On L_2-approximation in Hilbert spaces using function values
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We study L_2-approximation of functions from Hilbert spaces H in which function evaluation is a continuous linear functional, using function values as information. Under certain assumptions on H, we prove that the n-th minimal worst-case error e_n satisfies e_n ≲ a_n/(n), where a_n is the n-th minimal worst-case error for algorithms using arbitrary linear information, i.e., the n-th approximation number. Our result applies, in particular, to Sobolev spaces with dominating mixed smoothness H=H^s_ mix(T^d) with s>1/2 and we obtain e_n ≲ n^-s^sd(n). This improves upon previous bounds whenever d>2s+1.
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