Fourier–Jacobi harmonic analysis and approximation of functions

We use the methods of Fourier–Jacobi harmonic analysis to study problems of the approximation of functions by algebraic polynomials in weighted function spaces on $[-1,1]$. We prove analogues of Jackson's direct theorem for the moduli of smoothness of all orders constructed on the basis of Jacobi generalized translations. The moduli of smoothness are shown to be equivalent to $K$-functionals constructed from Sobolev-type spaces. We define Nikol'skii–Besov spaces for the Jacobi generalized translation and describe them in terms of best approximations. We also prove analogues of some inverse theorems of Stechkin.