Let $f$ be an $L^1$-function supported in $[-b, b]$, where $0 < b \le b_0 < 1$. Let $Q_n$ be the normalised Legendre polynomials and let $a_n = \int f(x) Q_n(x) dx$ be the Fourier-Legendre coefficients of $f$. Let $s > 0$ and denote the $H^s$ norm of $f$ by $\|f\|_s$, that is $\|f\|_s^2 = \int |\hat f(\xi)|^2 (1 + |\xi|^{2})^s d\xi$. I would like to know if there are constants $C_s$ (that may depend on $b_0$) such that \begin{equation*} \frac 1{C_s} \|f\|_s \le \Big(\sum_{n=0}^{\infty}(1 + n^{2s}) |a_n|^2\Big)^{1/2} \le C_s \|f\|_s . \end{equation*} I am particularly interested in the case $s = 1/2$.

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    $\begingroup$ Your definition of $H^s$ works on the whole line, but not on an interval. Moreover, your condition on the coefficients is equivalent to $f$ being in the domain of a power of the Legendre differential operator. These domains are not the same as Sobolev spaces. $\endgroup$ – Michael Renardy Feb 4 '15 at 15:10

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