  Paley's theorem about lacunary coefficients of functions in the classical
space $H^1$ on the unit circle is equivalent to the statement that certain
Hankel matrices define bounded operators on $\ell^2$ of the nonnegative
integers. Since that statement reduces easily to the case where the entries in
the matrix are all nonnegative, it must be provable by the Schur test. We give
such proofs with interesting patterns in the vectors used in the test, and we
recover the best constant in the main case. We use related ideas to reprove the
characterization of Paley multipliers from $H^1$ to $H^2$.
