Analogue of Men'shov's theorem ''on correction'' for discrete orthonormal systems (Q1812795)

Consider a unit vector in \(R^ n\) (\(n\) possibly large), with its components in the standard basis in \(R^ n\), and suppose that another orthonormal basis is given. The paper deals with the question of the possibility of changing a relatively small number of components of the vector in order that the ``partial sums'' of the vector with respect to the given basis be bounded in \(\ell^ n_ \infty\)-norm. The bound should be independent of the vector and the basis considered and scaled by \(n^{-1/2}\). Using non-trivial tools, the author shows that this is always possible, with the relative number of components on which the change is made not exceeding a prescribed arbitrarily small number. The result is an analogue of Men'shov's theorem on Fourier series which says that any continuous function on \([0,2\pi]\) can be changed on a set of small measure in order that the Fourier series of the new function be uniformly convergent.