Abstract. We prove that for any $0<\alpha < 2$, there exists a uniformly bounded complete orthonormal system $\Phi_{\alpha }=\{\phi_{n}\} _{n=1}^\infty $ of functions on $[-1, 1]$ and a continuous function $f$, such that the orthogonal expansion of any integrable function $g$ with $|\{x : g(x)=f(x)\} |>\alpha $ by the system $\Phi_{\alpha }$ does not converge in the $L^p_{[-1, 1]}$ metric for any $p>2$. We also prove that the analogous result is true for Köthe spaces with a certain property (E). Particularly this result holds for Orlicz spaces arbitrarily near to the space $L^2$ but with norm stronger than the $L^2$ norm.

AMS Subject Classification (1991): 42C15

Keyword(s): C-strong property, Köthe spaces

Received December 30, 1992 and in revised form February 24, 1993. (Registered under 5547/2009.)