Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis (AMS/MAA Textbooks)

Cover
Title page
Copyright
Contents
Introduction
Chapter 1. Overture
	1.1. Mathematical motivation: Series of functions
	1.2. Physical motivation: Acoustics
Part 1 Complex functions of a real variable
	Chapter 2. Real and complex numbers
		2.1. Axioms for the real numbers
		2.2. Complex numbers
		2.3. Metrics and metric spaces
		2.4. Sequences in C and other metric spaces
		2.5. Completeness in metric spaces
		2.6. The topology of metric spaces
	Chapter 3. Complex-valued calculus
		3.1. Continuity and limits
		3.2. Differentiation
		3.3. The Riemann integral: Definition
		3.4. The Riemann integral: Properties
		3.5. The Fundamental Theorem of Calculus
		3.6. Other results from calculus
	Chapter 4. Series of functions
		4.1. Infinite series
		4.2. Sequences and series of functions
		4.3. Uniform convergence
		4.4. Power series
		4.5. Exponential and trigonometric functions
		4.6. More about exponential functions
		4.7. The Schwartz space
		4.8. Integration on R
Part 2 Fourier series and Hilbert spaces
	Chapter 5. The idea of a function space
		5.1. Which clock keeps better time?
		5.2. Function spaces and metrics
		5.3. Dot products
	Chapter 6. Fourier series
		6.1. Fourier polynomials
		6.2. Fourier series
		6.3. Real Fourier series
		6.4. Convergence of Fourier series* of differentiable functions
	Chapter 7. Hilbert spaces
		7.1. Inner product spaces
		7.2. Normed spaces
		7.3. Orthogonal sets and bases
		7.4. The Lebesgue integral: Measure zero
		7.5. The Lebesgue integral: Axioms
		7.6. Hilbert spaces
	Chapter 8. Convergence of Fourier series
		8.1. Fourier series in ?²(?¹)
		8.2. Convolutions
		8.3. Dirac kernels
		8.4. Proof of the Inversion Theorem
		8.5. Applications of Fourier series
Part 3 Operators and differential equations
	Chapter 9. PDEs and diagonalization
		9.1. Some PDEs from classical physics
		9.2. Schrödinger’s equation
		9.3. Diagonalization
	Chapter 10. Operators on Hilbert spaces
		10.1. Operators on Hilbert spaces
		10.2. Hermitian and positive operators
		10.3. Eigenvectors and eigenvalues
		10.4. Eigenbases
	Chapter 11. Eigenbases and differential equations
		11.1. The heat equation on the circle
		11.2. The eigenbasis method
		11.3. The wave equation on the circle
		11.4. Boundary value problems
		11.5. Legendre polynomials
		11.6. Hermite functions
		11.7. The quantum harmonic oscillator
		11.8. Sturm-Liouville theory
Part 4 The Fourier transform and beyond
	Chapter 12. The Fourier transform
		12.1. The big picture
		12.2. Convolutions, Dirac kernels, and calculus on R
		12.3. The Fourier transform on schwartz
		12.4. Inversion and the Plancherel theorem
		12.5. The ?² Fourier transform
	Chapter 13. Applications of the Fourier transform
		13.1. A table of Fourier transforms
		13.2. Linear differential equations with constant coefficients
		13.3. The heat and wave equations on R
		13.4. An eigenbasis for the Fourier transform
		13.5. Continuous-valued quantum observables
		13.6. Poisson summation and theta functions
		13.7. Miscellaneous applications of the Fourier transform
	Chapter 14. What’s next?
		14.1. What’s next: More analysis
		14.2. What’s next: Signal processing and distributions
		14.3. What’s next: Wavelets
		14.4. What’s next: Quantum mechanics
		14.5. What’s next: Spectra and number theory
		14.6. What’s next: Harmonic analysis on groups
Appendices
	Appendix A. Rearrangements of series
	Appendix B. Linear algebra
	Appendix C. Bump functions
	Appendix D. Suggestions for problems
	Bibliography
	Index
	Index
	Back Cover