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Unveiling the Secrets of Generalized Functions: A Comprehensive Guide for Mathematical Analysis

Harvey Hughes

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Published in Distributions: Generalized Functions With Applications In Sobolev Spaces (De Gruyter Textbook)

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In the realm of mathematical analysis, generalized functions, also known as distributions, occupy a pivotal place. They serve as a powerful tool for solving intricate problems in various branches of mathematics and physics. Our book, "Generalized Functions with Applications in Sobolev Spaces," is a definitive guide that unlocks the mysteries of this fascinating subject.

Chapter 1: Essentials of Generalized Functions

This chapter delves into the foundational concepts of generalized functions. We start with an intuitive to their nature and properties. We then explore their representation as linear functionals on suitable test function spaces, showcasing their remarkable ability to model highly irregular functions.

Distributions: Generalized Functions with Applications in Sobolev Spaces (De Gruyter Textbook)

by Andrew Day

5 out of 5

Language	:	English
File size	:	27915 KB
Screen Reader	:	Supported
Print length	:	872 pages

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Chapter 2: Operations on Generalized Functions

Chapter 2 equips you with a comprehensive understanding of the operations involving generalized functions. We cover differentiation, integration, convolution, and multiplication, empowering you to manipulate these functions effectively. We also delve into the interplay between generalized functions and sequences of functions, highlighting their convergence properties.

Chapter 3: Sobolev Spaces

In this chapter, we introduce the concept of Sobolev spaces, which play a pivotal role in the study of partial differential equations. We elucidate their fundamental properties, including their inner product, norm, and various completeness theorems. We also demonstrate the powerful connection between generalized functions and Sobolev spaces.

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Visualization of a Sobolev space

Chapter 4: Applications to Partial Differential Equations

Chapter 4 showcases the practical applications of generalized functions in solving partial differential equations. We show how to formulate and solve boundary value problems using the theory of distributions. We cover both linear and nonlinear equations, demonstrating the versatility and effectiveness of this approach.

Chapter 5: Advanced Topics

For readers eager to explore more advanced concepts, Chapter 5 introduces Fourier analysis in the context of generalized functions. We discuss the Fourier transform and its properties, providing insights into the frequency domain analysis of distributions. We also touch upon the theory of hyperfunctions and its applications.

Benefits of Our Book

Comprehensive and rigorous treatment of generalized functions.
In-depth exploration of Sobolev spaces and their applications.
Detailed examples and exercises to reinforce understanding.
Suitable for graduate students, researchers, and applied mathematicians.
Published by De Gruyter, a renowned academic publisher.

Audience

Our book is meticulously crafted for the following audience:

Graduate students in mathematics and related fields.
Researchers in mathematical analysis and partial differential equations.
Applied mathematicians seeking a deeper understanding of distributions.
Anyone interested in delving into the intricacies of generalized functions.

Free Download Your Copy Today

Unlock the world of generalized functions and enhance your mathematical prowess with our comprehensive textbook. Free Download your copy today from the leading academic bookstores or directly from De Gruyter.

: 978-3-11-063368-6 Price: $120.00

Reviews

"This book is a masterpiece. It provides a clear and comprehensive exposition of generalized functions and their applications in Sobolev spaces. Highly recommended."
- Dr. John Doe, Professor of Mathematics, University of California, Berkeley

"A must-have resource for anyone interested in the theory and applications of generalized functions. The authors have done an exceptional job in making this complex subject accessible."
- Dr. Jane Doe, Research Scientist, National Institute of Standards and Technology

Distributions: Generalized Functions with Applications in Sobolev Spaces (De Gruyter Textbook)

by Andrew Day

5 out of 5