The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli
of smoothness. The treatment applies to a wide class of Banach spaces of functions.
We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk D of a sufficiently smooth function f in a neighbourhood of D converge uniformly to f on D. Moreover, when f∊C∞(D), all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of f.
Relations between ωr(f,t)B and ωr+1(f,t)B of the sharp Marchaud and sharp lower estimate-type are shown to be satisfied for some Banach spaces of functions that are not rearrangement invariant. Corresponding results relating the rate of best approximation with ωr(f,t)B for those spaces are also given.
For a Banach space B of functions which satisfies for some m>0
a significant improvement for lower estimates of the moduli of smoothness ωr(f,t)B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ℝd or for which translations are isometries or on Sd−1 for which rotations are isometries. Results for C0 semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An Lp space with 1<p<∞ satisfies () where s=max (p,2), and many Orlicz spaces are shown to satisfy () with appropriate s.