Universal functions in `correction' problems guaranteeing the convergence of Fourier–Walsh series

Abstract

We prove the existence of a function with monotone decreasing Fourier–Walsh coefficients which is universal in , , in the sense of modification with respect to the signs of the Fourier coefficients for the Walsh system. In other words, for every function and every one can find a function such that the measure is greater than , the Fourier series of in the Walsh system converges to in the -norm and , . We also prove that for every , , one can find a measurable set of measure and a function with , , such that for every function there is a function with the following properties: coincides with on , the Fourier–Walsh series of converges to in the norm of and the absolute values of all terms in the sequence of the Fourier–Walsh coefficients of the newly obtained function satisfy , .