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I wanted to prove that the Bessel function type 1 and 2 are a member of Sobolev space of $H^\infty (\Omega)$ where $\Omega=[-1,1]$. It is needed to prove that $(\sum\limits_{\alpha\leq m}\int_\Omega(D^\beta u)^2)^{1/2}<\infty$. It is obvious that they are members of $C^\infty (\Omega)$. I use the series expansion of the Bessel function but I got stuck when $\alpha$ is large (tends to infinity) and for higher order derivatives. Because when $\alpha$ is large then in $Y_\alpha$ the term $J_{-\alpha}$ will have $2^\alpha$ in the nominator which tends to infinity for small $m$. Also I can not consider the $\Gamma(m-\alpha+1)$ term since it is not obvious that what value it has when $m$ is small (it takes negative values in $\Gamma$ function). The Bessel functions are as follows:

$$J_\alpha(x) = \sum\limits_{m=0}^\infty\frac{(-1)^m}{m!\Gamma(m+\alpha+1)}(\frac{x}{2})^{2m+\alpha}$$ and $$Y_\alpha=\frac{J_\alpha(x)cos(\alpha\pi)-J_{-\alpha}(x)}{sin(\alpha\pi)}$$ I am so confused and I almost got to the point of doubt that they are members of $H^\infty(\Omega)$. Can anyone help me or show me another way?

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    It is probably simpler to consider that the Fourier transform of $J_\nu(x)$ is a nice function and apply some variation of the Paley-Wiener theorem (https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem), or directly work with Bessel's differential equation. – 2017-02-10
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    @JackD'Aurizio I searched it and it was very helpful. Would you please place your comment as an answer with a little more details? I don't want this question remains unanswered! Thank you. – 2017-02-10

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