ABSTRACT
In most articles on the subject it is not the transformation Tρ : f(x) = f(ρx) given in the last section but the transformation Sθf(x) that is used. Sθf(x) is given by
Sθf(x) = 1
m(θ)
f(y)dγ(y), Sθ1 = 1 (3.1)
where dγ(y) is the measure on the set {y : x·y = cos θ} induced by the Lebesgue measure . The smoothness is described by Sθf(x) − f(x) or combinations or iterations of it. It should be noted that because of its symmetry Sθf −f corresponds to the second modulus (not the first). It is not possible to mention all papers dealing with expressions using Sθf for describing smoothness and proving its relation to other concepts. (I believe there are at least three dozen.) I will highlight several of the articles and some of the related concepts. The description originated from works of Pawelke and Weherens students of Butzer (see [23] and [29]) and continued in many works of Lizorkin, Nikolskii, Teherin, Rustamov (see [20], [22], [25], [26] and [27]) and other Russian mathematicians. In Chapter 5 of [28] a flaw in Rustamov’s treatment is fixed. Some recent advances were made in [2], [7] and [13]. I will try to itemize the main different possibilities.
