Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 445 https://internationalpubls.com Approximation of Functions by 𝒏-Euler Product Means of Fourier Series and its Conjugate Md Hadish 𝟏, Rupesh Kumar Mishra 𝟐 and Shambhu Kumar Mishra πŸ‘ 1Assistant Professor (Mathematics), Department of Applied Science and Humanities Government Engineering College West Champaran, Bihar, India 2Research Scholar, Department of Mathematics, Patliputra University Patna Bihar, India. 3Professor, Department of Mathematics, Patliputra University Patna Bihar, India E-mail: 1h.ahmad043@gmail.com, 2rupeshmishra043@gmail.com, 3shambhumishra5@gmail.com Article History: Received: 22-08-2024 Revised: 07-10-2024 Accepted: 23-10-2024 Abstract: Our research in this work produced new theorems on approximation of functions by n- Eulers πΈπ‘ž product means of Fourier series and its conjugate Fourier series. This new results generalize the results of [5], [7] and [9]. Key Words: Approximation of function, Fourier series, Conjugate Fourier series, 𝐿𝑝 space, Lebesgue Integral. 1 Introduction Estimating functions using generalized Fourier series based on trigonometric polynomials has become increasingly important in the development of mathematics and engineering fields in recent years. For instance, a new 𝐿2-bassed technique for creating Finit Impulse Response digital filters to obtain an optimal approximation was devised by Psarakis and Moustakieds [2], utilising the features of approximation of functions. In the development of digital filters, 𝐿𝑝-space, 𝐿2-space, and 𝐿∞-space are also very important. In the past few years, numerous researchers have grown interested in the inaccuracy of approximation of periodic functions belonging to distinct classes using different summability methods. Several scholars, including Hadish [6], Sonkar and Singh[8], Saxena and Prabhakar [5], Sonkar and Sagwan[7], and Sachin [9], have studied the following topics: (𝐢, 1)(𝐸, π‘ž), double Euler summability, triple 𝐸1 Euler summability, and triple πΈπ‘ž-euler summability, respectively. We investigated the approximation of functions in this direction using the n-Euler πΈπ‘ž product summability approach. The outcomes of [7], [8], and [9] were generalized by our result. 2 Definitions and Notations Let 𝑔 be a Lebesgue integrable function with period 2πœ‹ on the interval [0,2πœ‹]. The Fourier series of a function 𝑔 is given by 𝑔(π‘₯) β‰ˆ π‘Ž0 2 + βˆ‘βˆž 𝑛=1 (π‘Žπ‘›π‘π‘œπ‘ π‘›π‘₯ + 𝑏𝑛𝑠𝑖𝑛𝑛π‘₯) (2.1) οΏ½ΜƒοΏ½(π‘₯) β‰ˆ βˆ‘βˆž π‘š=1 (π‘π‘šπ‘π‘œπ‘ π‘šπ‘₯ βˆ’ π‘Žπ‘šπ‘ π‘–π‘›π‘šπ‘₯) (2.2) with π‘›π‘‘β„Ž partial sum 𝑔𝑛(π‘₯). The 𝐿𝑝[0,2πœ‹] βˆ’ space can be defined as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 446 https://internationalpubls.com 𝐿𝑝[0,2πœ‹]: = {𝑔: [0,2πœ‹] β†’ 𝑅: ∫ 2πœ‹ 0 |𝑔(π‘₯)|𝑝𝑑π‘₯ < ∞} , 𝑝 β‰₯ 1. Let βˆ‘βˆž π‘Ÿ=0 π‘’π‘Ÿ be an infinite series and sequence {π‘ π‘Ÿ} is (π‘Ÿ + 1)π‘‘β„Ž partial sum of given series, then the series βˆ‘βˆž π‘Ÿ=0 π‘’π‘Ÿ is said to be (𝐸, π‘ž) summable[1] to s if π‘‘π‘Ÿ πΈπ‘ž = 1 (1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘—π‘ π‘— β†’ 𝑠 π‘Žπ‘  π‘Ÿ β†’ ∞. The series βˆ‘βˆž π‘Ÿ=0 π‘’π‘Ÿ is said to be (𝐸, π‘ž)(𝐸, π‘ž) summable to s, if π‘‘π‘Ÿ πΈπ‘žπΈπ‘ž = 1 (1 + π‘ž)π‘Ÿ βˆ‘ π‘Ÿ 𝑗=0 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Žπ‘ β„Ž β†’ 𝑠 π‘Žπ‘  π‘Ÿ β†’ ∞. The series βˆ‘βˆž π‘Ÿ=0 π‘’π‘Ÿ is said to be (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž) summable to s, if π‘‘π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž = 1 (1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž βˆ‘β„Ž 𝑙=0 ( β„Ž 𝑙 ) π‘žβ„Žβˆ’π‘™π‘ π‘™ β†’ 𝑠 π‘Žπ‘  π‘Ÿ β†’ ∞. Similarly, the series βˆ‘βˆž π‘Ÿ=0 π‘’π‘Ÿ is said to be (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) summable to s i.e. n-euler πΈπ‘ž summable to s. if π‘‘π‘Ÿ πΈπ‘žπΈπ‘ž...πΈπ‘ž = 1 (1 + π‘ž)π‘Ÿ βˆ‘ π‘Ÿ 𝑗=0 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž βˆ‘ β„Ž 𝑙=0 ( β„Ž 𝑙 ) π‘žβ„Žβˆ’π‘™. . . π‘žπ‘£βˆ’π‘– (1 + π‘ž)𝑖 βˆ‘ 𝑖 π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘šπ‘ π‘š β†’ 𝑠 π‘Žπ‘  π‘Ÿ β†’ ∞ We also write, π½π‘Ÿ(𝑑) = 1 2πœ‹(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž βˆ‘β„Ž 𝑙=0 ( β„Ž 𝑙 ) π‘žβ„Žβˆ’π‘™. . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š 𝑠𝑖𝑛(π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 }] π½π‘Ÿ(𝑑) = 1 2πœ‹(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž βˆ‘β„Ž 𝑙=0 ( β„Ž 𝑙 ) π‘žβ„Žβˆ’π‘™. . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š π‘π‘œπ‘ (π‘š+ 1 2 𝑑) 𝑠𝑖𝑛 𝑑 2 }] πœ™(𝑑) = 𝑔(π‘₯ + 𝑑) βˆ’ 2𝑔(𝑑) + 𝑔(π‘₯ βˆ’ 𝑑) and πœ“(𝑑) = 𝑔(π‘₯ + 𝑑) βˆ’ 𝑔(π‘₯ βˆ’ 𝑑) 2 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 447 https://internationalpubls.com 3 Main Results Saxena and Prabhakar[8] obtained novel findings in the domain of function approximation. The triple 𝐸1 Summability approach is introduced by Sonkar and Sangwan[7], who used it to generalise the findings of Saxena and Prabhakar.The triple πΈπ‘ž summability approach was recently introduced by Devaiya and Sriwastva[9], and they used it to generalise the findings of Sonkar and Sagwan[7]. In this work, we establish the more broad setting and generalise the findings of Sachin Devaiya and Shailesh Kumar Srivastava[9]. 3.1 Theorem let π‘π‘Ÿ be a none increasing positive sequence of real constant such that π‘π‘Ÿ = βˆ‘π‘Ÿ 𝑧=0 𝑝𝑧 β†’ 𝑠 π‘Žπ‘  π‘Ÿ β†’ ∞, (3.1) and given function πœ™(𝑑) satisfies, Ξ¦(𝑑) = ∫ 𝑑 0 |πœ™(𝑒)|𝑑𝑒 = π‘œ [ 𝑑 𝛼( 1 𝑑 )𝑝𝑑 ] , π‘Žπ‘  𝑑 β†’ 0 + , (3.2) where 𝛼(𝑑) is positrive, monotonic and none - increasing function of t. π‘™π‘œπ‘”π‘Ÿ = 𝑂[𝛼(π‘Ÿ). π‘π‘Ÿ], π‘Žπ‘  π‘Ÿ β†’ ∞ (3.3) The approximation of function g at π‘₯ = 𝑑 by n- Eular product means of its Fourier series is given by |π‘‘π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž...πΈπ‘ž βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 3.2 Theorem let π‘π‘Ÿ be a none-increasing positive sequence of real constant such that π‘π‘Ÿ = βˆ‘π‘Ÿ 𝑧=0 𝑝𝑧 β†’ 𝑠 π‘Žπ‘  π‘Ÿ β†’ ∞ and given function πœ“(𝑑) satisfies, Ξ¨(𝑑) = ∫ 𝑑 0 |πœ“(𝑒)|𝑑𝑒 = π‘œ [ 𝑑 𝛼( 1 𝑑 )𝑝𝑑 ] , π‘Žπ‘  𝑑 β†’ 0 +, (3.4) where 𝛼(𝑑) is positive, monotonic and none - increasing function of t. π‘™π‘œπ‘”π‘Ÿ = 𝑂[𝛼(π‘Ÿ). π‘π‘Ÿ], π‘Žπ‘  π‘Ÿ β†’ ∞, (3.5) then approximation of function οΏ½ΜƒοΏ½ at x=t by n- Eular product means of its conjugate fourier series is given by |οΏ½ΜƒοΏ½π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž...πΈπ‘ž βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 4 lemmas Here few lemmas are given, Which are useful to prove our theorem Lemma 4.1 |π½π‘Ÿ(𝑑)| = 𝑂(π‘Ÿ), for 0 ≀t≀ 1 π‘Ÿ . Proof. Using π‘ π‘–π‘›π‘Ÿπ‘‘ ≀ π‘Ÿπ‘‘ and 𝑠𝑖𝑛 𝑑 2 β‰₯ 𝑑 πœ‹ , we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 448 https://internationalpubls.com |π½π‘Ÿ(𝑑)| = 1 2πœ‹(1+π‘ž)π‘Ÿ |βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š 𝑠𝑖𝑛(π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 }]| ≀ 1 2πœ‹(1+π‘ž)π‘Ÿ |βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š (π‘š+ 1 2 )𝑑 𝑑 πœ‹ }]| ≀ 1 4πœ‹(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . (2𝑖 + 1) π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š}] ≀ 1 4(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž βˆ‘β„Ž 𝑙=0 ( β„Ž 𝑙 ) π‘žβ„Žβˆ’π‘™(2𝑙 + 1)] ≀ 1 4(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž(2β„Ž + 1)] ≀ 1 4(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘—(2𝑗 + 1)] ≀ 1 4 (2π‘Ÿ + 1) = 𝑂(π‘Ÿ) Lemma 4.2 |π½π‘Ÿ(𝑑)| = 𝑂 ( 1 𝑑 ) for 1 π‘Ÿ ≀ 𝑑 ≀ πœ‹ Proof. Using 𝑠𝑖𝑛(π‘Ÿπ‘‘) ≀ 1 and 𝑠𝑖𝑛( 𝑑 2 ) β‰₯ 𝑑 πœ‹ , We have |π½π‘Ÿ(𝑑)| = 1 2πœ‹(1+π‘ž)π‘Ÿ |βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š 𝑠𝑖𝑛(π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 }]| ≀ 1 2πœ‹(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š πœ‹ 𝑑 }] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š}] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘—] ≀ 1 2𝑑 = 𝑂 ( 1 𝑑 ). Lemma 4.3 |𝐽(𝑑)| =O( 1 𝑑 ), for 0 ≀ 𝑑 ≀ 1 π‘Ÿ . Proof. Using |cos(rt)| ≀ 1, we have |π½π‘Ÿ(𝑑)| = 1 2πœ‹(1+π‘ž)π‘Ÿ |βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š π‘π‘œπ‘ (π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 }]| Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 449 https://internationalpubls.com ≀ 1 2πœ‹(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š πœ‹ 𝑑 }] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š}] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘—] ≀ 1 2𝑑 = 𝑂 ( 1 𝑑 ). Lemma 4.4 |𝐽(𝑑)| =O( 1 𝑑 ), for 1 π‘Ÿ ≀t≀ πœ‹. Proof. Appling sin( 𝑑 2 ) β‰₯ 𝑑 πœ‹ , we have |π½π‘Ÿ(𝑑)| = 1 2πœ‹(1+π‘ž)π‘Ÿ |βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š π‘π‘œπ‘ (π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 }]| ≀ 1 2πœ‹(1 + π‘ž)π‘Ÿ |βˆ‘ π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž … π‘žπ‘£βˆ’π‘– (1 + π‘ž)𝑖 {βˆ‘ 𝑖 π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š π‘π‘œπ‘  (π‘š + 1 2 ) 𝑑 𝑑 πœ‹ }]| ≀ 1 2𝑑(1 + π‘ž)π‘Ÿ |βˆ‘ π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž … π‘žπ‘£βˆ’π‘– (1 + π‘ž)𝑖 𝑅𝑒 {βˆ‘ 𝑖 π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘šπ‘’π‘–π‘šπ‘‘}]| ≀ 1 2𝑑(1 + π‘ž)π‘Ÿ |βˆ‘ πœβˆ’1 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1 + π‘ž)𝑖 𝑅𝑒 {βˆ‘ 𝑖 π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘šπ‘’π‘–π‘šπ‘‘}]| + 1 2𝑑(1 + π‘ž)π‘Ÿ |βˆ‘ π‘Ÿ 𝑗=𝜏 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1 + π‘ž)𝑖 𝑅𝑒 {βˆ‘ 𝑖 π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘šπ‘’π‘–π‘šπ‘‘}]| ≀ 1 2𝑑(1 + π‘ž)π‘Ÿ βˆ‘ πœβˆ’1 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1 + π‘ž)𝑖 𝑅𝑒 {βˆ‘ 𝑖 π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š}] |π‘’π‘–π‘šπ‘‘| + 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=𝜏 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 π‘šπ‘Žπ‘₯0β‰€π‘šβ‰€π‘– βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘šπ‘’π‘–π‘šπ‘‘] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 450 https://internationalpubls.com ≀ 1 2𝑑(1 + π‘ž)π‘Ÿ βˆ‘ πœβˆ’1 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1 + π‘ž)𝑗 βˆ‘ 𝑗 β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1 + π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘–] + 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=𝜏 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž (1+π‘ž)β„Ž . . . π‘žπ‘£βˆ’π‘–] ≀ 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘πœβˆ’1 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž] + 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=𝜏 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) π‘žπ‘—βˆ’β„Ž] = 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘πœβˆ’1 𝑗=0 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— + 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=𝜏 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— = 1 2𝑑(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 ( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— Hence |π½π‘Ÿ(𝑑)| = 𝑂(1/𝑑). 5 Proof of Theorem 3.1 Proof. We have π‘ π‘Ÿ(𝑔; π‘₯) βˆ’ 𝑔(π‘₯) = 1 2πœ‹ ∫ πœ‹ 0 πœ™(𝑑)𝑠𝑖𝑛(π‘Ÿ+ 1 2 )𝑑 𝑠𝑖𝑛( 𝑑 2 ) 𝑑𝑑, and π‘‘π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž...πΈπ‘ž (𝑔; π‘₯) βˆ’ 𝑔(π‘₯) = 1 2πœ‹(1+π‘ž)π‘Ÿ | βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 ∫ πœ‹ 0 πœ™(𝑑) {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š 𝑠𝑖𝑛(π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 }]| = ∫ πœ‹ 0 πœ™(𝑑)π½π‘Ÿ(𝑑)𝑑𝑑 = [∫ 1 π‘Ÿ 0 πœ™(𝑑) + ∫ 𝛾 1 π‘Ÿ πœ™(𝑑) + ∫ πœ‹ 𝛾 πœ™(𝑑)] π½π‘Ÿ(𝑑)𝑑𝑑 = 𝑅1 + 𝑅2 + 𝑅3 (5.1) Applying Lemma 4.1, condion 3.2 and 3.3 and second mean value theorem is applying for second term integral , we have |𝑅1| ≀ ∫ 1 π‘Ÿ 0 |πœ™(𝑑)π½π‘Ÿ(𝑑)|𝑑𝑑 = 𝑂(π‘Ÿ) [∫ 1 π‘Ÿ π‘œ |πœ™(𝑑)|𝑑𝑑] = 𝑂(π‘Ÿ) [𝑂 ( 1 π‘Ÿ 𝛼(π‘Ÿ).π‘ƒπ‘Ÿ )] = 𝑂 ( 1 π‘™π‘œπ‘”π‘Ÿ ) = 𝑂(1) π‘Žπ‘  π‘Ÿ β†’ ∞ (5.2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 451 https://internationalpubls.com Apply Lemma 4.2, condition 3.2 and 3.3 and second mean value theorem is appling for second term integral, We have |𝑅2| ≀ ∫ 𝛾 1 π‘Ÿ |πœ™(𝑑)π½π‘Ÿ(𝑑)|𝑑𝑑 = 𝑂 [∫ 𝛾 1 π‘Ÿ |πœ™(𝑑)| ( 1 𝑑 )] = 𝑂 [{ 1 𝑑 πœ™(𝑑)} 1 π‘Ÿ 𝛾 + ∫ 𝛾 1 π‘Ÿ π‘œ { πœ™(𝑑) 𝑑2 } 𝑑𝑑] = 𝑂 [π‘œ { 1 𝛼( 1 𝑑 ).𝑃𝑑 } 1 π‘Ÿ 𝛾 + ∫ π‘Ÿ 1 π‘Ÿ π‘œ ( 1 𝑑 𝛼( 1 𝑑 ).𝑃𝑑 ) 𝑑𝑑] = 𝑂 [π‘œ { 1 𝛼(π‘Ÿ)π‘ƒπ‘Ÿ } + ∫ π‘Ÿ 1 𝛾 π‘œ ( 1 𝑒 𝛼(𝑒)𝑝𝑒 ) 𝑑𝑒] = 𝑂 ( 1 𝛼(π‘Ÿ)π‘ƒπ‘Ÿ ) + 𝑂 ( 1 π‘Ÿ 𝛼(π‘Ÿ)π‘π‘Ÿ ) ∫ π‘Ÿ 1 π‘Ÿ 1𝑑𝑒 = 𝑂 ( 1 π‘™π‘œπ‘”π‘Ÿ ) + 𝑂 ( 1 π‘™π‘œπ‘”π‘Ÿ ) = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞ (5.3) Applying Riemann-Lebesgue theorem and regularity condition of summability, we have |𝑅3| ≀ ∫ 𝛾 1 π‘Ÿ |πœ™(𝑑)π½π‘Ÿ(𝑑)|𝑑𝑑 = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. (5.4) Collecting (5.1)-(5.4), we get |π‘‘π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž...πΈπ‘ž(𝑔; π‘₯) βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. Hence complete the proof. 6 Proof of Theorem 3.2 Proof. We have οΏ½ΜƒοΏ½π‘Ÿ(𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯) = 1 2πœ‹ ∫ πœ‹ 0 πœ“(𝑑)π‘π‘œπ‘ (π‘Ÿ+ 1 2 )𝑑 𝑠𝑖𝑛( 𝑑 2 ) 𝑑𝑑 and οΏ½ΜƒοΏ½π‘Ÿ πΈπ‘žπΈπ‘ž...πΈπ‘ž (𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯) = 1 2πœ‹(1+π‘ž)π‘Ÿ βˆ‘π‘Ÿ 𝑗=0 [( π‘Ÿ 𝑗 ) π‘žπ‘Ÿβˆ’π‘— (1+π‘ž)𝑗 βˆ‘π‘— β„Ž=0 ( 𝑗 β„Ž ) . . . π‘žπ‘£βˆ’π‘– (1+π‘ž)𝑖 ∫ πœ‹ 0 πœ“(𝑑) {βˆ‘π‘– π‘š=0 ( 𝑖 π‘š ) π‘žπ‘–βˆ’π‘š π‘π‘œπ‘ (π‘š+ 1 2 )𝑑 𝑠𝑖𝑛 𝑑 2 } 𝑑𝑑] = ∫ πœ‹ 0 πœ“(𝑑)π½π‘Ÿ(𝑑)𝑑𝑑 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 452 https://internationalpubls.com = [∫ 1 π‘Ÿ 0 πœ“(𝑑) + ∫ 𝛾 1 π‘Ÿ πœ“(𝑑) + ∫ πœ‹ 𝛾 πœ“(𝑑)] π½π‘Ÿ(𝑑)𝑑𝑑 = οΏ½ΜƒοΏ½1 + οΏ½ΜƒοΏ½2 + οΏ½ΜƒοΏ½3, say. (6.1) Applying 4.3, condition 3.4 and 3.5 and second mean value theorem is applying for second term integral , we have |οΏ½ΜƒοΏ½1| ≀ ∫ 1 π‘Ÿ 0 |πœ“(𝑑)π½π‘Ÿ(𝑑)|𝑑𝑑 = [∫ 1 π‘Ÿ π‘œ |πœ“(𝑑)| 1 𝑑 𝑑𝑑] = 𝑂(π‘Ÿ) [∫ 1 π‘Ÿ π‘œ |πœ“(𝑑)|𝑑𝑑] = 𝑂(π‘Ÿ) [π‘œ ( 1 π‘Ÿ 𝛼(π‘Ÿ).π‘ƒπ‘Ÿ )] = 𝑂 ( 1 π‘™π‘œπ‘”π‘Ÿ ) = 𝑂(1), as π‘Ÿ β†’ ∞. (6.2) Apply Lemma 4.4, condition 3.4 and 3.5 and second mean value theorem is applying for second term integral, we have |οΏ½ΜƒοΏ½2| ≀ ∫ 𝛾 1 π‘Ÿ |πœ“(𝑑)π½π‘Ÿ(𝑑)|𝑑𝑑 = 𝑂 [∫ 𝛾 1 π‘Ÿ |πœ“(𝑑)| ( 1 𝑑 ) 𝑑𝑑] = 𝑂 [{ 1 𝑑 Ξ¨(𝑑)} 1 π‘Ÿ 𝛾 + ∫ 𝛾 1 π‘Ÿ π‘œ { πœ“(𝑑) 𝑑2 } 𝑑𝑑] = 𝑂 [π‘œ { 1 𝛼(π‘Ÿ) . π‘π‘Ÿ} + ∫ 𝛾 1 π‘Ÿ π‘œ ( 1 𝑑 𝛼( 1 𝑑 ).𝑝𝑑 ) 𝑑𝑑] = 𝑂 [π‘œ { 1 𝛼(π‘Ÿ)π‘ƒπ‘Ÿ } + ∫ π‘Ÿ 1 𝛾 π‘œ ( 1 𝑒 𝛼(𝑒)𝑝𝑒 ) 𝑑𝑒] = 𝑂 ( 1 𝛼(π‘Ÿ)π‘ƒπ‘Ÿ ) + 𝑂 ( 1 π‘Ÿ 𝛼(π‘Ÿ)π‘π‘Ÿ ) ∫ π‘Ÿ 1 π‘Ÿ 1𝑑𝑒 = 𝑂 ( 1 π‘™π‘œπ‘”π‘Ÿ ) + 𝑂 ( 1 π‘™π‘œπ‘”π‘Ÿ ) = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. (6.3) Applying Riemann-Lebesgue theorem and regularity condition of summability, we have |οΏ½ΜƒοΏ½3| ≀ ∫ 𝛾 1 π‘Ÿ |πœ“(𝑑)π½π‘Ÿ(𝑑)|𝑑𝑑 = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. (6.4) Collecting (6.1)-(6.4), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 453 https://internationalpubls.com |οΏ½ΜƒοΏ½π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž...πΈπ‘ž(𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. Hence complete the proof. 7 Corollaries Below some corollaries are given, which are derived from our theorems 3.1 and 3.2 7.1 Corollary If we take q=1 in theorem 3.1 then n- Eular product summability (𝐸, π‘ž) (𝐸, π‘ž)(𝐸, π‘ž)...(𝐸, π‘ž) reduce to (𝐸, 1) (𝐸, 1) (𝐸, 1)...(𝐸, 1), then |π‘‘π‘Ÿ 𝐸1𝐸1𝐸1...𝐸1(𝑔; π‘₯) βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 7.2 Corollary If we take q=1 in theorem 3.2, then n eular product summaility (𝐸, π‘ž) (𝐸, π‘ž)(𝐸, π‘ž)...(𝐸, π‘ž) reduce to (𝐸, 1) (𝐸, 1) (𝐸, 1)...(𝐸, 1), then |οΏ½ΜƒοΏ½π‘Ÿ 𝐸1𝐸1𝐸1...𝐸1(𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 7.3 Corollary If we take n=2 in our results 3.1 then 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, π‘ž)(𝐸, π‘ž) double Euler summability, then |π‘‘π‘Ÿ πΈπ‘žπΈπ‘ž (𝑔; π‘₯) βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 7.4 Corollary If we take n=2 in our results 3.2 then 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, π‘ž)(𝐸, π‘ž) double Euler summability, then |οΏ½ΜƒοΏ½π‘Ÿ πΈπ‘žπΈπ‘ž (𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞ 7.5 Corollary If we take n=2 and q=1 in our result 3.1, 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, 1)(𝐸, 1) then |π‘‘π‘Ÿ 𝐸1𝐸1 (𝑔; π‘₯) βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞ 7.6 Corollary If we take n=2 and q=1 in our result 3.2, 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, 1)(𝐸, 1) then |οΏ½ΜƒοΏ½π‘Ÿ 𝐸1𝐸1 (𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 454 https://internationalpubls.com 7.7 Corollary If we consider n=3 in our theorem 3.1, 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž) i.e. triple πΈπ‘ž summability then |π‘‘π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž (𝑔; π‘₯) βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 7.8 Corollary If we consider n= 3 in our theorem 3.2, 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž) i.e. triple πΈπ‘ž summability then |οΏ½ΜƒοΏ½π‘Ÿ πΈπ‘žπΈπ‘žπΈπ‘ž (𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 7.9 Corollary If we take n=3 and q=1 in our result 3.1, 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, 1)(𝐸, 1)(𝐸, 1) i.e. triple 𝐸1 summability then |π‘‘π‘Ÿ 𝐸1𝐸1𝐸1 (𝑔; π‘₯) βˆ’ 𝑔(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞. 7.10 Corollary If we take n=3 and q=1 in our result 3.2, 𝑛 βˆ’ Euler product summability (𝐸, π‘ž)(𝐸, π‘ž)(𝐸, π‘ž). . . (𝐸, π‘ž) reduces to (𝐸, 1)(𝐸, 1)(𝐸, 1) i.e. triple 𝐸1 summability then |οΏ½ΜƒοΏ½π‘Ÿ 𝐸1𝐸1𝐸1 (𝑔; π‘₯) βˆ’ οΏ½ΜƒοΏ½(π‘₯)| = 𝑂(1), π‘Žπ‘  π‘Ÿ β†’ ∞ 8 Particular case 8.1 In view of Corollary 7.3 and 7.4, theorem 1 and 2 [5] are particular cases of our Theorem 3.1 and 3.2 respectively. 8.2 In view of Corollary 7.5 and 7.6, theorem 1 and 2 [4] are particular cases of our Theorem 3.1 and 3.2 respectively. 8.3 In view of Corollary 7.7 and 7.8, theorem 3.1 and 3.2 [3] are particular cases of our Theorem 3.1 and 3.2 respectively. 8.4 In view of Corollary 7.9 and 7.10, theorem 3.1 and 3.2 [7] are particular cases of our Theorem 3.1 and 3.2 respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 455 https://internationalpubls.com 9 Conclusion The results of the paper are aimed to formulate the problem of approximation of function g and their conjugates οΏ½ΜƒοΏ½ by iterates of Euler sum of their Fourier series and conjugate Fourier series respectively. Acknowledgments The first author expresses his gratitude towards his parents for blessings. Second author exress his gratitude towards his parents for blessing. All the authors are also grateful to the Hon’ble vice- chancellor, Patliputra University, Patna, Bihar, India, for motivation to this work. Refrences [1] A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 3rd rev. ed., 2002. [2] E.Z. Psarakis and G.V. Moustakides, An 𝐿2-based method for the design of 1-D zero phase FIR digital filters, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 44(7), 551-601(1997). [3] Hare Krishna Nigam and Md. 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