Mathematics and Computer Science, Volume 1
Description
This first volume in a new multi-volume set gives readers the basic concepts and applications for diverse ideas and innovations in the field
of computing together with its growing interactions with mathematics.
This new edited volume from Wiley-Scrivener is the first of its kind to present scientific and technological innovations by leading academicians, eminent researchers, and experts around the world in the areas of mathematical sciences and computing. The chapters focus on recent advances in computer science, and mathematics, and where the two intersect to create value for end users through practical applications of the theory.
The chapters herein cover scientific advancements across a diversified spectrum that includes differential as well as integral equations with applications, computational fluid dynamics, nanofluids, network theory and optimization, control theory, machine learning and artificial intelligence, big data analytics, Internet of Things, cryptography, fuzzy automata, statistics, and many more. Readers of this book will get access to diverse ideas and innovations in the field of computing together with its growing interactions in various fields of mathematics. Whether for the engineer, scientist, student, academic, or other industry professional, this is a must-have for any library.
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Persons
Sharmistha Ghosh, PhD, is a professor at the Institute of Engineering and Management, India. She received her doctorate in mathematics from the Indian Institute of Technology, India. Her major field of study includes fuzzy and vague databases as well as computational fluid dynamics, and she has published several papers in scientific journals. She is also the editor of several scientific journals and works as a reviewer of journals as well as doctoral theses in India as well abroad.
M. Niranjanamurthy, PhD, is an assistant professor in the Department of Computer Applications, M S Ramaiah Institute of Technology, Bangalore, Karnataka. He earned his PhD in computer science at JJTU, Rajasthan, India. He has over 11 years of teaching experience and two years of industry experience as a software engineer. He has published several books, and he is working on numerous books for Scrivener Publishing. He has published over 60 papers for scholarly journals and conferences, and he is working as a reviewer in 22 scientific journals. He also has numerous awards to his credit.
Krishanu Deyasi is an associate professor in the Department of Basic Sciences and Humanities at the Institute of Engineering and Management, India. He earned his PhD from the Indian Institute of Science Education and Research, and he has postdoctoral experience from The Institute of Mathematical Sciences, India. He has written three books and has published papers in scientific journals. He is also an editor for several scientific journals.
Biswadip Basu Mallik is a senior assistant professor of mathematics in the Department of Basic Sciences and Humanities at the Institute of Engineering and Management, India. He has been involved in teaching and research for more than 21 years and has published several research papers in various scientific journals along with book chapters with various publishers. He has authored five books and has five patents to his credit. He is a managing editor of the Journal of Mathematical Sciences & Computational Mathematics and is an editorial board member and reviewer for several scientific journals.
Santanu Das is an assistant professor in the Department of Basic Sciences and Humanities at the Institute of Engineering and Management, India. He completed his undergraduate degree, and is pursuing his doctorate, from Jadavpur University.
Content
Preface xix
1 Error Estimation of the Function by (Z ru ,r = 1) Using Product Means (E,s)( N, pn ,,qn) n of) the Conjugate Fourier Series 1 Aradhana Dutt Jauhari and Pankaj Tiwar
1.1 Introduction 1
1.1.1 Definition 1 2
1.1.2 Definition 2 2
1.1.3 Definition 3 2
1.2 Theorems 5
1.2.1 Theorem 1 5
1.2.2 Theorem 2 5
1.3 Lemmas 6
1.3.1 Lemma 1 6
1.3.2 Lemma 2 6
1.3.3 Lemma 3 9
1.4 Proof of the Theorems 9
1.4.1 Proof of the Theorem 1 9
1.4.2 Proof of the Theorem 2 15
1.5 Corollaries 16
1.5.1 Corollary 1 16
1.5.2 Corollary 2 16
1.6 Example 16
1.7 Conclusion 18
References 18
2 Blow Up and Decay of Solutions for a Klein-Gordon Equation With Delay and Variable Exponents 21 Hazal Yüksekkaya and Erhan Piskin
2.1 Introduction 21
2.2 Preliminaries 23
2.3 Blow Up of Solutions 26
2.4 Decay of Solutions 36
Acknowledgment 43
References 43
3 Some New Inequalities Via Extended Generalized Fractional Integral Operator for Chebyshev Functional 45 Bhagwat R. Yewale and Deepak B. Pachpatte
3.1 Introduction 45
3.2 Preliminaries 46
3.3 Fractional Inequalities for the Chebyshev Functional 47
3.4 Fractional Inequalities in the Case of Extended Chebyshev Functional 53
3.5 Some Other Fracional Inequalities Related to the Extended Chebyshev Functional 57
3.6 Concluding Remark 63
References 64
4 Blow Up of the Higher-Order Kirchhoff-Type System With Logarithmic Nonlinearities 67 Nazli Irkil and Erhan Piskin
4.1 Introduction 67
4.2 Preliminaries 69
4.3 Blow Up for Problem for E (0) < d 78
4.4 Conclusion 84
References 85
5 Developments in Post-Quantum Cryptography 89 Srijita Sarkar, Saranya Kumar, Anaranya Bose and Tiyash Mukherjee
5.1 Introduction 90
5.2 Modern-Day Cryptography 90
5.2.1 Symmetric Cryptosystems 91
5.2.2 Asymmetric Cryptosystems 91
5.2.3 Attacks on Modern Cryptosystems 92
5.2.3.1 Known Attacks 93
5.2.3.2 Side-Channel Attacks 93
5.3 Quantum Computing 93
5.3.1 The Main Aspects of Quantum Computing 94
5.3.2 Shor's Algorithm 95
5.3.3 Grover's Algorithm 96
5.3.4 The Need for Post-Quantum Cryptography 96
5.4 Algorithms Proposed for Post-Quantum Cryptography 97
5.4.1 Code-Based Cryptography 97
5.4.2 Lattice-Based Cryptography 98
5.4.3 Multivariate Cryptography 99
5.4.4 Hash-Based Cryptography 99
5.4.5 Supersingular Elliptic Curve Isogeny Cryptography 100
5.4.6 Quantum-Resistant Symmetric Key Cryptography 100
5.5 Launching of the Project Called "Open Quantum Safe" 100
5.6 Algorithms Proposed During the NIST Standardization Procedure for Post-Quantum Cryptography 101
5.7 Hardware Requirements of Post-Quantum Cryptographic Algorithms 101
5.7.1 NTRUEncrypt 101
5.7.1.1 Polynomial Multiplication 102
5.7.1.2 Hardware to Accelerate NTRUEncrypt 103
5.7.2 Hardware-Software Design to Implement PCQ Algorithms 103
5.7.3 Implementation of Cryptographic Algorithms Using HLS 103
5.8 Challenges on the Way of Post-Quantum Cryptography 104
5.9 Post-Quantum Cryptography Versus Quantum Cryptography 105
5.10 Future Prospects of Post-Quantum Cryptography 106
References 107
6 A Statistical Characterization of MCX Crude Oil Price with Regard to Persistence Behavior and Seasonal Anomaly 111 Anindita Bhattacharjee, Jaya Mamta Prosad and M.K. Das
6.1 Introduction 111
6.2 Related Literature 113
6.3 Data Description and Methodology 114
6.3.1 Data 114
6.3.2 Methodology 115
6.3.2.1 Characterizing Persistence Behavior of Crude Oil Return Time Series Using Hurst Exponent 115
6.3.2.2 Zipf Plot 116
6.3.2.3 Seasonal Anomaly in Oil Returns 117
6.4 Analysis and Findings 117
6.4.1 Persistence Behavior of Daily Oil Stock Price 117
6.4.2 Detecting Seasonal Pattern in Oil Prices 121
6.5 Conclusion and Implications 123
References 125
Appendix 128
7 Some Fixed Point and Coincidence Point Results Involving Ga -Type Weakly Commuting Mappings 133 Krishna Kanta Sarkar, Krishnapada Das and Abhijit Pramanink
7.1 Introduction 133
7.2 Definitions and Mathematical Preliminaries 134
7.2.1 Definition: G-metric Space (G-ms) 134
7.2.2 Definition: t-norm 135
7.2.3 Definition: t-norm of Had?ic type (H-type) 135
7.2.4 Definition: G-fuzzy metric space (G-fms) 135
7.2.5 Definition 136
7.2.6 Lemma 136
7.2.7 Lemma 136
7.2.8 Definition 136
7.2.9 Definition 136
7.2.10 Definition: F-Function 136
7.2.11 Definition: ¿-Function 137
7.2.12 Lemma 137
7.2.13 Definition 138
7.2.14 Definition 138
7.2.15 Definition 138
7.2.16 Definition 138
7.2.17 Definition 139
7.2.18 Remarks 139
7.2.19 Lemma 139
7.3 Main Results 140
7.3.1 Theorem 140
7.3.2 Theorem 144
7.3.3 Definition ¿-Function 151
7.3.4 Theorem 152
7.3.5 Theorem 159
7.3.6 Corollary 167
7.3.7 Corollary 168
7.3.8 Example 169
7.3.9 Example 169
7.3.10 Example 170
7.3.11 Example 170
7.4 Conclusion 170
7.5 Open Question 171
References 171
8 Grobner Basis and Its Application in Motion of Robot Arm 173 Anjan Samanta
8.1 Introduction 173
8.1.1 Define Orderings in K[y1 , ., yn] 174
8.1.2 Introducing Division Rule in K[y1 , ., yn] 174
8.2 Hilbert Basis Theorem and Grobner Basis 175
8.3 Properties of Grobner Basis 175
8.4 Applications of Grobner Basis 176
8.4.1 Ideal Membership Problem 176
8.4.2 Solving Polynomial Equations 177
8.5 Application of Grobner Basis in Motion of Robot Arm 178
8.5.1 Geometric Elucidation of Robots 178
8.5.2 Mathematical Representation 179
8.5.3 Forward Kinematic Problem 179
8.5.4 Inverse Kinematic Problem 182
8.6 Conclusion 189
References 189
9 A Review on the Formation of Pythagorean Triplets and Expressing an Integer as a Difference of Two Perfect Squares 191 Souradip Roy, Tapabrata Bhattacharyya, Subhadip Roy, Souradeep Paul and Arpan Adhikary
9.1 Introduction 191
9.2 Calculation of Triples 193
9.2.1 Calculation for Odd Numbers 193
9.2.2 Calculation for Even Numbers 195
9.2.3 Code Snippet 199
9.2.4 Observation 200
9.3 Computing the Number of Primitive Triples 200
9.3.1 Calculation for Odd Numbers 200
9.3.2 Calculation for Even Numbers 203
9.3.3 Code Snippet 204
9.3.4 Observation 205
9.4 Representation of Integers as Difference of Two Perfect Squares 205
9.4.1 Calculation for Odd Numbers 205
9.4.2 Calculation for Even Numbers 206
9.4.3 Corollaries 208
9.4.4 Code Snippet 210
9.4.5 Output 210
9.5 Conclusion 211
References 211
10 Solution of Matrix Games With Pay-Offs of Single-Valued Neutrosophic Numbers and Its Application to Market Share Problem 213 Mijanur Rahaman Seikh and Shibaji Dutta
10.1 Introduction 213
10.2 Preliminaries 216
10.3 Matrix Games With SVNN Pay-Offs and Concept of Solution 218
10.4 Mathematical Model Construction for SVNNMG 219
10.4.1 Algorithm for Solving SVNNMG 223
10.5 Numerical Example 224
10.5.1 A Market Share Problem 224
10.5.2 The Solution Procedure and Result Discussion 226
10.5.3 Analysis and Comparison of Results With li and Nan's Approach 227
10.6 Conclusion 228
References 228
11 A Novel Score Function-Based EDAS Method for the Selection of a Vacant Post of a Company with q-Rung Orthopair Fuzzy Data 231 Utpal Mandal and Mijanur Rahaman Seikh
11.1 Introduction 231
11.2 Preliminaries 234
11.3 A Novel Score Function of q-ROFNs 236
11.3.1 Some Existing q-ROF Score Functions 236
11.3.2 A Novel Score Function of q-ROFNs 237
11.4 EDAS Method for q-ROF MADM Problem 240
11.5 Numerical Example 244
11.6 Comparative Analysis 246
11.7 Conclusions 247
Acknowledgments 248
References 248
12 Complete Generalized Soft Lattice 251 Manju John and Susha D.
12.1 Introduction 251
12.2 Soft Sets and Soft Elements-Some Basic Concepts 252
12.3 gs-Posets and gs-Chains 253
12.4 Soft Isomorphism and Duality of gs-Posets 257
12.5 gs-Lattices and Complete gs-Lattices 259
12.6 s-Closure System and s-Moore Family 264
12.7 Complete gs-Lattices From s-Closure Systems 266
12.8 A Representation Theorem of a Complete gs-Lattice as an s-Closure System 267
12.9 gs-Lattices and Fixed Point Theorem 268
References 269
13 Data Representation and Performance in a Prediction Model 271 Apurbalal Senapati, Soumen Maji and Arunendu Mondal
13.1 Introduction 272
13.1.1 Various Methods for Predictive Modeling 272
13.1.2 Problem Definition 275
13.2 Data Description and Representations 276
13.3 Experiment and Result 281
13.4 Error Analysis 282
13.5 Conclusion 283
References 284
14 Video Watermarking Technique Based on Motion Frames by Using Encryption Method 285 Praful Saxena and Santosh Kumar
14.1 Introduction 286
14.2 Methodology Used 287
14.2.1 Discrete Wavelet Transform 287
14.2.2 Singular-Value Decomposition 289
14.3 Literature Review 289
14.4 Watermark Encryption 290
14.5 Proposed Watermarking Scheme 292
14.5.1 Watermark Embedding 292
14.5.2 Watermark Extraction 294
14.6 Experimental Results 296
14.7 Conclusion 297
References 298
15 Feature Extraction and Selection for Classification of Brain Tumors 299 Saswata Das
15.1 Introduction 299
15.2 Related Work 301
15.3 Methodology 303
15.3.1 Contrast Enhancement 303
15.3.2 K-Means Clustering 303
15.3.3 Canny Edge Detection 305
15.3.4 Feature Extraction 308
15.3.5 Feature Selection 309
15.3.5.1 Genetic Algorithm for Feature Selection 309
15.3.5.2 Particle Swarm Optimization for Feature Selection 311
15.4 Results 313
15.5 Future Scope 313
15.6 Conclusion 314
References 315
16 Student's Self-Esteem on the Self-Learning Module in Mathematics 6 317 Ariel Gulla Villar and Biswadip Basu Mallik
16.1 Introduction 318
16.1.1 Research Questions 318
16.1.2 Scope and Limitation 319
16.1.3 Significance of the Study 319
16.2 Methodology 320
16.2.1 Research Design 320
16.2.2 Respondents of the Study 320
16.2.3 Sampling Procedure 320
16.2.4 Locale of the Study 320
16.2.5 Data Collection 321
16.2.6 Instrument of the Study 321
16.2.7 Validation of Instrument 321
16.3 Results and Discussion 322
16.4 Conclusion 329
16.5 Recommendation 330
References 331
17 Effects on Porous Nanofluid due to Internal Heat Generation and Homogeneous Chemical Reaction 333 Hiranmoy Mondal and Sharmistha Ghosh
Nomenclature 334
17.1 Introduction 334
17.2 Mathematical Formulations 336
17.3 Method of Local Nonsimilarity 341
17.4 Results and Discussions 342
17.5 Concluding Remarks 348
References 349
18 Numerical Solution of Partial Differential Equations: Finite Difference Method 353 Roushan Kumar, Rakhi Tiwari and Rashmi Prasad
18.1 Introduction 353
18.2 Finite Difference Method 356
18.2.1 Finite Difference Approximations to Derivatives 356
18.2.2 Discretization of Domain 356
18.2.3 Difference Scheme of Partial Differential Equation 358
18.3 Multilevel Explicit Difference Schemes 360
18.4 Two-Level Implicit Scheme 364
18.5 Conclusion 371
References 371
19 Godel Code Enciphering for QKD Protocol Using DNA Mapping 373 Partha Sarathi Goswami and Tamal Chakraborty
19.1 Introduction 374
19.2 Related Work 375
19.3 The DNA Code Set 376
19.4 Godel Code 376
19.5 Key Exchange Protocol 378
19.6 Encoding and Decoding of the Plain Text- The QKD Protocol 378
19.6.1 Plain Text to Encoded Text and Vice-Versa 379
19.6.2 The Proposed Message Passing Scheme 380
19.6.3 Illustration 381
19.7 Experimental Setup 388
19.8 Detection Probability and Dark Counts 389
19.9 Security Analysis of Our Algorithm 390
19.10 Conclusion 393
References 393
20 Predictive Analysis of Stock Prices Through Scikit-Learn: Machine Learning in Python 397 Vikash Kumar Mishra, Richa Binyala, Pratibha Sharma and Simran Singh
20.1 Introduction 397
20.2 Study Area and Dataset 398
20.3 Methodology 399
20.4 Results 401
20.5 Conclusion 402
References 403
21 Pose Estimation Using Machine Learning and Feature Extraction 405 J. Palanimeera and K. Ponmozhi
21.1 Introduction 406
21.2 Related Work 408
21.3 Proposed Work 409
21.3.1 Yoga Posture Identification 410
21.3.1.1 Deep Extraction of a Normal Image 410
21.3.1.2 Human Joints Identification 411
21.3.1.3 Extraction of L-DoD Features 411
21.3.1.4 Extraction of D-GoD Features 415
21.3.2 The Random Forest Classifier's Design 416
21.3.2.1 Construction of a Random Forest Model 416
21.3.2.2 Random Forest Two-Way Voting 417
21.3.3 Joint Positioning in Humans 418
21.4 Outcome and Discussion 420
21.5 Conclusion 422
References 423
22 E-Commerce Data Analytics Using Web Scraping 425 Vikash Kumar Mishra, Bosco Paul Alapatt, Aaditya Aggarwal and Divya Khemani
22.1 Introduction 425
22.1.1 Uses of Web Scraping 426
22.2 Research Objective 426
22.3 Literature Review 427
22.4 Feasibility and Application 428
22.4.1 Web Scrapers Process 428
22.5 Proposed Methodology 428
22.5.1 Coding Phase 429
22.5.2 Spreadsheet Analysis and Results 432
22.6 Conclusion 433
References 433
23 A New Language-Generating Mechanism of SNPSSP 435 Prithwineel Paul, Soumadip Ghosh and Anjan Pal
23.1 Introduction 436
23.2 Spiking Neural P Systems With Structural Plasticity ((SNPSSP) 437
23.3 Labeled SNPSSP (LSNPSSP) 440
23.3.1 Working of LSNPSSP 441
23.4 Main Results 442
23.5 Conclusion 450
References 450
24 Performance Analysis and Interpretation Using Data Visualization 455 Vikash Kumar Mishra, Iyyappan, M., Muskan Soni and Neha Jain
24.1 Introduction 455
24.2 Selecting Data Set 456
24.3 Proposed Methodology 457
24.4 Results 458
24.5 Conclusion 460
References 460
25 Dealing with Missing Values in a Relation Dataset Using the DROPNA Function in Python 463 Vikash Kumar Mishra, Shoney Sebastian, Maria Iqbal and Yashwin Anand
25.1 Introduction 464
25.2 Background 464
25.3 Study Area and Data Set 464
25.4 Methodology 466
25.5 Results 468
25.6 Conclusion 468
25.7 Acknowledgment 469
References 469
26 A Dynamic Review of the Literature on Blockchain-Based Logistics Management 471 C. Devi Parameswari and M. Ilayaraja
26.1 Introduction 471
26.2 Blockchain Concepts and Framework 473
26.3 Study of the Literature 475
26.3.1 Blockchain Technology and Supply Chain Trust 475
26.4 Challenges and Processes of Supply Chain Transparency 477
26.4.1 Motivation for Transparency in Data 478
26.5 Challenges in Security 478
26.6 Discussion: In Terms of Supply Chain Dynamics, Blockchain Technology and Supply Chain Integration 479
26.7 Conclusion 481
Acknowledgment 481
References 482
27 Prediction of Seasonal Aliments Using Big Data: A Case Study 485 K. Indhumathi and K. Sathesh Kumar
27.1 Introduction 486
27.2 Related Works 486
27.3 Conclusion 489
References 490
28 Implementation of Tokenization in Natural Language Processing Using NLTK Module of Python 493 Vikash Kumar Mishra, Abhimanyu Dhyani, Sushree Barik and Tanish Gupta
28.1 Introduction 493
28.2 Background 494
28.3 Study Area and Data Set 495
28.4 Proposed Methodology 495
28.5 Result 498
28.6 Conclusion 500
28.7 Acknowledgment 501
Conflicts of Interest/Competing Interests 501
Availability of Data and Material 503
References 503
29 Application of Nanofluids in Heat Exchanger and its Computational Fluid Dynamics 505 M. Appadurai, E. Fantin Irudaya Raj and M. Chithambara Thanu
29.1 Computational Fluid Dynamics 506
29.1.1 Continuity Equation 506
29.1.2 Momentum Equation 507
29.1.3 Energy Equation 508
29.1.4 Equations for Turbulent Flows 510
29.2 Nanofluids 510
29.2.1 Viscosity 511
29.2.2 Density 511
29.2.3 Heat Capacity 512
29.2.4 Thermal Conductivity 512
29.3 Preparation of Nanofluids 512
29.3.1 One-Step Method 513
29.3.2 Two-Step Method 514
29.3.3 Nanofluids Implementation in Heat Exchanger 515
29.4 Use of Computational Fluid Dynamics for Nanofluids 517
29.5 CFD Approach to Solve Heat Exchanger 518
29.6 Conclusion 522
References 522
About the Editors 525
Index 527
1
Error Estimation of the Function by Using Product Means of the Conjugate Fourier Series
Aradhana Dutt Jauhari* and Pankaj Tiwari
Division of Mathematics, Department of Basic Sciences, Galgotias University, Greater Noida, G. B. Nagar, U.P., India
Abstract
The purpose of the current chapter is to attain the best result on a new way and the best approximation of different classes of the work, which has been discussed by different mathematician under different summability means. Here, we are presenting the theorems established under means of the CFS of a signal, belongs to Some known and unknown results have been proven by many mathematicians. But this is a new and unique way of proving a new result.
Keywords: Generalized Zygmund class product means, Degree of Approximation (DoA), Conjugate Fourier Series (CFS)
1.1 Introduction
Let, us take San - an infinite series and sn - the sequence of partial sums. Also, {pn} and {qn} are the sequences of positive real numbers s.t.-
1.1.1 Definition 1
Sequence-to-sequence of the transformation:
(1.1)defines, mean of {sn}.
If the San is summable to s by (E, s) method regular ([12] Hardy, 1949).
1.1.2 Definition 2
Sequence-to-sequence transformation:
(1.2)where,
Rn = po qn + p1 qn - 1 + ? + pn qo (?), p-1 = q-1 = R-1 then the series San is summable to s. Regularity conditions [12]-
- where M > 0 and independent of n.
1.1.3 Definition 3
The transform (E, s) over summability and is given by [7]-
(1.3)and San is said to summable to s by the product means
Remark. If we put qn = 1 in equation (1.3) then summability mean reduces to mean and for pn = 1 it becomes to mean.
Now we take Fourier and its CFS (Conjugate Fourier series) as-
(1.4)The partial sum is given by,
Let known as space function which is 2p - periodic and also integrable. Norm ||.||r is defined as,
also,
again, C2p- The Banach space of 2p- periodic functions, defined on [0, 2p] under the sup. norm.
Where, 0 < a = 1,
the function space [11].
For g ? Lr [0, 2p], r = 1,
For g ? C2p and r = 8
also,
Now,
also, Banach space with the norm ||.||(a), r can be defined as the space Z(a), r = 1, 0 < a = 1
The class function
Where, w > 0, continuous function having sub-linear property, that is
- w (0) = 0,
- w (t1 + t2) = w (t1) + w (t2)
Let w: [0, 2p] Rs.t.w (t) > 0 for 0 = t < 2p.
Clearly is a norm on
As we know Lr (r = 1) is complete, the space is also complete Banach space under the norm
Used notations in this paper:
1.2 Theorems
Many mathematicians and the researcher studies and worked on Error Estimation of FS and its CFS in different Lip class, Weightage class as well as Zygmund class using various product means. Amongst them some are Nigam [1-3], Singh [4], Deer [8], Lal [5], Pradhan [6, 7] and Mishra [9, 10] etc.
We are going to prove a new result for conjugate Fourier series belonging to means.
1.2.1 Theorem 1
Let 2p-periodic function and Lebesgue integrable in [0, 2p] and belonging to then using the product mean and the DoA of the function is given by-
Where w(t) and v(t) denoted the Zygmund modulus of continuity s.t. positive.
1.2.2 Theorem 2
Let is 2p - periodic function and Lebesgue integrable in [0, 2p] and belonging to then using the product mean the DoA of is given by-
1.3 Lemmas
1.3.1 Lemma 1
Proof.
1.3.2 Lemma 2
Proof. Since, using Jordan lemma and sinnt = 1
(1.5)Taking, first term of (1.5)-
(1.6)Taking, next term of (1.5), by Abel's lemma-
(1.7)Combining (1.5), (1.6), and (1.7) we get
1.3.3 Lemma 3
Let, where 0 < t = p,
1.4 Proof of the Theorems
1.4.1 Proof of the Theorem 1
Using Riemann-Lebesgue theorem,
Further,
Using GMI, we have
(1.8)Now, using Lemma 1 and Lemma 3 with monotonicity of with the respect to t,
Using, 2nd MVT of integral, we have
(1.9)Using, Lemma 2 and Lemma 3-
(1.10)From (1.8), (1.9) and (1.10) we have
(1.11)Clearly,
Applying Minikowski's inequality,
Using Lemma 2 and Lemma 3, we get
(1.12)From (1.11) and (1.12), we have
(1.13)We write H1 in terms of H3 and H2, H3 in terms of H4.
In show of monotonicity of v (t) for 0 < t = p, we have
(1.14)Therefore, we can write (1.14)
Again, using monotonicity of v (t)
(1.15)By using the fact that we have
(1.16)Therefore, we write
(1.17)Now combining, (1.15) and (1.17) we have-
(1.18)Hence,
(1.19)Proof of the theorem 1 is completed.
1.4.2 Proof of the Theorem 2
We have,
(1.20)we have assumed that the is positive and decreasing in t. Hence,
(1.21)Proof of the theorem 2 is completed.
1.5 Corollaries
1.5.1 Corollary 1
If we put (E, 1), (C, 1) mean in place of mean, in the theorem-1, the DoA of the function (C, 1) mean-
Of the CFS is,
Proof: [8]
1.5.2 Corollary 2
If we put mean in place of mean in the Theorem 1, the DoA of the function by (E, 1), (C, 1) mean
of CFS is,
Proof: [8]
1.6 Example
In this example, we are showing (E, s) and (Nörlund (Np)) summability of the partial sums of a FS is better behaved than the sequence of partial sum sn (x).
Let
with f (x + 2p) = f (x), ? x.
Fourier series of
The nth partial sum sn (x) of FS is -
Figure 1.1 [10]: The graph of f (x) function and the Fourier series expansion for n = 10
If Nörland method Np has increasing weights {pn}, its means Np is given by
We observed that converge to f (x) faster than sn (x) in
Figure 1.1 Radial basis neural network architecture. Reprint with copyright permission from [11].
1.7 Conclusion
The proved results in this paper are an attempt to study the approximations of the functions in by using product means of conjugate Fourier series, which generalize several results. Further, the result can be extended for other functions belonging to
References
- 1. Nigam, H.K., Best approximation conjugate of a function in generalized Zygmund class. Tamkang J. Math., 50, 4, 417-428, Dec. 2019. doi:10.5556/j. tkjm.50.2019.3006.
- 2. Nigam, H.K. and Sharma, K., On (e, 1) (c, 1) summability of fourier series and its conjugate series. Int. J. Pure Appl. Math., 82, 3, 365-375, ISSN: 1311-8080, 2013.
- 3. Nigam, H.K., On approximation in generalized Zygmund class. De Gruyter. Demonstratio Mathematica, 52, 370-387, 2019. http://doi.org/10.1515/dem-2019-0022.
- 4. Singh, M.V., Mittal, M.L., Rhoades, B.E., Approximation of function in the generalized Zygmund class using hausdroff means. J. Inequal. Appl., 2017, 101, 2017. DOI 10.1186/s13660-017-1361-8.
- 5. Lal, S. and Shireen, Best approximations of functions of generalized Zygmund class by Matrix- Euler Summability means of fourier series. Bulletin of Mathematical Analysis & Applications, ISSN: 1821-1291, 5, 4, 1-13. 13p, 2013. URL: http//www.bmathaa.org.
- 6. Pradhan, T., Paikray, S., Das, A., Dutta, H., Approximation of signals in generalised Zygmund class via...
