A sharp upper bound for sampling numbers in L2 - Dolbeault, Matthieu; Krieg, David; Ullrich, Mario (Applied and computational harmonic analysis, 2023)
For a class F of complex-valued functions on a set D, we denote by gn(F) its sampling numbers, i.e., the minimal worst-case error on F, measured in L2, that can be achieved with a recovery algorithm based on n function evaluations. We prove that there is a universal constant c∈N such that, if F is the unit ball of a separable reproducing kernel Hilbert space, thengcn(F)2≤1n∑k≥ndk(F)2, where dk(F) are the Kolmogorov widths (or approximation numbers) of F in L2. We also obtain similar upper bounds for more general classes F, including all compact subsets of the space of continuous functions on a bounded domain D⊂Rd, and show that these bounds are sharp by providing examples where the converse inequality holds up to a constant. The results rely on the solution to the Kadison-Singer problem, which we extend to the subsampling of a sum of infinite rank-one matrices.