We consider a continuous version of the classical notion of Banach limits, i.e., normalized positive linear functionals on L∞(R+) invariant under translations f(x)↦f(x+s) of L∞(R+) for every s≥0. We give one of its characterizations in terms of the invariance under the operation of a certain linear transformation on L∞(R+). We also deal with invariant linear functionals under dilations f(x)↦f(rx), r≥1 and give a similar characterization via the Hardy operator. Applications to summability methods are presented in the last section.
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