Fourier Meets Hilbert and Riesz An Introduction to the Corresponding Transforms 1st Edition by René Erlin Castillo 3110784122 9783110784121

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Product details:

ISBN 10: 3110784122 

ISBN 13: 9783110784121

Author: René Erlin Castillo

This book provides an introduction into the modern theory of classical harmonic analysis, dealing with Fourier analysis and the most elementary singular integral operators, the Hilbert transform and Riesz transforms. Ideal for self-study or a one semester course in Fourier analysis, included are detailed examples and exercises.

Table of contents:

1 Fundamental concepts

1.1 Introduction

1.2 Inner products

1.3 Strong and weak convergence

1.4 The space L2(ℝ)

1.5 A glance at linear operators

1.6 The adjoint of an operator

2 Fourier series

2.1 Convolution

2.2 Fourier transform on ℝ

2.3 Fourier transform on L2(ℝ)

2.4 Definition of the Fourier transform in L2(ℝ)

2.5 Fourier transform on L1(ℝn)

2.6 Bessel functions

3 Schwartz spaces S(ℝn)

3.1 Definition of the Fourier transform in L2(ℝn)

3.2 Some applications of the Fourier transform to partial differential equations

Cauchy problem for the wave equation

Laplace and Poisson equations

4 Distribution functions

4.1 Extending the Fourier transform to Lp(ℝn)

4.2 Covering lemma

4.3 Maximal function

4.4 Weak- Lp space

4.5 Lebesgue differentiation theorem

4.6 Cube

4.7 Properties of dyadic subcubes

4.8 Calderón–Zygmund decomposition

4.9 Operators of type (p,q)

4.10 Marcinkiewicz interpolation theorem

4.11 The Cauchy principal value

5 Three-fold approach to the Hilbert transform

5.1 Derivation of the Hilbert transform on ℝ

5.2 The Hilbert transform

5.3 The Hilbert transform on the circle

5.4 Calderón–Zygmund singular integral operators

5.5 Some Hilbert transforms

5.5.1 Hilbert transform of a constant function

5.5.2 Hilbert transform of sinx and cosx

5.5.3 Hilbert transform of eix

5.6 Relationship between the Hilbert and Laplace transforms

5.6.1 The Hilbert transform of 1x+iα and the Poisson kernel

5.7 Hilbert transform of a periodic function

5.8 Properties

6 Hilbert transform in L2(ℝ)

6.1 Hilbert transform of a product

6.2 Differentiation property of the Hilbert transform in L2

6.3 Hilbert transform as an operator in L2

6.4 The Parseval-type form

6.5 A theorem due to E. M. Stein and G. Weiss

7 Embedding and strong Lp boundedness for the Hilbert transform

7.1 Strong Lp boundedness for the Hilbert transform on the circle

8 Riesz transform

8.1 Vectorial Riesz transform

A.1 Convergence in measure

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Tags: René Erlin Castillo, Fourier, Meets, Corresponding