EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 1302-1317 ISSN 1307-5543 – ejpam.com Published by New York Business Global Degree of convergence of a function in generalized Zygmund norm using Karamata-Matrix (KλA) product operator H. K. Nigam1, Manoj Kumar Sah1,∗ 1 Department of Mathematics, Central University of South Bihar, Gaya, Bihar, India Abstract. In the present paper, we obtain the results on the degree of convergence of a function of Fourier series in generalized Zygmund space using Karamata-Matrix (KλA) product operator. We also study an application of our main result. 2020 Mathematics Subject Classifications: 40C05, 40C10, 42A10 Key Words and Phrases: Degree of convergence, generalized Zygmund space (Z (η) r ; r ≥ 1), Karamata-Matrix (KλA) product operator, Fourier series, modulus of continuity. 1. Introduction Karamata ([3]) introduced Kλ-summability method for the first time. This method was again introduced by Lotosky ([8]) for λ = 1. A deep study on Kλ and their similar cases is studied after the publication of the work of Agnew ([1]). The degree of approxi- mation of a function in function spaces viz, Lipschitz, Hölder and generalized Hölder using different transforms of Fourier series, has been studied by the researchers [4–6, 9–12] etc. Therefore, in the present paper we study the degree of convergence of a function in gen- eralized Zygmund space (Z (η) r ; r ≥ 1) using Karamata-Matrix (KλA) product operator of Fourier series. 1.1. Fourier series Let g be a Lebesgue integrable function with period 2π on the interval [−π, π]. The Fourier series of a function g is given by g(t) ∼ a0 2 + ∞∑ ν=1 (aν cos νt+ bν sin νt), (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4756 Email addresses: hknigam@cusb.ac.in (H. K. Nigam), manojsah@cusb.ac.in (M. K. Sah) https://www.ejpam.com 1302 © 2023 EJPAM All rights reserved. H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1303 where a0, aν and bν are Fourier co-efficients. The νth partial sum of (1) is given ([15]) by sν(g; t) = sν(t)− g(t) = 1 2π ∫ π 0 ϕ(t, w)Dν(w)dw, (2) where ϕ(t, w) = g(t+ w) + g(t− w)− 2g(t), and Dν(w) (Dirichlet Kernal) is defined by Dν(w) = sin ( ν + 1 2 ) w sin ( w 2 ) . (3) 1.2. Summability operator Let u0 + u1 + u3 + · · · = ∞∑ ν=0 uν (4) be an infinite series with the sequence of its νth partial sum sν . 1.2.1. Karamata (Kλ) operator Let us define, for ν ∈ N ∪ {0}, the numbers [ ν k ] , for 0 ≤ k ≤ ν, by ν−1∏ p=0 (t+ p) = t(t+ 1) · · · (t+ ν − 1) = ν∑ k=0 [ ν k ] tk = Γ(t+ ν) Γt . The numbers [ ν k ] are said to be the absolute value of stirling number of first kind. Let {sν} be the seuqence of the partial sums of the series (4) and we write [3, 8] sλν = Γλ Γ(λ+ ν) ν∑ k=0 [ ν k ] λksk (5) to denote the νth Kλ-operator of order λ > 0. If sλν → s as ν → ∞, where s is a definite number, then the series (4) is said to be summable by Karamata metnhod (Kλ) of order λ > 0 to the sum s. Thus, sλν → s(Kλ) as ν → ∞. (6) H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1304 1.2.2. Matrix (A) operator Let A = (aν,k); ν, k = 0, 1, 2, · · · be an infinite lower triangular matrix satisfying the Silverman-Toeplitz [14] conditions of regularity i.e. ν∑ k=0 aν,k = 1 as ν → ∞, aν,k = 0 for k > ν, ν∑ k=0 |aν,k| ≤ M, a finite constant. The sequence to sequence transformation dAν := ν∑ k=0 aν,ksk (7) defines the sequence dAν of matrix operator of the sequence {sν} obtained by the sequence of co-efficient (aν,k). If d A ν → s as n → ∞, then (4) is said to be summable by matrix (A) method to a definite number s. 1.2.3. Karamata-Matrix (KλA) product operator Superimposing A operator onKλ, a Karamata-Matrix (KλA) product operator is obtained and is given by dK λA ν = Γλ Γν + λ ν∑ k=0 [ ν k ] λk(dAk ) = Γλ Γν + λ ν∑ k=0 [ ν k ] λk k∑ j=0 aν,jsj . (8) If dK λA ν → s as ν → ∞, then the series (4) is said to be summable to s by (KλA) product operator. Regularity of the Kλ and A methods implies the regularity of the KλA method. 1.3. Generalized Zygmund space Let C2π denotes the Banach space of all continuous and 2π-periodic functions defined on the interval [0, 2π] with the supremum norm. The function space for 0 < α < 1, Zα := {g ∈ C2π : |g(t+ w) + g(t− w)− 2g(t)| = O(|w|α)} (9) H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1305 is a Banach space with the norm ∥ · ∥(α) defined by ∥g∥(α) := sup 0≤t≤2π |g(t)|+ sup t,w w ̸=0 |g(t+ w) + g(t− w)− 2g(t)| |w|α The space of all Lebesgue integrable and periodic functions with period 2π be Lr := { g : [0, 2π] → R; ∫ 2π 0 |g(t)|rdt < ∞, r ≥ 1 } . (10) The norm of (10) is defined by ∥g∥r =  { 1 2π ∫ 2π 0 |g(t)|rdt } 1 r for 1 ≤ r < ∞, ess supt∈[0,2π] |g(t)| for r = ∞. We define Z(α),r := g ∈ Lr[0, 2π] : (∫ 2π 0 |g(t+ w) + g(t− w)− 2g(t)|rdt ) 1 r = O(|w|α)  . (11) The space Z(α),r, r ≥ 1, 0 < α ≤ 1 is a Banach space with the norm ∥ · ∥α,r: ∥g∥α,r := ∥g∥r + sup w ̸=0 ∥g(t+ w) + g(t− w)− 2g(t)∥r |w|α . ∥g∥α,r := ∥g∥r. The function space Z(η1) is a defined as Z(η1) := {g ∈ C2π : |g(t+ w) + g(t− w)− 2g(t)| = O(η1(w))} where η1 is a integral modulus of continuity, that is, η1 is a non-decreasing continuous function together with the property η1(0) = 0, η1(w1 + w2) ≤ η1(w1) + η1(w2). Let η1 : [0, 2π] → R be a real valued arbitrary function with η1(w) > 0 for 0 < w ≤ 2π and limn→0+ η1(w) = η1(0) = 0. Now, we define ([15]) Z(η1) r = { g ∈ Lr[0, 2π] : sup w ̸=0 ∥g(·+ w) + g(· − w)− 2g(·)∥r η1(w) < ∞, r ≥ 1 } , (12) with its norm given by ∥g∥(η1)r = ∥g∥r + sup w ̸=0 ∥g(·+ w) + g(· − w)− 2g(·)∥r η1(w) , r ≥ 1. (13) H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1306 Hence, the generalized Zygmund space (12) with (13) is a Banach space. The space ∥ · ∥(η1)r is complete in view of Lr(r ≥ 1) space. Note 1: η1(w) and η2(w) denote the moduli of continuity of order two ([15]). If η1(w) η2(w) be non-decreasing and positive, then ∥g∥(η2)r ≤ max ( 1, η1(2π) η2(2π) ) ∥g∥(η1)r < ∞. Note 2: We observe that Z(η1) r ⊂ Z(η2) r ⊂ Lr, r ≥ 1. Remark 1: (i) If r → ∞ in Z (η1) r then Z (η1) r reduces to Z(η1). (ii) If η1(w) = wα in Z(η1) then Z(η1) reduces to Zα. (iii) If η1(w) = wα in Z (η1) r then Z (η1) r reduces to Zα,r. (iv) If r → ∞ in Zα,r then Zα,r reduces to Zα. (v) If η1(w) = wα1 , η2(w) = wα2 , r → ∞ and α2 = 0 in Zα,r then Z (η1) r reduces to Lip(α). (vi) Let 0 ≤ δ2 < δ1 < 1, if η1(w) = wδ1 and η2(w) = wδ2 then η1(w) η2(w) is non-decreasing, while η1(w) wη2(w) is non-increasing. 1.4. Degree of convergence The degree of convergence of a summation method to a given function g is a measure that how fast wν converges to g, which is given by ([7]) ∥g − wν∥ = O ( 1 γν ) , where γν → ∞ as ν → ∞. We write Φ(w) = ϕ(t, w) = g(t+ w) + g(t− w)− 2g(t); Φ(t) = ∫ t 0 |ϕ(u)|du; Mν(w) = 1 2π ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j sin ( j + 1 2 ) w sin(w2 ) . The organization of the paper is as follows: In section 2, we give a motivation and propose our main results. In section 3, we establish two lemmas, which are used in the proofs of our main results. In section 4, we establish our main results. In section 5, we give applications of our main results and in section 6, we give a conclusion of the main results. H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1307 2. Main Results In this section, we state our main results: Theorem 1. Let g be a Lebesgue integrable function with period 2π then the degree of convergence of g of Fourier series in the generalized Zygmund space (Z (η1) r , r ≥ 1) using (KλA) operator, is given by ∥dKλA ν (g; ·)− g(·)∥(η2)r = O [( (1 + Γλ){2π(ν + 1)− 1} (ν + 1)Γλ{π(ν + 1)− 1} )∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] , (14) where η1(w) and η2(w) are as defined in Note 1 and η1(w) η2(w) is positive and non-decreasing. Theorem 2. Following the conditions of Theorem 1, if η1(w) wη2(w) is non-increasing, then the degree of convergence of g of Fourier series in the generalized Zygmund space (Z (η1) r , r ≥ 1) using (KλA) operator, is given by ∥dKλA ν (g; ·)− g(·)∥(η2)r = O [( (1 + Γλ){2π(ν + 1)− 1} Γλ{π(ν + 1)− 1} ) η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) log{(ν + 1)π} ] . (15) 3. Lemmas In this section, we prove the following lemmas: Lemma 1. ([6]) Let f ∈ Z (η1) r , then for 0 < w ≤ π. If η1(w) and η2(w) are as defined in Note 1, then ∥ϕ(·+ z, w) + ϕ(· − z, w)− 2ϕ(·, w)∥r = O ( η2(|z|) η1(w) η2(w) ) . Lemma 2. |Mν(w)| = O ( ν+1 Γλ ) for 0 < w ≤ 1 ν+1 . Proof. For 0 < w ≤ 1 ν+1 , sin( w 2 ) ≥ w π , | sin(νw)| ≤ νw. |Mν(w)| = 1 2π ∣∣∣∣∣∣ ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j sin ( j + 1 2 ) w Γ(n+ λ) sin(w2 ) ∣∣∣∣∣∣ ≤ 1 2π 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk j∑ q=0 ak,j | sin ( j + 1 2 ) w| | sin(w2 )| ≤ 1 2π 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j ( j + 1 2 ) w w π H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1308 ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j(2j + 1) ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk 2 k∑ j=0 jak,j + k∑ j=0 ak,j  ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk {2(ak,1 + 2ak,2 + · · ·+ kak,k) + 1} ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk {2(kak,1 + kak,2 + · · ·+ kak,k) + 1} ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk {2k(ak,0 + ak,1 + ak,2 + · · ·+ ak,k)− 2kak,0 + 1} ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk2k {(ak,0 + ak,1 + ak,2 + · · ·+ ak,k)− ak,0}+ 1 ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk2k {1− ak,0}+ 1 ≤ 1 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk(2k + 1) ≤ (2n+ 1) 4 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk ≤ (2n+ 1) 4 1 Γ(ν + λ) Γ(ν + λ) Γλ = O ( ν + 1 Γλ ) . Lemma 3. |Mν(w)| = O ( 1 w2(ν+1)Γλ ) for 1 ν+1 < w ≤ π. Proof. For 1 ν+1 < w ≤ π, sin(w2 ) ≥ w π . |Mν(w)| = 1 2π ∣∣∣∣∣∣ ν∑ k=0 [ ν k ] λk k∑ j=0 akbj sin ( j + 1 2 ) w Γ(ν + λ) sin(w2 ) ∣∣∣∣∣∣ ≤ 1 2π 1 Γ(ν + λ) ∣∣∣∣∣∣ ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j sin ( j + 1 2 ) w | sin(w2 )| ∣∣∣∣∣∣ ≤ 1 2π 1 Γ(ν + λ) ∣∣∣∣∣∣ ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j sin ( j + 1 2 ) w w π ∣∣∣∣∣∣ H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1309 ≤ 1 2w 1 Γ(ν + λ) ∣∣∣∣∣∣ ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j sin ( j + 1 2 ) w ∣∣∣∣∣∣ By Abel’s lemma, we get |Mν(w)| ≤ 1 2w 1 Γ(ν + λ) [ ν∑ k=0 [ ν k ] λk ∣∣∣∣∣∣ k−1∑ j=0 (ak,j − ak−1,j+1) j∑ p=0 sin ( p+ 1 2 ) w ∣∣∣∣∣∣ + ak,k k∑ j=0 sin ( j + 1 2 ) w ] ≤ 1 2w 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk ∣∣∣∣∣∣ k−1∑ j=0 ∆ak,j j∑ p=0 sin ( p+ 1 2 ) w ∣∣∣∣∣∣+ ak,k ∣∣∣∣∣∣ k∑ j=0 sin ( j + 1 2 ) w ∣∣∣∣∣∣ ≤ 1 2w 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk k−1∑ j=0 |∆ak,j |+ ak,k  max 0≤p≤m ∣∣∣∣∣∣ m∑ p=0 sin ( p+ 1 2 ) l ∣∣∣∣∣∣ ≤ 1 2w 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk [ O ( 1 k + 1 ) +O ( 1 k + 1 )] · 1 w ≤ 1 w2 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk ( 1 k + 1 ) . ≤ 1 w2(ν + 1) 1 Γ(ν + λ) ν∑ k=0 [ ν k ] λk ≤ 1 w2(ν + 1) 1 Γ(ν + λ) Γ(ν + λ) Γλ = O ( 1 w2(ν + 1)Γλ ) . 4. Proof of Main Results Proof. [Proof of Theorem 1] By using the integral representation ([13]) of sν(g; t), we have sν(g; t)− g(t) = 1 2π ∫ π 0 ϕ(t, w) sin(n+ 1 2)w sin(w2 ) dw. (16) Denoting KλA operator of sν(g; t) by dK λA ν , we get dK λA ν (g; t)− g(t) = Γλ Γν + λ ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j {sj(f ; t)− f(t)} H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1310 = Γλ Γν + λ ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j { 1 2π ∫ π 0 ϕ(t, w) sin(j + 1 2)w sin(w2 ) dw } = Γλ ∫ π 0 ϕ(t, w) 1 2π ν∑ k=0 [ ν k ] λk k∑ j=0 ak,j sin(j + 1 2)w Γν + λ · sin(w2 ) dw = Γλ ∫ π 0 ϕ(t, w)Mν(w)dw. Let ρν(t) := dK λA ν (g; t)− g(t) = Γλ ∫ π 0 ϕ(t, w)Mν(w)dw. (17) Now, ρν(t+ z) + ρν(t− z)− 2ρν(t) = Γλ ∫ π 0 {ϕ(t+ z, w) + ϕ(t− z, w)− 2ϕ(t, w)}Mν(w)dw. Using generalized Minkowski inequality ([2]), we can write ∥ρν(·+ z) + ρν(· − z)− 2ρν(·)∥r ≤ Γλ ∫ 1 ν+1 0 ∥ϕ(·+ z, w) + ϕ(· − z, w)− 2ϕ(·, w)∥r|Mν(w)|dw + Γλ ∫ π 1 ν+1 ∥ϕ(·+ z, w) + ϕ(· − z, w)− 2ϕ(·, w)∥r|Mν(w)|dw = I1 + I2. (18) Now, using Lemmas 1 and 2, we have I1 = [ Γλ ∫ 1 ν+1 0 η2(|z|) η1(w) η2(w) (ν + 1) Γλ dw ] = O [ (ν + 1)η2(|z|) ∫ 1 ν+1 0 η1(w) η2(w) dw ] = O (ν + 1)η2(|z|) η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) ∫ 1 ν+1 0 dw  = O η2(|z|)η1 ( 1 ν+1 ) η2 ( 1 ν+1 )  . (19) H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1311 Now, using Lemmas 1 and 3, we have I2 = O [ Γλ ∫ π 1 ν+1 η2(|z|) η1(w) η2(w) { 1 w2(ν + 1)Γλ } dw ] = O [ η2(|z|) (ν + 1) ∫ π 1 n+1 η1(w) η2(w) 1 w2 dw ] . (20) Combining (18)-(20), we have ∥ρν(·+ z)+ρν(· − z)− 2ρν(·)∥r = O η2(|z|)η1 ( 1 ν+1 ) η2 ( 1 ν+1 )  +O [ η2(|z|) (v + 1) ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . (21) Now, sup z ̸=0 ∥ρν(·+ z) + ρν(· − z)− 2ρν(·)∥r η2(|z|) = O η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) +O [ 1 (n+ 1) ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . (22) Now, ∥ρν(·)∥r ≤ ∫ π 0 ∥ϕ(·, w)∥r|Mν(w)|dw = O [∫ 1 ν+1 0 ∥ϕ(·, w)∥r|Mν(w)|dw ] + [∫ π 1 ν+1 ∥ϕ(·, w)∥r|Mν(w)|dw ] = J1 + J2. (23) Using Lemma 2, we get J1 = O [∫ 1 ν+1 0 ∥ϕ(·, w)∥r|Mν(w)|dw ] = O [ (ν + 1) Γλ ∫ 1 ν+1 0 η1(w)dw ] = O [ (ν + 1) Γλ η1 ( 1 ν + 1 )∫ 1 ν+1 0 dw ] = O [ 1 Γλ η1 ( 1 ν + 1 )] . (24) H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1312 Using Lemma 3, we get J2 = O [∫ π 1 ν+1 ∥ϕ(·, w)∥r|Mν(w)|dw ] = O [∫ π 1 ν+1 { 1 w2(ν + 1)Γλ } η1(w)dw ] = O [ 1 (ν + 1)Γλ ∫ π 1 ν+1 η1(w) w2 dw ] . (25) Combining (23)-(25), we have ∥ρν(·)∥r = O [ 1 Γλ η1 ( 1 ν + 1 )] +O [ 1 (ν + 1)Γλ ∫ π 1 ν+1 η1(w) w2 dw ] . (26) Now, we have ∥ρν(·)∥(η2)r = ∥ρν(·)∥r + sup z ̸=0 ∥ρν(·+ z) + ρν(· − z)− 2ρν(·)∥r η2(|z|) . From (22) and (26), we get ∥ρv(·)∥(η2)r = O [ 1 Γλ η1 ( 1 ν + 1 )] +O [ 1 (ν + 1)Γλ ∫ π 1 ν+1 η1(w) w2 dw ] = O η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) +O [ 1 (ν + 1) ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . In view of monotonicity of η2(w), we have η1(w) = η1(w) η2(w)η2(w) ≤ η2(π) η1(w) η2(w) = O ( η1(w) η2(w) ) for 0 < w ≤ π. Hence, ∥ρν(·)∥(η2)r = O  1 Γλ η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) +O [ 1 (ν + 1)Γλ ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] +O η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) +O [ 1 (ν + 1) ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . (27) Since η1 and η2 are as defined in Note 1 and η1(w) η2(w) is positive, non-decreasing, therefore, ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ≥ η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) ∫ π 1 ν+1 1 w2 dw H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1313 ≥ η1 ( 1 n+1 ) η2 ( 1 ν+1 ) [− 1 π + (ν + 1) ] ≥ η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) [π(ν + 1)− 1 π ] . Then, η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) = O [ π {π(ν + 1)− 1} ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . (28) From (27) and (28), we get ∥ρν(·)∥(η2)r = O [ 1 Γλ π {π(ν + 1)− 1} ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] +O [ 1 (ν + 1)Γλ ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] +O [ π {π(ν + 1)− 1} ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] +O [ 1 (ν + 1) ∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] = O [( π Γλ{π(ν + 1)− 1} + 1 Γλ(ν + 1) + π {π(ν + 1)− 1} + 1 (ν + 1) )∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] = O [( (1 + Γλ){2π(ν + 1)− 1} (ν + 1)Γλ{π(ν + 1)− 1} )∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . Proof. [Proof of Theorem 2] Following the proof of Theorem 1, we have Eν(g) = O [( (1 + Γλ){2π(ν + 1)− 1} (ν + 1)Γλ{π(ν + 1)− 1} )∫ π 1 ν+1 η1(w) η2(w) 1 w2 dw ] . Since η1(w) wη2(w) is non-increasing and positive, thus using second mean value theorem of the integral calculus, we have Eν(g) = O [( (1 + Γλ){2π(ν + 1)− 1} (ν + 1)Γλ{π(ν + 1)− 1} ) (ν + 1) η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) ∫ π 1 ν+1 1 w dw ] H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1314 = O [( (1 + Γλ){2π(ν + 1)− 1} Γλ{π(ν + 1)− 1} ) η1 ( 1 ν+1 ) η2 ( 1 ν+1 ) log{(ν + 1)π} ] . . 5. Application In this section, we study an application of our main result. We take η1( 1 ν+1) = ( 1 ν+1 )δ1 , η2( 1 ν+1) = ( 1 ν+1 )δ2 , δ1 = 1, δ2 = 0, and λ = 2 then from Theorem 2, we have Eν(g) = O [( (1 + Γλ){2π(ν + 1)− 1} Γλ{π(ν + 1)− 1} ) 1 (ν + 1) log{(ν + 1)π} ] . Table 1: Degree of convergence of g for different ν. ν Degree of convergence of g 1000 0.0322 10000 0.0041 50000 0.0009572 100000 0.0005063 500000 0.00011414 1000000 0.000005984 . . . . ∞ 0 H. K. Nigam, M. K. Sah / Eur. J. Pure Appl. Math, 16 (2) (2023), 1302-1317 1315 (a) For ν = 50000 (b) For ν = 100000 (c) For ν = 500000 (d) For ν = 1000000 Figure 1: Degree of convergence of function g. 6. Conclusion From the Table 1 and figures 1(a) to 1(d), we observed that the error estimation tends to zero rapidely as ν tends to infinity. 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