Best Approximation of Functions of Generalized Zygmund Class by Matrix-euler Summability Means of Fourier Series (communicated by Hüsein Bor)

The degree of approximation of a function f belonging to Lipschitz class by the Cesàro mean and f ∈ Hα by the Fejér means has been studied by Alexits [4] and Prössdorf [7] respectively. But till now no work seems to have been done to obtain best approximation of functions belonging to generalized Zygmund class, Z (w) r , (r ≥ 1) by product summability means of the form (∆.E1). Z (w) r class is a generalization of Zα, Zα,r, Z (w) classes. The Matrix-Euler (∆.E1) summability means includes (N, pn).E1, (N, pn, qn).E1 and (C, 1).E1 means as particular cases. In attempt to make an advance study in this direction, in this paper, best approximations of the function belonging to generalized Zygmund class Z (w) r , (r ≥ 1) have been obtained.