On the power of function values for the approximation problem in various settings
The paper provides a unified framework and new estimates for understanding the relative power of function values in approximation theory, but is primarily a survey with incremental contributions.
This expository paper defines and estimates the power function—the ratio of convergence rates using function values versus arbitrary linear functionals—for function approximation in Hilbert and Banach spaces under worst-case, average-case, and randomized settings, and presents eight open problems.
This is an expository paper on approximating functions from general Hilbert or Banach spaces in the worst case, average case and randomized settings with error measured in the $L_p$ sense. We define the power function as the ratio between the best rate of convergence of algorithms that use function values over the best rate of convergence of algorithms that use arbitrary linear functionals for a worst possible Hilbert or Banach space for which the problem of approximating functions is well defined. Obviously, the power function takes values at most one. If these values are one or close to one than the power of function values is the same or almost the same as the power of arbitrary linear functionals. We summarize and supply a few new estimates on the power function. We also indicate eight open problems related to the power function since this function has not yet been studied for many cases. We believe that the open problems will be of interest to a general audience of mathematicians.