Function Values Are Enough for L2 -Approximation

Abstract

We study the L2-approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number en is the minimal worst-case error that can be achieved with n function values, whereas the approximation number an is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that en≲1kn∑j≥knaj2,where kn≍ n/ log (n). This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces Hmixs(Td) with dominating mixed smoothness s> 1 / 2 and dimension d∈ N, and we obtain en≲n-slogsd(n).For d> 2 s+ 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.