Does order matter (Q2772770)

The author reviews the state of knowledge on the rearrangement-invariance of convergence of Fourier series, and more generally, of orthogonal basis expansions. Work of Kolmogorov, Ul'yanov, and Olevskij show that for any complete orthonormal system (or, more generally, a Riesz basis) in \(L^2([0,1]),\) there exist \(L^2\) series which after suitable rearrangement diverge almost everywhere. Indeed, one even has divergence for the Haar system. The author surveys this result and extensions to more general spaces, as well as a refinement of Kahane and Katznelson in which one can exactly specify the measure zero set of convergence. The author also mentions the convergence problem for hard thresholding or hard summation, in which only the coefficients larger than a certain threshold (which is eventually sent to zero) are retained. For Fourier series, it is a result of the author that even this method can diverge a.e. for certain \(L^2\) functions; on the other hand for wavelet systems such as the Haar basis, a result of Tao shows that one still has convergence a.e.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00019].