q-convolution and its q-Fourier transform

The functions on a lattice generated by the integer degrees of $q^2$ are considered, 0<q<1. The $q^2$-translation operator is defined. The multiplicators and the $q^2$-convolutors are defined in the functional spaces which are dual with respect to the $q^2$-Fourier transform. The $q^2$-analog of convolution of two $q^2$-distributions is constructed. The $q^2$-analog of an arbitrary (non integer) order derivative is introduced