An equality underlying Hardy's inequality

Jameson, Graham (2022) An equality underlying Hardy's inequality. American Mathematical Monthly, 129 (6). pp. 582-586. ISSN 0002-9890

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Abstract

A classical inequality of G. H. Hardy states that Cx≤2x for x in l2, where C is the Cesàro (alias averaging) operator. This inequality has been strengthened to (C−I)x≤x. It has also been shown that CTx≤Cx for x in l2. We present equalities that imply these inequalities, together with the reverse inequalities (C−I)x≥(1/√2)x and Cx≤√2CTx. We also present companion results involving the shift operator.