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∞(sinx)/x dx
=π/2; the formula
f ∧∨=f for differentiable and b.v. f;
convolutions; exp(-x2/2);
∫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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