u − x − t) − f (u − x) | Hence, by Minkowiski’s inequality, 2π β r {|φ (x + t) − φ (x)} sin x| dx r 2π ≤ β r | f (u + x + t) − f (u + x) sin x| dx r 2π + β | f (u − x − t) − f (u − x) sin x| dx = O {ξ (t)} Then f Ỵ W (Lr,ξ(t))⇒  Ỵ W(Lr, ξ (t)) Now we consider, |I2.1 | ≤ 0n + |φ (t) ||Kn (t) |dt Using Lemma 2, n + |φ (t) |dt I2.1 = O λ log (n + 1) + O (1) Using Hölder’s inequality and the fact that  (t) Ỵ W (Lr, ξ (t)), ⎡ I2.1 = O ⎢ λ log (n + 1) + ⎣ ⎡ ⎢ ·⎣ n+1 ⎤ ξ (t) β tsin t s ⎥ dt ⎦ n+1 ⎤ ⎥ dt ⎦ r s ⎡ = O λ log (n + 1) e t|φ (t) |sinβ (t) ξ (t) r ⎢ ⎣ n+1 n+1 ⎤ ξ (t) β tsin t s ⎥ dt ⎦ s by (3.4) r r Nigam and Sharma Journal of Inequalities and Applications 2011, 2011:85 http://www.journalofinequalitiesandapplications.com/content/2011/1/85 Page 10 of 21 Since sin t ≥ 2t/π, ⎤ ⎡ I2.1 log (n + 1) e ⎢ n + =O ⎣ n+1 s ξ (t) t1+β s ⎥ dt ⎦ Since ξ (t) is a positive increasing function and using second mean value theorem for integrals, ⎡ log (n + 1) e ξ n+1 I2.1 = O ⎢ ⎣ n+1 n+1 ∈ ⎤ ⎥ dt⎦ t(1+β)s s for some