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Description of Bessel potential spaces

Hi, let $1 < p <\infty $, $ 0 < \alpha < 1$, and $ \mathscr{L}^p_\alpha(R^n) $ be the usual Bessel potential space defined by $$ \mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n). ...
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Sum of two random variables following K0 (modified 2nd kind Bessel) distributions

Hello, If X and Y follow independently a density distribution represented by the function $\tfrac{1}{\pi} K_0\left(\tfrac{|x|}{a^2}\right)$ (a modified Bessel function of the second kind), then the ...
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Bessel identities

Please help me prove the following identity $$ a(J_1(a)Y_0(a)-J_0(a)Y_1(a))=\frac{2}{\pi} $$ for any $a$. $J$ and $Y$ are bessel functions of the first and second kind respectively. Thank you.
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Bessel Potential Space inequality

The Bessel Potential Space is defined for $s\in\mathbb{R}$ as, $H^s(\mathbb{R}^d) = \{f\in L_2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L_2(\mathbb{R}^n)\}. $ This defines a Hilbert space ...