"Asymptotic relations between density of sampling and oscillation estimates: An exploration using low discrepancy point sets"Holighaus, NickiOscillation estimates are a central tool for deriving sampling results for integral transforms induced by continuous frames, most commonly in the context of coorbit spaces. In most prior work, oscillation estimates are merely used to prove the existence of a density of point evaluations that ensures the sampling property. However, the decay of the oscillation estimate in the number of point evaluations, which is of great interest for potential applications of such discretizations, has attracted considerably less attention. It is tempting to suggest that decay is slow and scales poorly in the dimension of the phase space (i.e., the "index set" of the continuous frame). We will present some work-in-progress results, providing explicit rates of decay for certain special cases by adapting results for Quasi-Monte Carlo integration to the setting of estimating the Schur-class norm of oscillations. In particular, we show that the estimates only depend weakly on the dimension, for point evaluations with respect to low discrepancy sets. |
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