Universality properties of Walsh–Fourier series

On the universal and strong $(L^{1},L^{\infty})$ -property related to Fourier–Walsh series

“…For every measurable, almost everywhere finite function f on [0, 2π] and every > 0, there is a continuous function In 1988 we were able to show that the trigonometric system possesses the L-strong property for integrable functions; that is, for each > 0 there exists a (measurable) set E ⊂ [0, 2π] of measure |E| > 2π − such that for each function f ∈ L 1 [0, 2π] there exists a function g ∈ L 1 [0, 2π] equal to f (x) on E and with Fourier series with respect to the trigonometric system convergent to g(x) in the L 1 [0, 2π]-norm (see [5]). After Menchoff's proof of the C-strong property, many "correction" type theorems were proved for different systems (see [1], [2], [8], [9], [15], [17], [18], [20]; we refrain from providing a complete survey of all research done in this area). A number of papers (see [7], [5], [16]) have been devoted to correction theorems in which the absolute values of nonzero Fourier coefficients (by the Haar and Walsh systems) of the corrected function are monotonically decreasing.…”