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Preliminaries Read chapter Read first chapter

2019 | OriginalPaper | Chapter

2. Distributions, Sobolev Spaces and the Fourier Transform

Authors : Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

Published in: Problems on Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

This chapter begins with the introduction and properties of the Fourier transform, and then distributions and weak derivatives are introduced. Second half of the chapter is devoted to Sobolev spaces and their properties.

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previous chapter Preliminaries
next chapter Common Methods
Footnotes
1
Once we know that W m, p is complete, we can alternately talk about the completion of a subset of W m, p in the Sobolev norm or about the closure of that subset in W m, p—both constructions lead to the same set.
 
2
This means that the boundary is locally a graph of a Lipschitz continuous function.
 
3
A function is smooth up to the boundary of Ω if it extends to a smooth function on some open \(V\supset \overline {\Omega }\).
 
4
Called also Rellich’s Theorem; F. Rellich proved it for p = 2, V.I. Kondrachov (spelled also Kondrashov or Kondrashev) for all other p.
 
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L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.MATH
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L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015.
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S. Krantz. Explorations in Harmonic Analysis: With Applications to Complex Function Theory and the Heisenberg Group Birkhäuser Boston, Springer 2009.CrossRef
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G. Leoni. A first course in Sobolev spaces. volume 181 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2017.CrossRef
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W. Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.MATH
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E. Stein and R. Shakarchi. Fourier analysis. An introduction. Volume 1 of Princeton Lectures in Analysis, Princeton University Press, Princeton, NJ, 2003.MATH
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R. S. Strichartz. A guide to distribution theory and Fourier transforms. Reprint of the 1994 original. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
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M. E. Taylor. Partial differential equations Vol. I-III, volume 115–117 of Applied Mathematical Sciences. Springer, New York, second edition, 2011.
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F. Trèves. Topological vector spaces, distributions and kernels. Dover Publications, Inc., Mineola, NY, 2006 (unabridged replication of the 1967 original).
Metadata
Title
Distributions, Sobolev Spaces and the Fourier Transform
Authors
Maciej Borodzik
Paweł Goldstein
Piotr Rybka
Anna Zatorska-Goldstein
Copyright Year
2019
Book
Problems on Partial Differential Equations

Print ISBN: 978-3-030-14733-4

Electronic ISBN: 978-3-030-14734-1

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-14734-1_2

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