Spaces of generalised smoothness in summability problems for Phi-means of spectral decomposition

Let GsubsetBbb R^n be an arbitrary domain. The authors introduce a non-negative function phi on Bbb R with some special properties on (0,1] and then with the help of its Riemann-Liouville fractional integral of order s (s>0) define the notion of Phi-means of spectral decomposition of the space L_2(G) with respect to a self-adjoint non-negative extension of the Laplace operator in G. The problem of localization of Phi-means of spectral decomposition is considered in the article. For Riesz means a similar problem has been studied by V.~A. Ilʹin [Differencialʹnye Uravnenija {bf 7} (1971), 670--710; [msn] MR0284867 [/msn]] and S.~A. Alimov and Ilʹin [Differencialʹnye Uravnenija {bf 7} (1971), 851--882; [msn] MR0284868 [/msn]].