This research project is concerned with the following three topics in approximation theory and Fourier analysis: 1) Simultaneous approximation of functions and their derivatives in Lp, 00. We expect to obtain sufficient conditions of the boundedness for such operators in terms of the simultaneous behavior of a multiplier and its derivatives in different functional spaces, and to apply such conditions for solving problems from this proposal. In our approaches we will combine the methods from approximation theory and Fourier analysis simultaneously, contrary to the previous research concerning the mentioned tasks. Moreover, by using and developing the newly introduced concepts of families of multiplier operators in Lp, 0