Means and families of operators generated by Bochner-Riesz kernels are studied. Some sharp results on their convergence are achieved. The equivalence of the approximation errors of these methods to smoothness quantities related to the Laplacian is proved.Keywords Bochner-Riesz kernels and means · Families of operators · Necessary and sufficient conditions of convergence · K-functional · Realizations and moduli of smoothness related to the Laplacian Mathematics Subject Classification (2000) 42A10 · 42A15

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It is shown that best approximation by trigonometric polynomials is achieved in average by families of linear polynomial operators in the L p-metric for all p, 0 < p ≤ ∞. This is compared with approximation by Fourier means and interpolation means which is restricted to 1 ≤ p ≤ ∞ and p = ∞, respectively.