Optimal nonlinear sampling recovery of mixed-smoothness classes H^r_q is at least c m^{-r+1/q-1/p} (log m)^{(d-1)/p}, a logarithmic factor not captured by previous lower-bound techniques.
On optimal recovery in $L_2$
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abstract
We prove that the optimal error of recovery in the $L_2$ norm of functions from a class $\bF$ can be bounded above by the value of the Kolmogorov width of $\bF$ in the uniform norm. We demonstrate on a number of examples of $\bF$ from classes of functions with mixed smoothness that the obtained inequality provides a powerful tool for estimating errors of optimal recovery.
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Some lower bounds for optimal sampling recovery of functions with mixed smoothness
Optimal nonlinear sampling recovery of mixed-smoothness classes H^r_q is at least c m^{-r+1/q-1/p} (log m)^{(d-1)/p}, a logarithmic factor not captured by previous lower-bound techniques.