About twenty years ago the measure of smoothness ω r ϕ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time.
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 Lp spaces on T d , R d and S d−1 sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on T d , R d and S d−1) for which M (u 1/q) is convex for some q, 1 < q ≤ 2, where M (u) is the Orlicz function. Sharp converse inequalities for such spaces are deduced.
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)
The rate of convergence of Poisson sums and their combinations are shown to be equivalent to appropriate K-functionals. For a function f that has the expansion (1) f (x) ∼ ∞ k=0 P k (f) the Poisson sum is given by (2) A r f = ∞ k=0 r k P k (f), 0 < r < 1. For A r f to be written as a semi-group, we set r = e −t and obtain (3) T (t)f = ∞ k=0 e −kt P k (f).… (More)