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## Hot answers tagged fourier-analysis

5

A more complete list of particular self-reciprocal Fourier functions, i.e. eigenfunctions of the cosine Fourier transform: $1.$ $\displaystyle \frac{1}{\sqrt{x}}$ $2.$ $\displaystyle e^{-x^2/2}$ (and more generally $e^{-x^2/2}H_{2n}(x)$, where $H_n$ is Hermite polynomial) $3.$ $\displaystyle\frac{1}{\cosh\sqrt{\frac{\pi}{2}}x}$ $4.$ $\displaystyle ... 3 Fourier dimension doesn't directly say anything about lower bounds for the mass of balls. Fourier dimension is smaller or equal than Hausdorff dimension, and in order to give a bound of the form$\mu(B(x,r))\ge r^s$, the exponent$s$needs to be large. This suggests that one should seek a notion of dimension that gives larger values than Hausdorff ... 3 This is closely related to the so-called metric of strict convergence which is $$d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)|$$ where$TV(u)$denoted the total variation of$u$. This is indeed a metric on the space$BV(\Omega)$(also for$\Omega\subset\mathbb{R}^n$). Hence, strict convergence of$u_n$to$u$is nothing else than saying$\$ u_n\to ...

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