Description
Book Details
Title: Wavelet Transform Involving Index Integral Transforms
Author: Dr. Upain Kumar Mandal
Affiliation: Assistant Professor, Department of Mathematics, Nalanda College, Biharsharif, Bihar, India
Publisher: Cogniverse Press, Jorhat, Assam, India
First Edition: June 2026
DOI: 10.5281/zenodo.20502516
ISBN: 978-93-47652-35-6 (Print Edition)
e-ISBN: 978-93-47652-31-8 (Digital Edition)
Cover Design: Cogniverse Press Digital Team
Copyright: © Author
Publisher Information
Cogniverse Press
Nakari Gaon, Borigaon Siding, Jorhat – 1, Assam, India
Phone: +91-9101730579
Website: cogniversepress.com
Email: cogniversepress@gmail.com
Preface
Integral transformations have long occupied a central position in both pure and applied mathematics. Among these analytical tools, wavelet analysis has emerged as a powerful framework capable of providing localized time-frequency representations beyond the capabilities of many classical transforms.
This volume presents a comprehensive study of the Continuous Wavelet Transform (CWT), its structural properties, and its interaction with important classes of index integral transforms. Special emphasis is placed on combining wavelet analysis with various integral transforms to develop new analytical perspectives and mathematical techniques.
A major theme of the book is the investigation of Parseval-type and Plancherel-type relations within composite transform frameworks. These relations establish important correspondences between function spaces while preserving energy and orthogonality properties, thereby enriching the understanding of harmonic analysis.
The book also explores wave packet transforms, which offer greater flexibility than classical wavelets through enhanced localization in both spatial and frequency domains. Their formulation through multiple index transforms expands the theoretical foundation and application potential of wavelet methods.
Particular attention is devoted to the Kontorovich–Lebedev Transform, the Generalized Kontorovich–Lebedev Transform, and the Lebedev–Skalskaya Transform. By integrating these transforms within wavelet frameworks, the book develops new representations, inversion techniques, and operational methods applicable to spectral theory, differential equations, and special-function analysis.
The volume seeks both to consolidate existing research into a coherent presentation and to introduce new methodologies that may inspire future investigations. Beginning with foundational concepts and progressing toward advanced theoretical developments, the book provides a structured pathway for researchers and students interested in modern harmonic analysis.
It is hoped that this work will contribute meaningfully to the advancement of wavelet theory and index integral transforms while encouraging new research directions through a unified analytical framework.
Dr. Upain Kumar Mandal
Acknowledgement
The author expresses sincere gratitude to Almighty God for His blessings, wisdom, and guidance throughout the completion of this work.
Deep appreciation is extended to the author’s parents for their unwavering support, sacrifices, and encouragement. Their faith and guidance have been fundamental to every academic achievement.
Special thanks are offered to the author’s brothers for their continual encouragement and support, and to his nephew, Adhrit, whose cheerful presence provided inspiration and moments of joy during the preparation of the manuscript.
The author is profoundly grateful to his wife, Priya, for her unconditional love, patience, and constant motivation throughout the journey.
Sincere appreciation is also extended to Prof. Akhilesh Prasad for his valuable guidance, scholarly insights, and encouragement, which played a significant role in the successful completion of this work.
Finally, the author thanks the Principal, colleagues, contributors, and well-wishers whose direct and indirect support contributed to the completion of this book.
Dr. Upain Kumar Mandal
Key Themes
- Continuous Wavelet Transform (CWT)
- Index Integral Transforms
- Kontorovich–Lebedev Transform
- Generalized Kontorovich–Lebedev Transform
- Lebedev–Skalskaya Transform
- Wave Packet Transforms
- Parseval-Type Relations
- Plancherel-Type Theorems
- Harmonic Analysis
- Spectral Theory and Special Functions
- Transform Composition Techniques
- Mathematical Methods in Applied Analysis
Table of Contents
Chapter 1: Kontorovich–Lebedev Transforms and Involving Continuous Waveforms
- 1.1 Introduction
- 1.2 Preparatory Results
- 1.3 Kontorovich–Lebedev Continuous Wavelet Transformation (KLCWT)
- 1.4 Plancherel and Parseval’s Relation for KLCWT
Chapter 2: Composition and Wave Packet Transform Associated with the KL Transform
- 2.1 Introduction
- 2.2 Composition of KLCWT
- 2.3 Wave Packet Transform Involving the KL Transform
- 2.4 Plancherel and Parseval’s Relation for WPT Involving the KL Transform
Chapter 3: Wavelet Transform Involving the Kontorovich–Lebedev–Clifford Transform
- 3.1 Introduction
- 3.2 Continuous KLC Wavelet Transform
- 3.3 Plancherel and Parseval’s Relation for CKLCWT
Chapter 4: The Generalized Kontorovich–Lebedev Transform and Associated Wavelet Transform
- 4.1 Introduction
- 4.2 Continuous Wavelet Transform
- 4.3 Plancherel and Parseval’s Relation for Generalized CWT
Chapter 5: Wavelet Transform Involving the LS Transform
- 5.1 Introduction
- 5.2 Preparatory Results
- 5.3 Continuous Wavelet Transform Associated with the LS Transform
- 5.4 Parseval and Plancherel Type Relations for LSCWT
- 5.5 Composition Associated with LSCWT
Chapter 6: Lebedev–Skalskaya Transforms and Associated Wave Packet Transform
- 6.1 Introduction
- 6.2 Continuous LS-Wavelet Transform (LSCWT)
- 6.3 Wave Packet Transform Involving the LS Transform
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