Chapter 28: Fourier analysis

281
The Stone-Weierstrass theorem

Approximating a function on a compact set by members of a given
lattice or algebra of functions; real and complex cases;
approximation by polynomials and trigonometric functions; Weyl's
Equidistribution Theorem in [0,1]^{r}.

282
Fourier series

Fourier and Fejér sums; Dirichlet and Fejér kernels;
Riemann-Lebesgue lemma; uniform convergence of Fejér sums of a
continuous function; a.e. and ∥ ∥_{1}-convergence of Fejér
sums of an integrable function; ∥ ∥_{2}-convergence of Fourier
sums of a square-integrable function; convergence of Fourier sums of
a differentiable or b.v. function; convolutions and Fourier
coefficients.

283
Fourier transforms I

Fourier and inverse Fourier transforms; elementary properties;
∫_{0}^{∞}(sin*x*)/*x dx*
=π/2; the formula
*f* ^{∧∨}=*f* for differentiable and b.v. *f*;
convolutions; exp(-*x*^{2}/2);
∫*f* ×*g*^{∧}=∫*f* ^{∧}×*g*.

284
Fourier transforms II

Test functions; *h*^{∧∨}=*h*; tempered
functions; tempered functions which represent each other's
transforms; convolutions; square-integrable functions; Dirac's
delta function.

285
Characteristic functions

The characteristic function of a distribution; independent r.vs;
the normal distribution; the vague topology on the space of
distributions, and sequential convergence of characteristic
functions; Poisson's theorem; convolutions of distributions.

*286
Carleson's theorem

The Hardy-Littlewood Maximal Theorem; the Lacey-Thiele proof of
Carleson's theorem for square-integrable functions on **R** and
]-π,π].

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Revised 8.4.2016