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Rate of approximation of combinations of averages on the spheres is shown to be equivalent to K-functionals yielding higher degree of smoothness. Results relating combinations of averages on rims of caps of spheres are also achieved.

For trigonometric polynomials on [−, ] ≡ T , the classical Jackson inequality E n (f) p C r (f, 1/n) p was sharpened by M. Timan for 1 < p < ∞ to yield n −r n k=1 k sr−1 E k (f) s p 1/s C r (f, n −1) p where s = max(p, 2). In this paper a general result on the relations between systems or sequences of best approximation and appropriate measures of… (More)