Lebesgue constants for Hadamard matrices (Q1762624)

The behaviour of the Lebesgue constant \(L(E)=n^{-1} \sup_{1\leqslant i\leqslant m} \sum^n_{j=1} | \sum ^m_{k=1} e_{ki} e_{kj}| \) of Hadamard matrices \(E=E(e_{ij})\) or their submatrices is studied. Let \(E_m\) denote the first \(m\)~rows of a Hadamard matrix~\(E\). It is shown that \(L(E_m)\) is bounded above by \(c_1 \log m\) for a reasonable wide class of recursively generated Hadamard matrices~\(E\). On the other hand, there are constructed Hadamard matrices \(E\) for which \(L(E_m)\) grows at least as fast as \(c_2\sqrt m\). The authors also give heuristic arguments suggesting that the \(\sqrt m\)~growth of~\(L(E_m)\) is typical rather than exception.