Functions almost universal in the sense of signs with respect to the trigonometric system and the Walsh system (Q6576789)

This paper is a continuation of the author's recently published investigations.\N\NIt is established that there exists an integrable function \(U \in L^1[0, 1]\) with Fourier series converging with respect to the \(L^1[0, 1]\)-norm with monotone decreasing Fourier-Walsh coefficients that is universal in the sense of signs with respect to the Walsh system for the class \(M[0, 1]\) in the case of convergence with respect to the measure and almost universal in the sense of signs with respect to the Walsh system for the class \(L^0[0, 1]\) in the case of convergence almost everywhere.\N\NIt is also stated that any measurable almost everywhere finite function, by changing its values on a set of arbitrarily small measure, can be turned into an almost universal function in the sense of signs with respect to the trigonometric system, as well as with respect to the Walsh system for the class \(M[0, 1]\).