Fourier–Jacobi harmonic analysis and some problems of approximation of functions on the half-axis in L2 metric: Jackson's type direct theorems

We use the methods of Fourier–Jacobi harmonic analysis to study problems of the approximation of functions in weighted L2 function spaces on the half-axis [0,+∞). We prove analogues of Jackson's direct theorem for the moduli of smoothness of all orders constructed on the basis of Jacobi generalized translations. As a tool for approximation, we use functions with bounded spectrum.