Fourier Meets Hilbert and Riesz: An Introduction to the Corresponding Transforms

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.--taken From Back Cover.,This Book Is Based On Lecture Notes From A Special Topics Course On Advanced Analysis Offered At Universidad Nacional De Colombia During The Spring Semester Of 2019. Prerequisites Are A First-year Course On Measure And Integration Theory And Functional Analysis, As Well As Some Basics Of Metric Spaces And Complex Variables. Our Intention In Writing These Notes Is To Prepare Students To Do Research On Fourier Analysis. The Lecture Notes Were Rather Informal With A Conversational Tone. While Emphasizing Formality, These Notes Keep Pedagogy In Mind As We Aim To Serve Students Who Might Need To Teach The Material To Themselves Because They Do Not Have Access To An Equivalent Course At Their Institutions. We Would Like To Emphasize That This Manuscript Is Meant As A Textbook, Not As A Reference Book. The Expository Paper [1] Was An Early Byproduct Of This Effort. We Have Made Every Effort To Keep This Book Self-contained. To That End, The First Chapter Includes All Fundamental Concepts Needed For The Rest Of The Book Where, Indeed, Fourier Meets Hilbert And Riesz. The Hilbert Transform Is Essentially The Only Singular Operator In One Dimension, In The Sense That, Without A Doubt, It And Its Generalizations Are Some Of The Most Important Operators In Harmonic Analysis. We Present And Elementary Introduction To The Hilbert And Riesz Transforms In Chapters 5,6, 7, And 8. An Appendix Is Also Given For The Sake Of Completeness. Exercises Are Included Throughout The Text To Help Challenge And Motive Students As They Gauge Their Progress. While Most Problems Are Of A Routine Nature, A Few Problems Will Likely Be Challenging To Beginning Students.--taken From Preface, Page [vii].-- Source Other Than The Library Of Congress.