foreword
preface
chapter 1. the genesis of fourier analysis
1 the vibrating string
1.1 derivation of the wave equation
1.2 solution to the wave equation
1.3 example： the plucked string
2 the heat equation
2.1 derivation of the heat equation
2.2 steady-state heat equation in the disc
3 exercises
4 problem
chapter 2. basic properties of fourier series
1 examples and formulation of the problem
1.1 main definitions and some examples
2 uniqueness of fourier series
3 convolutions
4 good kernels
5 cesaro and abel summability： applications to fourier
series
.5.1 cesaro means and snmmation
5.2 fejer''s theorem
5.3 abel means and s-ruination
5.4 the poisson kernel and dirichlet''s problem in the unit
disc
6 exercises
7 problems
chapter 3. convergence of fourier series
1 mean-square convergence of fourier series
1.1 vector spaces and inner products
1.2 proof of mean-square convergence
2 return to pointwise convergence
2.1 a local result
2.2 a continuous function with diverging fourier
series
3 exercises
4 problems
chapter 4. some applications of fourier series
1 the isoperimetric inequality
2 weyl''s equidistribution theorem
3 a continuous but nowhere differentiable function
4 the heat equation on the circle
5 exercises
6 problems
chapter 5. the fourier transform on r
1 elementary theory of the fourier transform
1.1 integration of functions on the real line
1.2 definition of the fourier transform
1.3 the schwartz space
1.4 the fourier transform on 3
1.5 the fourier inversion
1.6 the plancherel formula
1.7 extension to functions of moderate decrease
1.8 the weierstrass approximation theorem
2 applications to some partial differential equations
2.1 the time-dependent heat equation on the real line
2.2 the steady-state heat equation in the upper
half-plane
3 the poisson summation formula
3.1 theta and zeta functions
3.2 heat kernels
3.3 poisson kernels
4 the heisenberg uncertainty principle
5 exercises
6 problems
chapter 6. the fourier transform on ra
1 preliminaries
1.1 symmetries
1.2 integration on ra
2 elementary theory of the fourier transform
3 the wave equation in rd ×r
3.1 solution in terms of fourier transforms
3.2 the wave equation in r3× r
3.3 the wave equation in r2 × r： descent
4 radial symmetry and bessel functions
5 the radon transform and some of its applications
5.1 the x-ray transform in r2
5.2 the radon transform in r3
5.3 a note about plane waves
6 exercises
7 problems
chapter 7. finite fourier analysis
1 fourier analysis on z（n）
1.1 the group z（n）
1.2 fourier inversion theorem and plancherel identity on
z（n）
1.3 the fast fourier transform
2 fourier analysis on finite abelian groups
2.1 abelian groups
2.2 characters
2.3 the orthogonality relations
2.4 characters as a total family
2.5 fourier inversion and plancherel formula
3 exercises
4 problems
chapter 8. dirichlet''s theorem
1 a little elementary number theory
1.1 the fundamental theorem of arithmetic
1.2 the infinitude of primes
2 dirichlet''s theorem
2.1 fourier analysis， dirichlet characters， and reduc-tion of
the theorem
2.2 dirichlet l-functions
3 proof of the theorem
3.1 logarithms
3.2 l-functions
3.3 non-vanishing of the l-function
4 exercises
5 problems
appendix： integration
1 definition of the riemann integral
1.1 basic properties
1.2 sets of measure zero and discontinuities of inte-grable
functions
2 multiple integrals
2.1 the riemann integral in rd
2.2 repeated integrals
2.3 the change of variables formula
2.4 spherical coordinates
3 improper integrals. integration over rd
3.1 integration of functions of moderate decrease
3.2 repeated integrals
3.3 spherical coordinates
notes and references
bibliography
symbol glossary
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