In Iranian Islam Volume III: Followers of Shi'ism and Sufism 3

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. Book I Cover Half-Title Title Copyright Authors’ Dedications Foreword Preface to Book I Contents Chapter 1. The Genesis of Fourier Analysis 1.1 The vibrating string Simple harmonic motion Standing and traveling waves Harmonics and superposition of tones 1.1.1 Derivation of the wave equation 1.1.2 Solution to the wave equation Traveling waves Superposition of standing waves 1.1.3 Example: the plucked string 1.2 The heat equation 1.2.1 Derivation of the heat equation 1.2.2 Steady-state heat equation in the disc 1.3 Exercises 1.4 Problem Chapter 2. Basic Properties of Fourier Series 2.1 Examples and formulation of the problem Everywhere continuous functions Piecewise continuous functions Riemann integrable functions Functions on the circle 2.1.1 Main definitions and some examples 2.2 Uniqueness of Fourier series 2.3 Convolutions 2.4 Good kernels 2.5 Cesàro and Abel summability: applications to Fourier series 2.5.1 Cesàro means and summation 2.5.2 Fejér’s theorem 2.5.3 Abel means and summation 2.5.4 The Poisson kernel and Dirichlet’s problem in the unit disc 2.6 Exercises 2.7 Problems Chapter 3. Convergence of Fourier Series 3.1 Mean-square convergence of Fourier series 3.1.1 Vector spaces and inner products Preliminaries on vector spaces Two important examples 3.1.2 Proof of mean-square convergence 3.2 Return to pointwise convergence 3.2.1 A local result 3.2.2 A continuous function with diverging Fourier series 3.3 Exercises 3.4 Problems Chapter 4. Some Applications of Fourier Series 4.1 The isoperimetric inequality Curves, length and area Statement and proof of the isoperimetric inequality 4.2 Weyl’s equidistribution theorem The reals modulo the integers 4.3 A continuous but nowhere differentiable function 4.4 The heat equation on the circle 4.5 Exercises 4.6 Problems Chapter 5. The Fourier Transform on R 5.1 Elementary theory of the Fourier transform 5.1.1 Integration of functions on the real line 5.1.2 Definition of the Fourier transform 5.1.3 The Schwartz space 5.1.4 The Fourier transform on S The Gaussians as good kernels 5.1.5 The Fourier inversion 5.1.6 The Plancherel formula 5.1.7 Extension to functions of moderate decrease 5.1.8 The Weierstrass approximation theorem 5.2 Applications to some partial differential equations 5.2.1 The time-dependent heat equation on the real line 5.2.2 The steady-state heat equation in the upper half-plane 5.3 The Poisson summation formula 5.3.1 Theta and zeta functions 5.3.2 Heat kernels 5.3.3 Poisson kernels 5.4 The Heisenberg uncertainty principle 5.5 Exercises 5.6 Problems Chapter 6. The Fourier Transform on Rd 6.1 Preliminaries 6.1.1 Symmetries 6.1.2 Integration on Rd Polar coordinates 6.2 Elementary theory of the Fourier transform 6.3 The wave equation in Rd × R 6.3.1 Solution in terms of Fourier transforms 6.3.2 The wave equation in R3 × R Huygens principle 6.3.3 The wave equation in R2 × R: descent 6.4 Radial symmetry and Bessel functions 6.5 The Radon transform and some of its applications 6.5.1 The X-ray transform in R2 6.5.2 The Radon transform in R3 6.5.3 A note about plane waves 6.6 Exercises 6.7 Problems Chapter 7. Finite Fourier Analysis 7.1 Fourier analysis on Z(N) 7.1.1 The group Z(N) 7.1.2 Fourier inversion theorem and Plancherel identity on Z(N) 7.1.3 The fast Fourier transform 7.2 Fourier analysis on finite abelian groups 7.2.1 Abelian groups Examples of abelian groups The group Z∗(q) 7.2.2 Characters 7.2.3 The orthogonality relations 7.2.4 Characters as a total family 7.2.5 Fourier inversion and Plancherel formula 7.3 Exercises 7.4 Problems Chapter 8. Dirichlet’s Theorem 8.1 A little elementary number theory 8.1.1 The fundamental theorem of arithmetic 8.1.2 The infinitude of primes The zeta function and its Euler product 8.2 Dirichlet’s theorem 8.2.1 Fourier analysis, Dirichlet characters, and reduction of the 8.2.2 Dirichlet L-functions Historical digression 8.3 Proof of the theorem 8.3.1 Logarithms 8.3.2 L-functions 8.3.3 Non-vanishing of the L-function Case I: complex Dirichlet characters Case II: real Dirichlet characters 8.4 Exercises 8.5 Problems Appendix : Integration A.1 Definition of the Riemann integral A.1.1 Basic properties A.1.2 Sets of measure zero and discontinuities of integrable func- A.2 Multiple integrals A.2.1 The Riemann integral in Rd Definitions A.2.2 Repeated integrals A.2.3 The change of variables formula A.2.4 Spherical coordinates A.3 Improper integrals. Integration over Rd A.3.1 Integration of functions of moderate decrease A.3.2 Repeated integrals A.3.3 Spherical coordinates Notes and References Bibliography Symbol Glossary Index This First Volume, A Three-part Introduction To The Subject, Is Intended For Students With A Beginning Knowledge Of Mathematical Analysis Who Are Motivated To Discover The Ideas That Shape Fourier Analysis. It Begins With The Simple Conviction That Fourier Arrived At In The Early Nineteenth Century When Studying Problems In The Physical Sciences--that An Arbitrary Function Can Be Written As An Infinite Sum Of The Most Basic Trigonometric Functions. The Genesis Of Fourier Analysis -- The Vibrating String -- Derivation Of The Wave Equation -- Solution To The Wave Equation -- Example: The Plucked String -- The Heat Equation -- Derivation Of The Heat Equation -- Steady-state Heat Equation In The Disc -- Exercises -- Problem -- Basic Properties Of Fourier Series -- Examples And Formulation Of The Problem -- Main Definitions And Some Examples -- Uniqueness Of Fourier Series -- Convulusions -- Good Kernels -- Cesaro And Abel Summability: Applications To Fourier Series -- Cesaro Means And Summation -- Fejer's Theorem -- Abel Means And Summation -- The Poisson Kernel And Dirichlet's Problem In The Unit Disc -- Exercises -- Problems -- Convergence Of Fourier Series -- Mean-square Convergence Of Fourier Series -- Vector Spaces And Inner Products -- Proof Of Mean-square Convergence -- Return To Pointwise Convergence -- A Local Result -- A Continuous Function With Diverging Fourier Series -- Exercises -- Problems --^ Some Applications Of Fourier Series -- The Isoperimetric Inequality -- Weyl's Equidistribution Theorem -- A Continuous But Nowhere Differentiable Function -- The Heat Equation On The Circle -- Exercises -- Problems -- The Fourier Transform On R -- Elementary Theory Of The Fourier Transform -- Integration Of Functions On The Real Line -- Definition Of The Fourier Transform -- The Schwartz Space -- The Fourier Transform On S -- The Fourier Inversion -- The Plancherel Formula -- Extension To Functions Of Moderate Decrease -- The Weierstrass Approximation Theorem -- Applications To Some Partial Differential Equations -- The Time-dependent Heat Equation On The Real Line -- The Steady-state Heat Equation In The Upper Half-plane -- The Poisson Summation Formula -- Theta And Zeta Functions -- Heat Kernels -- Poisson Kernels -- The Heisenberg Uncertainty Principle -- Exercises -- Problems -- The Fourier Transform On Rd -- Preliminaries -- Symmetries -- Integration On Rd --^ Elementary Theory Of The Fourier Transform -- The Wave Equation In Rd X R -- Solution In Terms Of Fourier Transforms -- The Wave Equation In R3 X R -- The Wave Equation In Ir2 X R: Descent -- Radial Symmetry And Bessel Functions -- The Radon Transform And Some Of Its Applications -- The X-ray Transform In R2 -- The Radon Transform In R3 -- A Note About Plane Waves -- Exercises -- Problems -- Finite Fourier Analysis -- Fourier Analysis On Z(n) -- The Group Z(n) -- Fourier Inversion Theorem And Plancherel Identity On Z(n) -- The Fast Fourier Transform -- Fourier Analysis On Finite Abelian Groups -- Abelian Groups -- Characters -- The Orthogonality Relations -- Characters As A Total Family -- Fourier Inversion And Plancherel Formula -- Exercises -- Problems -- Dirichlet's Theorem -- A Little Elementary Number Theory -- The Fundamental Theorem Of Arithmetic -- The Infinitude Of Primes -- Dirichlet's Theorem -- Fourier Analysis, Dirichlet Characters, And Reduction Of The Theorem --^ Dirichlet L-functions -- Proof Of The Theorem -- Logarithms -- L-functions -- Non-vanishing Of The L-function -- Exercises -- Problems -- Appendix: Integration -- Definition Of The Riemann Integral -- Basic Properties -- Sets Of Measure Zero And Discontinuities Of Integrable Functions -- Multiple Integrals -- The Riemann Integral In Rd -- Repeated Integrals -- The Change Of Variables Formula -- Spherical Coordinates -- Improper Integrals. Integration Over Rd -- Integration Of Functions Of Moderate Decrease -- Repeated Integrals -- Spherical Coordinates -- Notes And References -- Bibliography -- Symbol Glossary. Elias M. Stein & Rami Shakarchi. Includes Bibliographical References (p. [301]-303) And Index. This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences—that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.High quality vector PDF with bookmarks.