On the universal functions

In this article a function U∈L1([−π,π]) and a set E⊂[−π,π] with 0<μE<2π are constructed, such that for each function f∈L1(E) one can find a sequence {δk=±1}k=1∞ and a function g∈L1([−π,π]) which coincides with f(x) on E and satisfies ck(g)=δkck(U)(k=0,±1,±2,…), where {ck(g)}k∈Z and {ck(U)}k∈Z are the Fourier coefficients of functions g and U, respectively.