ON FOURIER COEFFICIENTS WITH RESPECT TO THE SYSTEM OF DAUBECHIES WAVELETS { ψ(2)jk } j,k ∈ Z
The work is devoted to investigation of behavior of Fourier coefficients of positive functions from rearrangement invariant spaces with respect to the system of Daubechies simplest wavelets with four coefficients in the mask. Nonnegativity of Fourier coefficients of such functions is investigated and the criterion of their infinitesimality in rearrangement invariant spaces satisfying the special condition of concavity is obtained. In addition, dependence of function norm on behavior of its Fourier coefficients and equivalence of the above mentioned system of Daubechies wavelets of zero scale with integer shifts to the standard basis of the space l p, 1 < p < то, are studied. These properties of the Fourier coefficients are considered for rearrangement invariant spaces defined on the interval [- 1,2], however, they are naturally transferred to the rearrangement invariant spaces defined on any finite interval of the number scale.