8_620_nigam.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 3, 2011, 276-286 ISSN 1307-5543 – www.ejpam.com Approximation of Conjugate of Functions Belonging to Lipα Class and W � Lr ,ξ (t) � Class by Product Means of Conjugate Fourier Series H. K. Nigam∗, Kusum Sharma Department of Mathematics, Faculty of Engineering and Technology, Mody Institute of Technology and Science (Deemed University), Laxmangarh-332311, Sikar (Rajasthan), India Abstract. In this paper, two quite new theorems on degree of approximation of conjugate of functions f ∈ Lipα class and f ∈W � Lr ,ξ (t) � class using (E, 1) (C , 1) product summability means of conjugate Fourier series have been established. 2000 Mathematics Subject Classifications: 42B05, 42B08 Key Words and Phrases: Degree of approximation, Lipα Class, W (Lr ,ξ(t)) class of functions, (E, 1) summability, (C , 1) summability, (E, 1) (C , 1) product summability, Fourier series, conjugate Fourier series, Lebesgue integral. 1. Introduction A good amount of work to determine the degree of approximation of functions belong- ing to the classes Lipα, Lip (α, r), Lip (ξ (t) , r) and W � Lr ,ξ (t) � using Cesàro, Nörlund and generalized Nörlund single summability methods has been done by several researchers like Alexits [1], Sahney and Goel [12], Qureshi and Neha [8], Qureshi [9, 10], Chandra [2], Khan [4], Leindler [6] and Rhoades [11] . But nothing seems to have been done so far to obtain degree of approximation using different class of functions by product summability method. Therefore, in present work, two theorems on degree of approximation of the conjugate of functions f ∈ Lipα and f ∈ W � Lr ,ξ (t) � , (r ≥ 1) using (E, 1) (C , 1) summability means of conjugate Fourier series have been proved. Let ∑∞ n=0 un be a given infinite series with sequence of its nth partial sum � sn . ∗Corresponding author. Email addresses: harekrishnan�yahoo. om (H. Nigam), kusum31sharma�rediffmail. om (K. Sharma) http://www.ejpam.com 276 c© 2011 EJPAM All rights reserved. H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 277 If (E, 1) transform is defined as the nth partial sum of (E, 1) summability and it can be denoted by E1 n , which is given by E1 n = 1 2n n ∑ k=0 � n k � sk→ s as n→∞ (1) then the infinite series ∑∞ n=0 un is summable (E, 1) to a definite number s (Hardy [3]). If tn = s0 + s1 + s2 + . . .+ sn n+ 1 = 1 n+ 1 n ∑ k=0 sn→ s as n→∞ (2) then the infinite series ∑∞ n=0 un is summable to the definite number s by (C , 1) method. The (E, 1) transform of (C , 1) transform defines (E, 1)(C , 1) product transform and we denote it by (EC)1n. Thus if (EC)1n = 1 2n n ∑ k=0 � n k � C1 k → s as n→∞ (3) then the infinite series ∑∞ n=0 un is said to be summable by (E, 1) (C , 1) method or summable (E, 1) (C , 1) to a definite number s. Let f (x) be a 2π-periodic function and Lebesgue integrable. The Fourier series of f (x) is given by f (x)∼ a0 2 + ∞ ∑ n=1 � an cos nx + bn sin nx � (4) with nth partial sum sn � f ; x � . The conjugate series of Fourier series (4) is given by ∞ ∑ n=1 � an cos nx − bn sin nx � , (5) and we shall call it as conjugate Fourier series. A function f ∈ Lipα if f (x + t)− f (x) = O � |t|α � for 0< α≤ 1. (6) H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 278 f ∈ Lip (α, r ) for 0≤ x ≤ 2π, if [definition 5.38 of McFadden, 7] ∫ 2π 0 � � f (x + t)− f (x) � � r d x ! 1 r = O � |t|α � , 0 < α≤ 1, r ≥ 1. (7) Given a positive increasing function ξ (t) and an integer r ≥ 1, f ∈ Lip (ξ (t) , r) if ∫ 2π 0 � � f (x + t)− f (x) � � r d x ! 1 r = O (ξ (t)) (8) and that f ∈W � Lr ,ξ (t) � if ∫ 2π 0 � � � f (x + t)− f (x) sinβ x � � r d x ! 1 r = O (ξ (t)) ,β ≥ 0. (9) where ξ(t) is a positive increasing function of t. If β = 0 then W � Lr ,ξ (t) � reduces to the class Lip (ξ (t) , r) and if ξ(t) = tα then Lip(ξ(t), r) class coincides with the class Lip(α, r) and if r → ∞ then Lip(α, r) class re- duces to the class Lipα. L∞-norm of a function f : R→ R is defined by f ∞ = sup ¦� � f (x) � � : x ∈ R © (10) Lr -norm is defined by f r = ∫ 2π 0 � � f (x) � � r d x ! 1 r , r ≥ 1. (11) The degree of approximation of a function f : R→ R by a trigonometric polynomial tn of degree n under sup norm ‖·‖∞ is defined as [Zygmund, 13] tn − f ∞ = sup ¦� �tn (x)− f (x) � � : x ∈ R © (12) and En � f � of a function f ∈ Lr is given by En � f � = min tn tn − f r (13) We use the following notations throughout this paper: ψ (t) = f (x + t) + f (x − t) K̄n (t) = 1 π 2n+1 n ∑ k=0 ( � n k � 1 (1+ k) k ∑ ν=0 cos � ν + 1 2 � t sin t 2 ) τ = � 1 t � , where τ denotes the greatest integer not greater than 1 t . H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 279 2. Main Theorems We prove the following theorems: Theorem 1. If a function f , conjugate to a 2π-periodic function f , belongs to Lipα class, then its degree of approximation by (E, 1) (C , 1) means of conjugate Fourier series is given by sup 0 0, 1 r + 1 s = 1, conditions (17) and (18) hold uniformly in x and (EC)1n, as defined in (3), is (E, 1) (C , 1) means of the series (5) and f (x) = − 1 2π ∫ 2π 0 ψ (t) cot � t 2 � d t (19) 3. Lemmas For the proof of our theorems, following lemmas are required: Lemma 1. � �Gn (t) � � = O � 1 t � , for 0≤ t ≤ 1 n+ 1 H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 280 Proof. For 0≤ t ≤ 1 n+1 , sin � t 2 � ≥ t π and |cos nt| ≤ 1 � �Gn (t) � � ≤ 1 π 2n+1 � � � � � n ∑ k=0 ( � n k � 1 (1+ k) k ∑ ν=0 cos � ν + 1 2 � t sin t 2 )� � � � � ≤ 1 π 2n+1 n ∑ k=0    � n k � 1 (1+ k) k ∑ ν=0 � � �cos � ν + 1 2 � t � � � � � �sin t 2 � � �    = 1 t 2n+1 n ∑ k=0 ( � n k � � 1 1+ k � k ∑ ν=0 1 ) = 1 t 2n+1 n ∑ k=0 ¨� n k �« = 1 t 2n+1 2n = O � 1 t � since n ∑ k=0 � n k � = 2n Lemma 2. For 0≤ a ≤ b ≤∞, 0≤ t ≤ π and any n, we have � �Gn (t) � �= O � 1 t � Proof. For 0≤ 1 n+1 ≤ t ≤ π, sin � t 2 � ≥ t π . � �Gn (t) � � ≤ 1 π 2n+1 � � � � � n ∑ k=0 ( � n k � 1 (1+ k) k ∑ ν=0 cos � ν + 1 2 � t sin t 2 )� � � � � ≤ 1 2n+1 t � � � � � n ∑ k=0   � n k � 1 (1+ k) Re ( k ∑ ν=0 ei � ν+ 1 2 � t )  � � � � � ≤ 1 2n+1 t � � � � � n ∑ k=0   � n k � 1 (1+ k) Re ( k ∑ ν=0 eiν t )  � � � � � � � �e i t 2 � � � ≤ 1 2n+1 t � � � � � n ∑ k=0   � n k � 1 (1+ k) Re ( k ∑ ν=0 eiν t )  � � � � � ≤ 1 2n+1 t � � � � � τ−1 ∑ k=0   � n k � 1 (1+ k) Re ( k ∑ ν=0 eiν t )  � � � � � + 1 2n+1 t � � � � � n ∑ k=τ   � n k � 1 (1+ k) Re ( k ∑ ν=0 eiν t )  � � � � � (20) H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 281 Now considering the first term of (20), 1 2n+1 t � � � � � τ−1 ∑ k=0   � n k � 1 (1+ k) Re ( k ∑ ν=0 eiν t )  � � � � � ≤ 1 2n+1 t � � � � � τ−1 ∑ k=0   � n k � 1 (1+ k) ( k ∑ ν=0 1 )  � � � � � � �eiν t � � ≤ 1 2n+1 t � � � � � τ−1 ∑ k=0 �� n k �� � � � � � (21) Now considering the second term of (20) and using Abel’s lemma 1 2n+1 t � � � � � n ∑ k=τ   � n k � 1 (1+ k) Re ( k ∑ ν=0 eiν t )  � � � � � ≤ 1 2n+1 t n ∑ k=τ � n k � 1 (1+ k) max 0≤ m≤ k � � � � � m ∑ ν=0 eiν t � � � � � ≤ 1 2n+1 t n ∑ k=τ � n k � 1 (1+ k) (1+ k) = 1 2n+1 t n ∑ k=τ � n k � (22) Combining (20), (21) and (22), we get � �Gn (t) � � ≤ 1 2n+1 t τ−1 ∑ k=0 � n k � + 1 2n+1 t n ∑ k=τ � n k � = O � 1 t � 4. Proof of Theorems 4.1. Proof of Theorem 1 Let sn � f ; x � denote the partial sum of series (5). Then following Lal [5], we have sn (x)− f (x) = 1 2π ∫ π 0 ψ (t) cos � n+ 1 2 � t sin � t 2 � d t Using (5) the (C , 1) transform C1 n of sn � f ; x � is given by C1 n − f (x) = 1 2π (n+ 1) ∫ π 0 ψ (t) n ∑ k=0 cos � k+ 1 2 � t sin t 2 d t (23) H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 282 Now denoting (E, 1) (C , 1) transform of sn by (EC)1n, we write (EC)1n − f (x) = 1 2n+1 π n ∑ k=0   � n k �∫ π 0 ψ (t) sin t 2 � 1 k+ 1 � ( k ∑ ν=0 cos � ν + 1 2 � t ) d t   = ∫ π 0 ψ (t) Gn (t) d t =    ∫ 1 n+1 0 + ∫ π 1 n+1   ψ (t) Gn (t) d t = I1.1 + I1.2 (say) (24) Now using Lemma 1, we have � �I1.1 � � ≤ ∫ 1 n+1 0 � �ψ (t) � � � �Gn (t) � �d t = ∫ 1 n+1 0 |tα| t d t = ∫ 1 n+1 0 tα−1d t = � tα α � 1 n+1 0 = O � 1 (n+ 1)α � (25) Using Lemma 2, we have � �I1.2 � � = ∫ π 1 n+1 � �ψ (t) � � � �Gn (t) � � d t = ∫ π 1 n+1 |tα| |t| d t = ∫ π 1 n+1 tα−1d t = � tα α �π 1 n+1 = O � 1 (n+ 1)α � (26) H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 283 Combining (24), (25) and (26), we get (EC)1n− f ∞ = §� � �(EC)1n − f � � � : x ∈ [0,2π] ª = O � 1 (n+ 1)α � This completes the proof of Theorem 1. 4.2. Proof of Theorem 2 Following the proof of Theorem 1, (EC)1n− f (x) =    ∫ 1 n+1 0 + ∫ π 1 n+1   ψ (t) Gn (t) d t = I2.1 + I2.2 (say) (27) Applying Hölder’s inequality and the fact that ψ (t) ∈W � Lr ,ξ (t) � , condition (17), Lemma 1 and second mean value theorem for integrals, we have � �I2.1 � � ≤    ∫ 1 n+1 0 ( t � �ψ (t) � � sinβ t ξ (t) )r d t    1 r    ∫ 1 n+1 0 ( ξ (t) � �Gn (t) � � t sinβ t )s d t    1 s = O � 1 n+ 1 �    ∫ 1 n+1 0 � ξ (t) t2+β �s d t    1 s = O �� 1 n+ 1 � ξ � 1 n+ 1 ��    ∫ 1 n+1 ∈ d t t(2+β)s    1 s for some 0<∈< 1 n+ 1 = O    � 1 n+ 1 � ξ � 1 n+ 1 � ( t−(2+β)s+1 − � 2+ β � s+ 1 ) 1 n+1 ∈    1 s = O �� 1 n+ 1 � ξ � 1 n+ 1 � (n+ 1)2+β− 1 s � = O � (n+ 1)β+1− 1 s ξ � 1 n+ 1 �� = O � (n+ 1)β+ 1 r ξ � 1 n+ 1 �� since 1 r + 1 s = 1,1≤ r ≤∞. (28) Now using Hölder’s inequality, |sin t| < 1, sin t ≥ � 2t π � , conditions (16) and (18), Lemma 2 and second mean value theorem for integrals, we have � �I2.2 � �≤    ∫ π 1 n+1 ( t−δ � �ψ (t) � � sinβ t ξ (t) )r d t    1 r    ∫ π 1 n+1 ( ξ (t) � �Gn (t) � � t−δ sinβ t )s d t    1 s H. Nigam, K. Sharma / Eur. J. Pure Appl. Math, 4 (2011), 276-286 284 = O ¦ (n+ 1)δ ©    ∫ π 1 n+1 � ξ (t) t1−δ+β �s d t    1 s = O ¦ (n+ 1)δ ©      ∫ n+1 1 π    ξ � 1 y � yδ−1−β    s d y y2      1 s = O � (n+ 1)δ ξ � 1 n+ 1 ��   ∫ n+1 1 π d y ys(δ−1−β)+2   1 s = O � (n+ 1)δ ξ � 1 n+ 1 ��   (n+ 1)s(1+β−δ)−1 −πs(δ−1−β)+1 s � 1+ β − δ � − 1   1 s = O � (n+ 1)δ ξ � 1 n+ 1 �� h (n+ 1)(1+β−δ)− 1 s i = O � (n+ 1)β+1− 1 s ξ � 1 n+ 1 �� = O � (n+ 1)β+ 1 r ξ � 1 n+ 1 �� since 1 r + 1 s = 1 (29) Now combining (27) to (29), we get � � �(EC)1n− f � � � = O � (n+ 1)β+ 1 r ξ � 1 n+ 1 �� (EC)1n − f r = ( ∫ 2π 0 � � �(EC)1n− f � � � r d x ) 1 r = ( ∫ 2π 0 � (n+ 1)β+ 1 r ξ � 1 n+ 1 ��r d x ) 1 r = O � (n+ 1)β+ 1 r ξ � 1 n+ 1 ��    ( ∫ 2π 0 d x ) 1 r    = � (n+ 1)β+ 1 r ξ � 1 n+ 1 �� This completes the proof of the Theorem 2. 5. Applications Following corollaries can be derived from our main theorem: REFERENCES 285 Corollary 1. If β = 0 and ξ (t) = tα, then the degree of approximation of a function f , conjugate to 2π-periodic function f ∈ Lip (α, r) , 1 r ≤ α ≤ 1, is given by (EC)1n − f r = O ( 1 (n+ 1)α− 1 r ) Corollary 2. If r → ∞ in Corollary 1, then Lip (α, r) reduces to Lipα for 0 < α < 1, and we have (EC)1n− f r = O � 1 (n+ 1)α � Remark 1. An independent proof of Corollary 1 can be obtained along the same lines of our Theorem 2. ACKNOWLEDGEMENTS The first Author is thankful to his parents for their encouragement and support to this work. References [1] G Alexits, Convergence problems of orthogonal series, Pergamon Press, London, 1961. [2] P Chandra, Trigonometric approximation of functions in Lp norm, J. Math. Anal. Appl. 275 No. 1, 13-26. 2002. [3] G Hardy, Divergent series, first edition, Oxford University Press, 70. 1949. [4] H Khan, On degree of approximation of functions belonging to the class Lip(α, p), Indian J. Pure Appl. Math. 5 No. 2, 132-136. 1974. 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