Sharp Jackson inequalities

For trigonometric polynomials on [− , ] ≡ T , the classical Jackson inequalityEn(f )p C r (f, 1/n)p was sharpened by M. Timan for 1<p<∞ to yield n−r { n ∑ k=1 ksr−1Ek(f )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 smoothness is given. Approximation by algebraic polynomials on [−1, 1], by spherical harmonic polynomials on the unit sphere, and by functions of exponential type on Rd are among the systems for which the present treatment yields sharp Jackson inequalities. Analogous sharper versions of the inequality r+1(f, t)p C r (f, t)p are also achieved. © 2007 Elsevier Inc. All rights reserved. MSC: 41A17; 41A10; 41A63