Abstract:
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.
Keywords:Fourier–Jacobi harmonic analysis, approximation of functions, generalized
translations, Jacobi polynomials, function spaces.
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