An orthonormal basis for $C[0,1]$ that is not an unconditional basis for $L^ p[0,1],\;1<p\not = 2$

Abstract:

In a recent article, Kazaryan has employed an orthonormal system constructed by Olevskii in order to obtain a negative answer to the following question posed by Ulyanov: Is an orthonormal basis for $C\left [ {0,1} \right ]$ necessarily an unconditional basis for each space ${L^p}\left [ {0,1} \right ],1 < p < \infty$? The elements of the Olevskii system, however, are finite linear combinations of Haar functions, and thus, most of them are not continuous on $\left [ {0,1} \right ]$. For this reason, the example is mildly unsatisfying, since one generally requires the members of a Schauder basis for a given space to belong to that space. In the present work, the author shows that this minute defect can be removed if one modifies the Olevskii system by replacing the Haar functions involved therein with corresponding members of the Franklin system.