Reconstruction of the Potential of the Sturm–Liouville Operator from a Finite Set of Eigenvalues and Normalizing Constants

It is well known that the potential $q$ of the Sturm–Liouville operator
$$ Ly=-y''+q(x)y $$
on the finite interval $[0,\pi]$ can be uniquely reconstructed from the spectrum $\{\lambda_k\}_1^\infty$ and the normalizing numbers $\{\alpha_k\}_1^\infty$ of the operator $L_D$ with the Dirichlet conditions. For an arbitrary real-valued potential $q$ lying in the Sobolev space $W^\theta_2[0,\pi]$, $\theta>-1$, we construct a function $q_N$ providing a $2N$-approximation to the potential on the basis of the finite spectral data set $\{\lambda_k\}_1^N\cup\{\alpha_k\}_1^N$. The main result is that, for arbitrary $\tau$ in the interval $-1\le\tau <\theta$, the estimate
$$ \|q-q_N\|_\tau \le CN^{\tau-\theta} $$
is true, where $\|\cdot\|_\tau$ is the norm on the Sobolev space $W^\tau_2$. The constant $C$ depends solely on $\|q\|_\theta$.