EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 5, 2020, 1325-1336 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday On approximation of signals in the generalized Zygmund class using (E, r)(N, qn) mean of conjugate derived Fourier series Anwesha Mishra1, Birupakhya Prasad Padhy1,∗, Umakanta Misra3 1 Department of Mathematics, School of Applied Sciences, KIIT, Deemed to be University, Bhubaneswar, Odisha, India 2 Department of Mathematics, National Institute of Science and Technology, Pallur Hills, Berhampur, Odisha, India. Abstract. In the present article, we have established a result on degree of approximation of function in the generalized Zygmund class Zl (m),(l ≥ 1) by (E, r)(N, qn)- mean of conjugate derived Fourier series. 2020 Mathematics Subject Classifications: 42A10, 41A10, 42B05, 42B08 Key Words and Phrases: Degree of approximation, Generalized Zygmund class, Fourier series, Conjugate Fourier Series, Conjugate Derived Fourier series, (E, r)-Summability mean, (N, qn) - summability mean, (E, r)(N, qn)-summability mean 1. Introduction Signal Analysis describes the field of study whose objective is to collect, understand and deduce information and intelligence from various signals. Now-a-days the analysis of signals is a fundamental problem for many engineers and scientists. In the recent past, we have seen the applications of mathematical methods such as Probability theory, Mathematical statistics etc. in the analysis of signals. Very recently, approximation ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3714 Email addresses: m.anwesha17@gmail.com (A. Mishra), birupakhya.padhyfma@kiit.ac.in (B. P. Padhy), umakanta misra@yahoo.com (U. K. Misra) https://www.ejpam.com 1325 c© 2020 EJPAM All rights reserved. A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1326 theory has got a large popularity as it has given a new dimension in approximating the signals. The estimation of error functions in Lipschitz and Zygmund space using different summability techniques of Fourier series and conjugate Fourier series have been of great interest among the researchers in the last decades. For details see [3, 7, 9, 12, 13] and [15] to [16]. Also, the generalized Zygmund class Zl (m),(l ≥ 1) was investigated by Leindler [8], Moricz [4], Moricz and Nemeth [6] etc. Very recently Das et al.[1], Nigam [7], Pradhan et al.[11, 14] and Singh et al.[10] proved approximation of functions in the generalized Zygmund class by using different summability means. In the present paper, we investigate on the degree of approximation of a function in the generalized Zygmund class Zl (m),(l ≥ 1) by (E, r)(N, qn) product mean of the conjugate derived Fourier series. 2. Definitions and Notations Let h be a function, which is periodic in [0, 2π] such that ∫ 2π 0 |h(x)|ldx <∞. Let us denote Ll[0, 2π] = { h : [0, 2π]→ R : ∫ 2π 0 |h(x)|ldx <∞ } , l ≥ 1. The Fourier series of h(x) is given by ∞∑ n=0 un(x) = a0 2 + ∞∑ n=1 ( ancosnx+ bnsinnx ) (1) Also,the conjugate Fourier series and derived conjugate Fourier series of h(x) are respec- tively ∞∑ n=1 ( bncosnx− ansinnx ) and − ∑ n un(x). Let us define ‖h‖l = ( 1 2π ∫ 2π 0 |h(x)|ldx ) 1 l , 1 ≤ l <∞ and ‖h‖l = ess sup 0≤x≤2π |h(x)|, l =∞ Let S′p ( h;x ) denotes the p-th partial sum of conjugate derived Fourier series and is given by S′p ( h;x ) − h′(x)− = − 2 π ∫ π 0 Ψ(x; v) 4sinv2 ( k + 1 2 ) sin ( k + 1 2 ) v dv − 1 π ∫ π 0 Ψ(x; v) 4sinv2 sin ( k + 1 2 ) v tanv2 dv A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1327 Where h′ is the conjugate derived function of 2π periodic function ’h’, which is given by h′(x) = − 1 π ∫ π 0 Ψ(x; v) cosec2 v 2 dv Let the Zygmund modulus of continuity of h(x) be: m(h; r) = sup 0≤r,x∈R |h(x+ v) + h(x− v)|(see [2] Let B represents the Banach space of all 2π periodic functions which are continuous and defined over [0, 2π] under the supremum norm. Clearly, Z(α) = { h ∈ B : |h(x+ v) + h(x− v)| = O ( |v|α ) , 0 < α ≤ 1 } is a Banach space under the norm ‖.‖(α) defined by ‖h‖(α) = sup 0≤x≤2π |h(x)|+ sup x,t6=0 |h(x+ v) + h(x− v)| |v|α For h ∈ Ll[0, 2π], (l ≥ 1), the integral Zygmund modulus of continuity is defined by ml(h; r) = sup 0 0 for 0 ≤ v < 2π and lim v→0+ m(v) = m(0) = 0. Define Z (m) l = { h ∈ Ll : 1 ≤ l <∞, sup v 6=0 ‖h(.+ v) + h(.− v)‖l m(v) <∞ } where ‖h‖(m) l = ‖h‖l + sup v 6=0 ‖h(.+ v) + h(.− v)‖l m(v) , l ≥ 1. Clearly, ‖.‖(m) l is a norm Z (m) l . Also, Z (m) l is complete since Ll, (l ≥ 1) is complete. So, Z (m) l is a Banach space under ‖.‖(m) l . Let m(v) and µ(v) represents the Zygmund moduli of continuity such that ( m(v) µ(v) ) is positive and non-decreasing then ‖h‖(µ) l ≤ max . ( 1, m(2π) µ(2π) ) ‖h‖(m) l ≤ ∞ (2) Clearly, Z (m) l ⊆ Z(µ) l ⊆ Ll, (l ≥ 1). Let ∑ un be an infinite series with sequence of partial sums {sn}. Let {qk} represents the sequence of non-negative integers such that Qn = n∑ k=0 qk →∞ as n→∞. (3) Let τNn = 1 Qn n∑ k=0 qn−ksk, n = 0, 1, 2, ... (4) represents the (N, qn) mean of {sn} generated by the sequence {qn}. By (N, qn) method, the series ∑ un is said to be summable to ′s′ if lim n→∞ τNn → s. A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1329 We know, (N, qn) method is regular [5]. The (E, r) transform of {sn} is given by Ern = 1 (1 + r)n n∑ k=0 C(n, k) rn−ksk (5) If Ern → s as n → ∞ then ∑ un is summable to ’s’ by (E, r) summability. Also, (E, r) method is regular [5]. The (E, r)(N, qn) transform of {sn} is given by τEr,N n = 1 (1 + r)n n∑ k=0 C(n, k) { 1 Qk k∑ ν=0 qk−νsν } (6) The series ∑ un is summable to s by the (E, r)(N, qn) transform if τEr,N n → s as n→∞. Also we have used the following notation in the rest part of our paper. Ψ(x, v) = h(x+ v) + h(x− v) 3. Known Result Using Hausdorff mean, Nigam [7] proved the following theorem: Theorem 1. Error approximation of a conjugate derived function h′ of a 2π periodic function h ∈ Z(m) l , l ≥ 1, using H = ( θj,α ) of conjugate derived Fourier series is given by ‖M ′H j (h; .)− h′(.)‖(m) l = O ( 1 j + 1 ∫ π 1 j+1 (v + 1)m(v) v3 µ(v) dv ) , where m(v) and µ(v) are Zygmund moduli of continuity, provided∫ π 0 m(v) v2µ(v) dv = O ( m(η) η µ(η) ) , 0 < η < π. 4. Main Theorem Theorem 2. The degree of approximation of a conjugate derived function h′ of a 2π periodic function h ∈ Z(m) l , l ≥ 1, using (E, r)(N, qn)- mean of conjugate derived Fourier series is given by En(h) = inf n ‖χn ′ (.)‖µl = O (∫ π 1 n+1 m(v) v2 µ(v) dv ) A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1330 where m(v) and µ(v) are the Zygmund moduli of continuity and m(v) v µ(v) is positive and non-decreasing, provided ∫ η 0 m(v) v µ(v) dv = O (m(η) µ(η) ) . We require the below mentioned lemmas to prove our main theorem: 5. Lemmas Lemma 1. |Y1 ′ (v)| = O(n2) for 0 < v ≤ π. Lemma 2. |Y2 ′ (v)| = O ( 1 v2 ) for 0 < v ≤ π. Lemma 3. Let h ∈ Z(m) l then for 0 < v ≤ π, (i) ‖Ψ(., v)|l = O ( m(v) ) (ii) ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l = O ( m(v) ) or O ( m(y) ) (iii) If m(v) and µ(v) are as defined in the main theorem, then ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l = O ( µ(y) m(v) µ(v) ) where Ψ(x, v) = h(x+ v) + h(x− v). 6. Proof of the Lemmas 6.1. Proof of Lemma-1 For v ∈ ( 0, 1 n+1 ] and sin nv ≤ n sin v, we have |Y1 ′ (v)| = ∣∣∣ −2 4π(1 + r)n n∑ k=0 k C(n, k)rn−k { 1 Qk k∑ ν=0 qk−ν sin(ν + 1 2)v sinv2 }∣∣∣ ≤ 1 2π(1 + r)n ∣∣∣ n∑ k=0 k C(n, k) rn−k { 1 Qk k∑ ν=0 qk−ν (2ν + 1))sin ( ν + 1 2 ) v sinv2 }∣∣∣ ≤ 1 2π(1 + r)n ∣∣∣ n∑ k=0 k C(n, k) rn−k (2k + 1) { 1 Qk k∑ ν=0 qk−ν }∣∣∣ A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1331 = O(n2) For v ∈ [ 1 n+1 , π ] , 1 sin ( v 2 ) ≤ π v , sinv ≤ v |Y1 ′ (v)| = ∣∣∣ −2 4π(1 + r)n n∑ k=0 k C(n, k)rn−k { 1 Qk k∑ ν=0 qk−ν sin(ν + 1 2)v sinv2 }∣∣∣ ≤ 1 4π(1 + r)n ∣∣∣ n∑ k=0 k C(n, k) rn−k { 1 Qk k∑ ν=0 π v (2ν + 1)v qk−ν} ∣∣∣ ≤ 1 2π(1 + r)n ∣∣∣ n∑ k=0 k C(n, k) rn−k (2k + 1) { 1 Qk k∑ ν=0 qk−ν }∣∣∣ = O(n2) 6.2. Proof of Lemma-2 We know, 1 sin ( v 2 ) ≤ π v , ( 0 < v ≤ π ) ; sin v ≤ v, v > 0; and |sinv| ≤ 1, |cosv| ≤ 1 for all t. Clearly, for v ∈ (0, π], |Y2 ′ (v)| = ∣∣∣ −1 4π(1 + r)n n∑ k=0 C(n, k) rn−k { 1 Qk k∑ ν=0 qk−ν cos νv sin2 v 2 }∣∣∣ = O ( 1 v2 ) 6.3. Proof of Lemma-3 See [12]. 7. Proof of the Main Theorem Let Sk ′ (h;x) denotes the k-th partial sum of the conjugate derived Fourier series, we have Sk ′ (h;x)− h′(x) = −2 π ∫ π 0 Ψ(x, v) 4sinv2 ( k + 1 2 ) sin ( k + 1 2 ) v dv − 1 π ∫ π 0 Ψ(x, v) 4sinv2 cos ( k + 1 2 ) v tanv2 dv where h′ is the conjugate derived function of 2π periodic function h, which is given by h′(x) = 1 4π ∫ π 0 Ψ(x; v) cosec2 v 2 dv A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1332 and the (N, qn) transform of it is given by 1 Qn n∑ k=0 qn−k{Sk ′ (h;x)− h′(x)} = −2k π ∫ π 0 ρ(x, v) 1 Qn n∑ k=0 qn−ksin ( k + 1 2 ) v dv − 1 π ∫ π 0 ρ(x, v) 1 Qn n∑ k=0 qn−k coskv sinv2 dv where ρ(x, v) = Ψ(x;v) 4sin v 2 . Denoting the (E, r)(N, qn) transform of Sk ′ (h;x) by τn′ Er,N . Then, τn′ Er,N − h′(x) = −2k π(1 + r)n ∫ π 0 ρ(x, v) n∑ k=0 C(n, k) rn−k { 1 Qk k∑ ν=0 qk−νsin ( ν + 1 2 ) v } dv − 1 π(1 + r)n ∫ π 0 ρ(x, v) n∑ k=0 C(n, k) rn−k { 1 Qk k∑ ν=0 qk−ν cosνv sinv2 } dv τn′ Er,N − h′(x) = −2k 4π(1 + r)n ∫ π 0 Ψ(x, v) n∑ k=0 C(n, k) rn−k { 1 Qk k∑ ν=0 qk−ν sin ( ν + 1 2 ) v sinv2 } dv − 1 4π(1 + r)n ∫ π 0 Ψ(x, v) n∑ k=0 C(n, k) rn−k { 1 Qk k∑ ν=0 qk−ν cosνv sin2 v 2 } dv = ∫ π 0 Ψ(x, v){Y ′1 (v) + Y ′ 2 (v)} dv = χn ′ (x), (say) Then, χn ′ (x+ y) + χn ′ (x− y) = ∫ π 0 { Ψ(x+ y, v) + Ψ(x− y, v) } {Y ′1 (v) + Y ′ 2 (v)} dv Using Minkowski’s inequality, we have ‖χn ′ (.+ y) + χn ′ (.− y)‖l = { 1 2π ∫ 2π 0 |χn ′ (x+ y) + χn ′ (x− y)|l dx } 1 l ≤ ∫ π 0 { 1 2π ∫ 2π 0 |Ψ(x+ y, v) + Ψ(x− y, v)|ldx } 1 l |Y ′1 (v) + Y ′ 2 (v)| dv = ∫ π 0 ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l |Y ′ 1 (v) + Y ′ 2 (v)| dv = ∫ 1 n+1 0 ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l |Y ′ 1 (v) + Y ′ 2 (v)| dv + ∫ π 1 n+1 ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l |Y ′ 1 (v) + Y ′ 2 (v)| dv A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1333 = I ′ 1 + I ′ 2, (say) (7) Further, |Ψ(x+ y; v) + Ψ(x− y; v)| ≤ ‖h(x+ y + v) + h(x+ y − v)‖+ ‖h(x− y + v) + h(x− y − v)‖ By Minkowski’s inequality, we have |Ψ(.+ y; v) + Ψ(.− y; v)|l ≤ ‖h(.+ y + v) + h(.+ y − v)‖l + ‖h(.− y + v) + h(.− y − v)‖l = O(m(v)) or O(m(y)) Again, by using lemma-1, lemma-3 and monotonicity of m(v) µ(v) , we get I ′ 1 = ∫ 1 n+1 0 ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l|Y ′ 1 (v) + Y ′ 2 (v)| dv ≤ O (∫ 1 n+1 0 µ(y) m(v) µ(v) n2 dv ) +O (∫ 1 n+1 0 µ(y) m(v) µ(v) 1 v2 dv ) = O ( n2µ(y) m ( 1 n+1 ) µ ( 1 n+1 ) )+O ( µ(y) ∫ 1 n+1 0 m(v) µ(v) 1 v2 dv ) (8) Similarly, by using lemma-2, lemma-3 and monotonicity of m(v) µ(v) , we get I ′ 2 = ∫ π 1 n+1 ‖Ψ(.+ y, v) + Ψ(.− y, v)‖l|Y ′ 1 (v) + Y ′ 2 (v)| dv = O ( n2µ(y) ∫ 1 n+1 0 m(v) dv µ(v) ) +O ( (n+ 1)µ(y) m ( 1 n+1 ) µ ( 1 n+1 ) ) (9) By, (7), (8) and (9) ‖χn ′ (.+ y) + χn ′ (.− y)|l = O ( n2µ(y) m ( 1 n+1 ) µ ( 1 n+1 ) )+O ( µ(y) ∫ 1 n+1 0 m(v) µ(v) 1 v2 dv ) +O ( n2µ(y) ∫ 1 n+1 0 m(v) dv µ(v) ) +O ( (n+ 1)µ(y) m ( 1 n+1 ) µ ( 1 n+1 ) ) Therefore, we have sup y 6=0 ‖χn ′ (.+ y) + χn ′ (.− y)‖l µ(y) A. Mishra, B. P. Padhy, U. K. Misra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1325-1336 1334 = O ( n2 m ( 1 n+1 ) µ ( 1 n+1 ) )+O (∫ 1 n+1 0 m(v) µ(v) 1 v2 dv ) +O ( n2 ∫ 1 n+1 0 m(v) dv µ(v) ) +O ( (n+ 1) m ( 1 n+1 ) µ ( 1 n+1 ) ) (10) Since, h ∈ Z(m) l and Ψ(x; v) = |h(x+ v) + h(x− v)|, by Minkowski’s inequality, we have ‖Ψ(x, v)‖l = ‖h(x+ v) + h(x− v)‖l = O ( m(v) ) Therefore, ‖χn ′ (.)‖l ≤ (∫ 1 n+1 0 + ∫ π 1 n+1 ) ‖Ψ(., v)‖l| Y1 ′ (v) + Y2 ′ (v)| dv = O ( n2 ∫ 1 n+1 0 m(v) dv ) +O (∫ 1 n+1 0 m(v) v2 dv ) +O ( n2 ∫ π 1 n+1 m(v) dv ) +O (∫ π 1 n+1 m(v) v2 dv ) (11) From (10) , (11) and by the monotonicity of µ(v) we have ‖χn ′ (.)‖µl = ‖χn ′ (.)‖l + sup y 6=0 ‖χn′(.+ y) + χn ′ (.− y)‖l µ(y) = O (∫ π 1 n+1 m(v) v2 µ(v) dv ) provided ∫ η 0 m(v) v µ(v) dv = O (m(η) µ(η) ) Hence, En(h) = inf n ‖χn ′ (.)‖µl = O (∫ π 1 n+1 m(v) v2 µ(v) dv ) This completes the proof of our main theorem. 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