Chapter 28: Fourier analysis
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.
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.
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.
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.
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.
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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