傅立叶分析导论（英文影印版）

.5.2 fejer's theorem
5.3 abel means and summation
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 ir
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 s
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 rd
1 preliminaries
1.1 symmetries
1.2 integration on rd
2 elementary theory of the fourier transform
3 the wave equation in rd x
3.1 solution in terms of fourier transforms
3.2 the wave equation in r3 x
3.3 the wave equation in r2 x 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 reduction 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 integrable functions
multiple integrals
2.1 the riemann integral in rd
2.2 repeated integrals
2.3 the change of variables formula
2.4 spherical coordinates
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