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 FourierLegendre 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 $s = 1/2$.
Asked
Viewed
117 times
1

2$\begingroup$ 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