A Note on Varying Cardinality in the Average Case Setting

We study how much information with varying cardinality can be better than information with fixed cardinality for approximating linear operators in the average case setting with Gaussian measure. It has been known that adaptive choice of functionals forming information is not better than nonadaptive, and that the only gain may be obtained by using varying cardinality. We prove that the lower bounds from Traub (J. F. Traub, G. W. Wasilkowski, and H. Wo?niakowski, "Information-Based Complexity," Academic Press, San Diego, 1988) et al. on the efficiency of varying cardinality are sharp. In particular, we show that information whose cardinality assumes at most two different values can significantly help in approximating any linear operator with infinite dimensional domain space.