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Mostrando las entradas de abril, 2019

Problem set 11, Harmonic Analysis

Due May 3 Problem 1 Let $\gamma$ be the lower semicircle of radius $N$ around the origin, and $\xi > 0.$ Then $\displaystyle \int_\gamma f(z) dz = 2\pi i\text{Res}_{z=-it} f(z) = i e^{-2\pi t\xi},$ where $f(z)$ is the function defined by $f(z) = \dfrac{1}{\pi} \dfrac{z}{z^2+t^2} e^{-2\pi iz\xi}.$ Problem 2 If $f\in L^1(\mathbb R)$ and diferentiable at $x\in\mathbb R$, then the limit $\displaystyle \lim_{t\to 0} \int_{|y|\ge t} \frac{f(x-y)}{y} dy$ exists. ( Hint:  Use the identity, for any $\delta_n>0$, $\displaystyle \int_{t\le|y|<\delta_n} \frac{f(x-y)}{y} dy = \int_{t\le|y|<\delta_n} \frac{f(x-y) - f(x)}{y} dy + \int_{t\le|y|<\delta_n} \frac{f(x)}{y} dy,$ and take $\delta_n\to 0$.) Problem 3 If $f_n, g_n$ are sequences in $C_c^\infty(\mathbb R)$ that converge in $L^1(\mathbb R)$ to $f$, then $Hf_n$ and $Hg_n$ converge in measure t

Harmonic Analysis: Week 5 notes

Notes on pointwise and uniform convergence of Fourier series of continuous functions:  Chapter05.pdf

Harmonic Analysis: Weeks 9 and 10 notes

Notes on the Hardy-Littlewood maximal functions and boundary limits of harmonic functions:  Chapter09.pdf Notes on the Fourier transform, the inversion formula and square means convergence:  Chapter10.pdf