Sharp Jackson inequalities

Abstract

For trigonometric polynomials on [ - π , π ] ≡ T , the classical Jackson inequality E n ( f ) p ⩽ C ω r ( f , 1 / n ) p was sharpened by M. Timan for 1 < p < ∞ to yield n - r ∑ k = 1 n k sr - 1 E k ( f ) p s 1 / s ⩽ C ω 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 R d are among the systems for which the present treatment yields sharp Jackson inequalities. Analogous sharper versions of the inequality ω r + 1 ( f , t ) p ≤ C ω r ( f , t ) p are also achieved.