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Paley's theorem about lacunary coefficients of functions in the classicalspace $H^1$ on the unit circle is equivalent to the statement that certainHankel matrices define bounded operators on $\ell^2$ of the nonnegativeintegers. Since that statement reduces easily to the case where the entries inthe matrix are all nonnegative, it must be provable by the Schur test. We givesuch proofs with interesting patterns in the vectors used in the test, and werecover the best constant in the main case. We use related ideas to reprove thecharacterization of Paley multipliers from $H^1$ to $H^2$.

Paley's theorem for Hankel matrices via the Schur test