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Fourier analysis

Author: Javier Duoandikoetxea Zuazo
Publisher: Providence, R.I. : American Mathematical Society, ©2001.
Series: Graduate studies in mathematics, v. 29.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Fourier analysis encompasses a variety of perspectives and techniques. This book presents the real variable methods of Fourier analysis introduced by Calderon and Zygmund. It includes topics such as  Read more...

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Details

Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Javier Duoandikoetxea Zuazo
ISBN: 0821821725 9780821821725
OCLC Number: 44811910
Description: xviii, 222 pages ; 27 cm.
Contents: Chapter 1. Fourier Series and Integrals 1 --
1. Fourier coefficients and series 1 --
2. Criteria for pointwise convergence 2 --
3. Fourier series of continuous functions 6 --
4. Convergence in norm 8 --
5. Summability methods 9 --
6. Fourier transform of L[superscript 1] functions 11 --
7. Schwartz class and tempered distributions 12 --
8. Fourier transform on L[superscript p], 1 <p [less than or equal] 2 15 --
9. Convergence and summability of Fourier integrals 17 --
Chapter 2. Hardy-Littlewood Maximal Function 25 --
1. Approximations of the identity 25 --
2. Weak-type inequalities and almost everywhere convergence 26 --
3. Marcinkiewicz interpolation theorem 28 --
4. Hardy-Littlewood maximal function 30 --
5. Dyadic maximal function 32 --
6. Weak (1, 1) inequality for the maximal function 35 --
7. A weighted norm inequality 37 --
Chapter 3. Hilbert Transform 49 --
1. Conjugate Poisson kernel 49 --
2. Principal value of 1/x 50 --
3. Theorems of M. Riesz and Kolmogorov 51 --
4. Truncated integrals and pointwise convergence 55 --
5. Multipliers 58 --
Chapter 4. Singular Integrals (I) 69 --
1. Definition and examples 69 --
2. Fourier transform of the kernel 70 --
3. Method of rotations 73 --
4. Singular integrals with even kernel 77 --
5. An operator algebra 80 --
6. Singular integrals with variable kernel 83 --
Chapter 5. Singular Integrals (II) 91 --
1. Calderon-Zygmund theorem 91 --
2. Truncated integrals and the principal value 94 --
3. Generalized Calderon-Zygmund operators 98 --
4. Calderon-Zygmund singular integrals 101 --
5. A vector-valued extension 105 --
Chapter 6. H[superscript 1] and BMO 115 --
1. Space atomic H[superscript 1] 115 --
2. Space BMO 117 --
3. An interpolation result 121 --
4. John-Nirenberg inequality 123 --
Chapter 7. Weighted Inequalities 133 --
1. A[subscript p] condition 133 --
2. Strong-type inequalities with weights 137 --
3. A[subscript 1] weights and an extrapolation theorem 140 --
4. Weighted inequalities for singular integrals 143 --
Chapter 8. Littlewood-Paley Theory and Multipliers 157 --
1. Some vector-valued inequalities 157 --
2. Littlewood-Paley theory 159 --
3. Hormander multiplier theorem 163 --
4. Marcinkiewicz multiplier theorem 166 --
5. Bochner-Riesz multipliers 168 --
6. Return to singular integrals 172 --
7. Maximal function and the Hilbert transform along a parabola 178 --
Chapter 9. T1 Theorem 195 --
1. Cotlar's lemma 195 --
2. Carleson measures 197 --
3. Statement and applications of the T1 theorem 201 --
4. Proof of the T1 theorem 205.
Series Title: Graduate studies in mathematics, v. 29.
Other Titles: Análisis de Fourier.
Responsibility: Javier Duoandikoetxea ; translated and revised by David Cruz-Uribe.

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