Mean convergence of orthogonal Fourier series of modified functions

Abstract:

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the $L^{p}$-metric for any $p > 2.$ We prove also that if $\Phi$ is a uniformly bounded ONS which is complete in all the spaces $L _ {[0,1]} ^{p} , 1 \leq p < \infty$, then there exists a rearrangement $\sigma$ of the natural numbers $\mathbf {N}$ such that the system $\Phi _{\sigma }= \{ \phi _{\sigma (n)} \}_{n=1}^{\infty }$ has the strong $L^{p}$-property for all $p>2$; that is, for every $2 \leq p < \infty$ and for every $f \in L _ {[0,1]} ^{p}$ and $\epsilon > 0$ there exists a function $f_ \epsilon \in L _ {[0,1]} ^{p}$ which coincides with $f$ except on a set of measure less than $\epsilon$ and whose Fourier series with respect to the system $\Phi _{\sigma }$ converges in $L _ {[0,1]} ^{p} .$